# Real Number Systems and Number Theory

## Irrational Numbers

### Learning Objectives

After completing this section, you should be able to:
1. Define and identify numbers that are irrational.
2. Simplify irrational numbers and express in lowest terms.
3. Add and subtract irrational numbers.
4. Multiply and divide irrational numbers.
5. Rationalize fractions with irrational denominators.
The Pythagoreans were a philosophical sect in ancient Greece. Their philosophy included reincarnation and purifying the mind through the study and contemplation of mathematics and science. One of their principles was the cosmos is ruled by order, specifically mathematics and music. They even held mystic beliefs about specific numbers and figures. For example, the number 1 was associated with the mind and essence. Four represented justice, as it is the first product of two even numbers. Most famously, though, is the association with the Pythagorean Theorem, which states that in a right triangle, the sum of the squares of the shorter sides of the triangle (the legs) equals the square of the longer side (the hypotenuse). Even the ancient Egyptians used this relationship, as triangles with side measures 3, 4, and 5 were often used in surveying following the flooding of the Nile.
There is a story of a Pythagorean, Hippasus, discovering that not all numbers could be expressed as fractions. In other words, not all numbers were rational numbers. The story ends with Hippasus, who shared this, or in some versions discovered it, put to death by drowning for sharing this fact, that not all quantities could be expressed as the ratio of two natural numbers.
As colorful as that story may be, it is most likely false, as there are no contemporary sources to corroborate it. But it does seem to mark the discovery that not all quantities or measures were fractions of numbers. And so, irrational numbers were discovered.

### Defining and Identifying Numbers That Are Irrational

We defined rational numbers in the last section as numbers that could be expressed as a fraction of two integers. Irrational numbers are numbers that cannot be expressed as a fraction of two integers. Recall that rational numbers could be identified as those whose decimal representations either terminated (ended) or had a repeating pattern at some point. So irrational numbers must be those whose decimal representations do not terminate or become a repeating pattern.
One collection of irrational numbers is square roots of numbers that aren’t perfect squares.  is the square root of the number , denoted , if . The number  is the perfect square of the integer  if . The rational number  is a perfect square if both  and  are perfect squares.
One method of determining if an integer is a perfect square is to examine its prime factorization. If, in that factorization, all the prime factors are raised to even powers, the integer is a perfect square. Another method is to attempt to factor the integer into an integer squared. It is possible that you recognize the number as a perfect square (such as 4 or 9). Or, if you have a calculator at hand, use the calculator to determine if the square root of the integer is an integer.
Another collection of irrational numbers is based on the special number, pi, denoted by the Greek letter , which is the ratio of the circumference of the diameter of the circle ().
Any multiple or power of  is an irrational number.
Any number expressed as a rational number times an irrational number is an irrational number also. When an irrational number takes that form, we call the rational number the rational part, and the irrational number the irrational part. It should be noted that a rational number plus, minus, multiplied by, or divided by any irrational number is an irrational number.

### Simplifying Square Roots and Expressing Them in Lowest Terms

To simplify a square root means that we rewrite the square root as a rational number times the square root of a number that has no perfect square factors. The act of changing a square root into such a form is simplifying the square root.
The number inside the square root symbol is referred to as the radicand. So in the expression  the number  is referred to as the radicand.
Before discussing how to simplify a square root, we need to introduce a rule about square roots. The square root of a product of numbers equals the product of the square roots of those number. Written symbolically, .
Using this formula, we can factor an integer inside a square root into a perfect square times another integer. Then the square root can be applied to the perfect square, leaving an integer times the square root of another integer. If the number remaining under the square root has no perfect square factors, then we’ve simplified the irrational number into lowest terms. To simplify the irrational number into lowest terms when  is an integer:
Step 1: Determine the largest perfect square factor of , which we denote .
Step 2: Factor  into .
Step 3: Apply .
Step 4: Write  in its simplified form, .
When a square root has been simplified in this manner,  is referred to as the rational part of the number, and  is referred to as the irrational part.

### Adding and Subtracting Irrational Numbers

Just like any other number we’ve worked with, irrational numbers can be added or subtracted. When working with a calculator, enter the operation and a decimal representation will be given. However, there are times when two irrational numbers may be added or subtracted without the calculator. This can happen only when the irrational parts of the irrational numbers are the same.
To add or subtract two irrational numbers that have the same irrational part, add or subtract the rational parts of the numbers, and then multiply that by the common irrational part.

### Multiplying and Dividing Irrational Numbers

Just like any other number that we’ve worked with, irrational numbers can be multiplied or divided. When working with a calculator, enter the operation and a decimal representation will be given. Sometimes, though, you may want to retain the form of the irrational number as a rational part times an irrational part. The process is similar to adding and subtracting irrational numbers when they are in this form. We do not need the irrational parts to match. Even though they need not match, they do need to be similar, such as both irrational parts are square roots, or both irrational parts are multiples of pi. Also, if the irrational parts are square roots, we may need to reduce the resulting square root to lowest terms.
When multiplying two square roots, use the following formula. It is the same formula presented during the discussion of simplifying square roots.
When dividing two square roots, use the following formula.
To multiply or divide irrational numbers with similar irrational parts, do the following:
Step 1: Multiply or divide the rational parts.
Step 2: If necessary, reduce the result of Step 1 to lowest terms. This becomes the rational part of the answer.
Step 3: Multiply or divide the irrational parts.
Step 4: If necessary, reduce the result from Step 3 to lowest terms. This becomes the irrational part of the answer.
Step 5: The result is the product of the rational and irrational parts.

### Rationalizing Fractions with Irrational Denominators

Fractions often represent that some amount is being equally divided into some number of parts. But to conceptualize a fraction in that manner, the denominator needs to be an integer. An irrational number in the denominator interferes with that interpretation of a fraction. Fractions that have denominators that are just the square root of an integer can be altered into fractions with integer denominators using a process called rationalizing the denominator. The process relies on the following property of square roots:  and the following property of fractions:  for any non-zero number .
Using these two properties, when a fraction has a square root in the denominator, we can eliminate that square root. Multiply the numerator and denominator by that square root from the denominator, . Then apply  to the denominator, yielding . Notice that there is no longer a square root in the denominator, which allows for interpreting the fraction as dividing a whole into equal parts.
There are occasions when the denominator is irrational but is the sum of two numbers where one or both involve square roots. For instance, . The process used earlier required that the denominator was the square root of a number and would not work here. However, this type of denominator can be rationalized. In order to rationalize such a denominator, we will multiply the numerator and denominator of the fraction by the conjugate of the denominator. The conjugate of  is . We say that  and  are conjugate numbers.
So, the conjugate of  is just . But why is this of interest? The reason is because it leads to the difference of squares formula, which is used to factor the difference of two squares. Or, for our purposes, in reverse it allows us to eliminate a square root.
Looking at that formula, you should see that the two factors on the right-hand side of the equals sign are conjugates of one another. So, for our purposes, we’re interested in . This tells us that when we multiply  by its conjugate, we get  squared minus  squared, or . But how is this useful? Let’s return to the fraction above, . The denominator is . Its conjugate is . According to the formula, and letting  and , we see that . But  is just 3. That means the product is  or 13. This no longer has a square root. We use this to rationalize the denominator.
We will also need the distributive property of numbers.

### Check Your Understanding

### Key Terms

1. conjugate numbers
2. difference of squares
3. irrational numbers
4. lowest terms
5. rationalize the denominator

### Key Concepts

1. Irrational numbers are numbers that cannot be written as an integer divided by another integer. One example is pi, denoted . Another collection of irrational numbers are natural numbers that are not perfect squares.
2. Some irrational numbers can be written as a rational part multiplied by an irrational part. If two irrational numbers have the same irrational parts, they can be added or subtracted.
3. When irrational numbers are similar, on can multiply and divide the numbers without a calculator.
4. Since  , and , products and quotients of square roots can be determined.
5. Because  and , it is possible to simplify square root expressions so the radicand contains no perfect square factors.
6. When a fraction has an irrational number as its denominator, it is possible to convert the denominator into a rational number using its conjugate. Doing so involves multiplying the numerator and denominator by the conjugate of the denominator, and then applying the difference of squares formula.
7. With a single square root term
8. Using conjugate numbers for two term denominators

### Formulas

1.
2.
3.
4.

### Videos

1. The Philosophy of the Pythagoreans
2. Using Desmos to Find the Square Root of a Number
3. Simplifying Square Roots
4. Rationalizing the Denominator


# Real Number Systems and Number Theory

## Real Numbers

### Learning Objectives

After completing this section, you should be able to:
1. Define and identify numbers that are real numbers.
2. Identify subsets of the real numbers.
3. Recognize properties of real numbers.
Have you ever been impressed by the speed at which someone can do math in their head? Most of us at one time or another have witnessed a person speed through mental math, an impressive feat that often bests calculators. One such person is Neelkantha Bhanu Prakash. As of September 20, 2020, he is considered the world’s fastest human calculator. He currently holds four world records. How does someone do that, though? Have they memorized lots of arithmetic facts? Are they simply brilliant?
The answer isn't simple so much as it is about knowledge. Real numbers behave in some very regular ways, following rules that can be learned. In this section, those rules are explored.
Watch the video of Arthur Benjamin’s TED Talk to learn about another mathematician with remarkable mental abilities.

### Defining and Identifying Real Numbers

Real numbers are the rational and irrational numbers combined. The real numbers represent the collection of all physical distances that exist, along with 0 and the negatives of those physical distances. For example, if you take a measure of three units, and divide that distance into eight (8) equal lengths, the distance you have formed is  units. Also, if you draw a right triangle (a triangle with one angle equal to 90 degrees) with one side length of 1, and the other side length of 3, the long side of the triangle will have length  units, as shown in .
Of course, if we name something the real numbers, there must be numbers that aren't real. Otherwise, they’d just be called the numbers. One such not real number, one that cannot be a length, is . It is part of a collection of numbers called the complex numbers, it is denoted with the letter . As an extension, the square root of any negative number is not a real number, but instead a complex number.
To determine if a number is real, check to see if there are any negatives under a square root or any . If there are any present, the number is not real.

### Identifying Subsets of Real Numbers

The real numbers were built out of pieces, including integers, rational numbers, and irrational numbers. As such, the real numbers have named subsets, as shown in the table below.
When we categorize numbers using these sets, we use the smallest set that they belong to. For instance, −7 is an integer, and a rational number, and a real number. The smallest set to which −7 belongs is integer, so we’d say it belongs to the integers.
We can also represent the relationships between the different sets of real numbers using set notation. All natural numbers are integers, but there are integers that are not natural numbers, so . Similarly, every integer is a rational number, but there are rational numbers that are not integers, so . The same is true of the rational numbers and the real numbers, so .
There is no agreed-upon symbol for the irrational numbers. If we represent the irrationals as the set , we should note that the following are true:  and . Recall that this means the irrationals are the complement of the rational numbers in the universal set of real numbers.

### Recognizing Properties of Real Numbers

The real numbers behave in very regular ways. These behaviors are called the properties of the real numbers. Knowing these properties helps when evaluating formulas, working with equations, or performing algebra. Being familiar with these properties is helpful in all settings where numbers are used and manipulated. For example, when multiplying , you could multiply the 4 and 25 first. If you know that product is 100, it makes the multiplication easier.
The table below is a partial list of properties of real numbers.
The names of the properties are suggestive. The commutative properties, for example, suggest commuting, or moving. Associative properties suggest which items are associated with others, or if order matters in the computation. The distributive property addresses how a number is distributed across parentheses.
Using these properties to perform arithmetic quickly relies on spotting easy numbers to work with. Look for numbers that add to a multiple of 10, or multiply to a multiple of 10 or 100.

### Videos

1. Properties of the Real Numbers 1
2. Properties of the Real Numbers 2
3. Properties of the Real Numbers 3

### Check Your Understanding

### Key Terms

1. complex number
2. imaginary number
3. real number

### Key Concepts

1. Real numbers is the collection of all rational and irrational numbers. Conceptually, it is the collection of all values that can be represented on a number line, or, as a length along with sign.
2. The subsets of the real numbers include the natural numbers, integers, rational numbers and irrational numbers. The natural numbers are a subset of the integers, which is a subset of the rational numbers. The rational and irrational numbers are disjoint sets.
3. The real numbers, due to order of operation rules and that performing arithmetic operations on real number always results in a real number, have arithmetic properties that apply in all cases. There include the distributive property, the commutative property, and the associative property. Also, every real number has an additive inverse and, except for zero (0), have a multiplicative inverse.

### Formulas

1.
2.
3.
4.
5.
6.
7.
8.
9.

### Videos

1. Arthur Benjamin TED talk, Faster than a Calculator
2. Identifying Sets of Real Numbers
3. Properties of the Real Numbers #1
4. Properties of the Real Numbers #2
5. Properties of the Real Numbers #3


# Real Number Systems and Number Theory

## Clock Arithmetic

### Learning Objectives

After completing this section, you should be able to:
1. Add, subtract, and multiply using clock arithmetic.
2. Apply clock arithmetic to calculate real-world applications.
Online shopping requires you to enter your credit card number, which is then sent electronically to the vendor. Using an ATM involves sliding your bank card into a reader, which then reads, sends, and verifies your card. Swiping or tapping for a purchase in a brick–and-mortar store is how your card sends its information to the machine, which is then communicated to the store’s computer and your credit card company. This information is read, recorded, and transferred many times. Each instance provides one more opportunity for error to creep into the process, a misrecorded digit, transposed digits, or missing digits. Fortunately, these card numbers have a built-in error checking system that relies on modular arithmetic, which is often referred to as clock arithmetic. In this section, we explore clock, or modular, arithmetic.

### Adding, Subtracting, and Multiplying Using Clock Arithmetic

When we do arithmetic, numbers can become larger and larger. But when we work with time, specifically with clocks, the numbers cycle back on themselves. It will never be 49 o’clock. Once 12 o’clock is reached, we go back to 1 and repeat the numbers. If it's 11 AM and someone says, “See you in four hours,” you know that 11 AM plus 4 hours is 3 PM, not 15 AM (ignoring military time for now). Math worked on the clock, where numbers restart after passing 12, is called clock arithmetic.
Clock arithmetic hinges on the number 12. Each cycle of 12 hours returns to the original time (). Imagine going around the clock one full time. Twelve hours pass, but the time is the same. So, if it is 3:00, 14 hours later and two hours later both read the same on the clock, 5:00. Adding 14 hours and adding 2 hours are identical. As is adding 26 hours. And adding 38 hours.
What do 2, 14, 26, and 38 have in common in relation to 12? When they are divided by 12, they each have a remainder of 2. That's the key. When you add a number of hours to a specific time on the clock, first divide the number of hours being added by 12 and determine the remainder. Add that remainder to the time on the clock to know what time it will be.
A good visualization is to wrap a number line around the clock, with the 0 at the starting time. Then each time 12 on the number line passes, the number line passes the starting spot on the clock. This is referred to as modulo 12 arithmetic. Even though the process says to divide the number being added by 12, first perform the addition; the result will be the same if you add the numbers first, and then divide by 12 and determine the remainder.
In general terms, let  be a positive integer. Then  modulo 12, written ( mod 12), is the remainder when  is divided by 12. If that remainder is , we would write  (mod 12).
Subtracting time on the clock works in much the same way as addition. Find the value of the number of hours being subtracted modulo 12, then subtract that from the original time.
Recall that clock arithmetic was referred to as modulo 12 arithmetic. Multiplying in modulo 12 also relies on the remainder when dividing by 12. To multiply modulo 12 is just to multiply the two numbers, and then determine the remainder when divided by 12.

### Calculating Real-World Applications with Clock Arithmetic

Clock arithmetic processes can be applied to days of the week. Every 7 days the day of the week repeats, much like every 12 hours the time on the clock repeats. The only difference will be that we work with remainders after dividing by 7. In technical terms, this is referred to as modulo 7. More generally, let  be a positive integer. Then  modulo 7, written  mod 7, is the remainder when  is divided by 7. If that value is , we may write  (mod 7).

### Check Your Understanding

### Key Terms

1. clock arithmetic
2. modulo 7
3. modulo 12

### Key Concepts

1. Clock arithmetic uses the idea that after 12 o’clock comes 1 o’clock. For clock arithmetic, this means that every time 12 is passed in an arithmetic process, the next number is 1, not 13.
2. To determine the clock result of an arithmetic operation, divide the final result by 12 and keep the remainder. If the remainder is 0, then the time is 12 o’clock.
3. Clock arithmetic is technically called modulo 12 arithmetic. To perform modulo 12 arithmetic, calculate the expression, then divide the result by 12. The modulo 12 result is the remainder.
4. Days, in our system, pass in groups of seven. To calculate in day arithmetic, modulo 7 is used. To perform modulo 7 arithmetic, calculate the expression, then divide the result by 7. The modulo 7 result is the remainder.

### Videos

1. Determining the Day of the Week for Any Date in History
2. Clock Arithmetic


# Real Number Systems and Number Theory

## Exponents

### Learning Objectives

After completing this section, you should be able to:
1. Apply the rules of exponents to simplifying expressions.
Sometimes, we look for shorthand when writing or expressing something that simply takes too long. The use of LOL and tl;dr. This shorthand only works if everyone reading the shorthand knows what it stands for. Using exponents is a similar instance. Writing out a long string of a number times itself over and over takes too much time, and eventually one would forget how many of the value has been written or read. For example, . There has to be a shorter and more efficient way to write 8 times itself 1, 2, 3….hmmmm, 19 times.
And that’s the role that exponents play in mathematics. They are shorthand for multiplying a number by itself a number of times. Without it, calculations would become a mess and we’d have to write a lot more.

### Applying the Rules of Exponents to Simplify Expressions

Squaring a number is multiplying it by itself, and has that name because it is the area of a square with that side length. Cubing a number is finding the volume of a cube with that length of sides. That’s why we refer to  as five squared, or  as ten cubed. Exponents represent that multiplication.
Let’s remind ourselves of the terminology associated with exponents and what exponents represent. Suppose you want to multiply a number, let’s label that number , by itself some number of times. Let’s label the number of times . We denote that as . We say  raised to the th power. When we write or see , we call the 7 the base and we call 5 the exponent. What it represents is 7 multiplied by itself 5 times. This means exponents are used as a shorthand for repeated multiplications, where we write . We would write  and say seven to the fifth power.
The definitions of base and exponent make it possible to understand the exponent rules.

### Product Rule for Exponents

The first rule we examine is the product rule, . This rule means that when we multiply a base raised to a power times the same base to another power, the result is the base raised to the sum of the powers. To demonstrate, consider . If we apply the product rule to that we get . This can be tested by looking at the multiplications that are represented. The  is 9 times itself 3 times, while  is 9 times itself 5 times. Substituting those into  we see , which is what the formula told us would happen.
These rules can be applied to unknowns too.

### Quotient Rule for Exponents

The next rule we examine is the quotient, or division, rule.
This rule means that when we divide a base raised to a power by the same base to another power, the result is the base raised to the difference of the powers. To demonstrate, consider . If we apply the quotient rule to that, we get . This can be tested by looking at the division that is represented. Remember,  is 14 multiplied to itself 13 times, while  is 14 multiplied to itself 6 times. Substituting those into  gives the following:
We see here that there are a LOT of fours to be divided out.
What remains is 4 to the 7th power, .
All of the work above confirmed what the formula told us would be the result.
A natural consequence of the quotient rule is what it means to raise a non-zero number to the zeroth power. Let’s look at the simplification when the exponents are equal.

We know that a number divided by itself is 1, so . From that is must be that . This provides the rule for a number raised to the power 0: .

### Distributive Rule for Exponents

The next rule we look to is a distributive rule for exponents.
This means that when we have two numbers multiplied together, and that is raised to a power, it is the same as raising each of the numbers to the same power first, then multiplying. For example, . This can be explained using the definition of exponents and multiplying all the factors.

We may change the order in which numbers are multiplied. This is the commutative property of the real numbers. This can be written as . Using exponents, that shortens to .
This also works in the other direction, . Read this way, if we have one base raised to an exponent, and another base raised to the same exponent, we can multiply the bases and raise that product to the shared exponent. For instance, .
This distribution also works for quotients. A fraction raised to an exponent equals the numerator raised to the exponent divided by the denominator raised to the exponent. For example, . Demonstrating this is similar to the previous rule.

### Power Rule

In the previous two sets of rules, we’ve seen exponents applied to products and quotients. Now we look to exponents applied to other exponents. For example, . This can be explained by examining what the outer exponent does. We raise  to the fourth power, so we multiply  by itself 4 times, . Now if we apply the product rule for exponents, this becomes .

### Negative Exponent Rule

Up until now, we’ve only looked at positive exponents. The last exponent rule we look at is what negative exponents represent. Recall the quotient rule: . What would happen if the exponent in the denominator was larger than that in the numerator? For example, . If we apply the quotient rule, we obtain . We need to make sense of that negative exponent. To do so, we can expand the quotient and see what happens: . When we divide out common factors, only two factors of 4 are left in the denominator, as we see here:. Using exponent notation, this is . Since  and  represent the same number, , they are equal. This demonstrates how negative exponents are defined.
The table below shows a summary of the exponent rules from this section.
These rules often occur in tandem with each other, but it requires that you carefully apply the rules.

### Check Your Understanding

### Key Terms

1. base
2. exponent

### Key Concepts

1. Exponents are used to express multiplying a number by itself a number of times. The number being multiplied by itself is the base. The number of times it is multiplied by itself is the exponent, which is often referred to as the power.
2. Understanding that exponents represent repeated multiplication of a base makes it possible to establish some rules for combining exponential expressions, using the product rule, the quotient rule, and the power rule. Additionally, it allows us to formulate distributive rules for exponents.
3.  Any non-zero number raised to the 0th power is 1. This makes the definition of the 0th power consistent with the division rule for exponents.
4. For consistency, negative exponents represent the reciprocal of the base raised to the power, so that , provided that .

### Formulas

1.
2.
3. , provided that
4.
5.
6.
7. , provided that

### Videos

1. Exponential Notation
2. Product and Quotient Rule for Exponents
3. Fraction Raised to a Power
4. Simplifying Expressions with Exponents


# Real Number Systems and Number Theory

## Scientific Notation

### Learning Objectives

After completing this section, you should be able to:
1. Write numbers in standard or scientific notation.
2. Convert numbers between standard and scientific notation.
3. Add and subtract numbers in scientific notation.
4. Multiply and divide numbers in scientific notation.
5. Use scientific notation in computing real-world applications.
The amount of information available on the Internet is simply incomprehensible. One estimate for the amount of data that will be on the Internet by 2025 is 175 Zettabytes. A single zettabyte is one billion trillion. Written out, it is 1,000,000,000,000,000,000,000. One estimate is that we’re producing 2.5 quintillion bytes of data per day. A quintillion is a trillion trillion, or, written out, 1,000,000,000,000,000,000. To determine how many days it takes to increase the amount of information that is on the Internet by 1 zettabyte, divide these two numbers, a zettabyte being 1,000,000,000,000,000,000,000, and 2.5 quintillion, being 2,500,000,000,000,000,000, shows it takes 400 days to generate 1 zettabyte of information. But doing that calculation is awkward with a calculator. Keeping track of the zeros can be tedious, and a mistake can easily be made.
On the other end of the scale, a human red blood cell has a diameter of 7.8 micrometers. One micrometer is one millionth of a meter. Written out, 7.8 micrometers is 0.0000078 meters. Smaller still is the diameter of a virus, which is about 100 nanometers in diameter, where a nanometer is a billionth of a meter. Written out, 100 nanometers is 0.0000001 meters. To compare that to engineered items, a single transistor in a computer chip can be 14 nanometers in size (0.000000014 meters). Smaller yet is the diameter of an atom, at between 0.1 and 0.5 nanometers.
Sometimes we have numbers that are incredibly big, and so have an incredibly large number of digits, or sometimes numbers are incredibly small, where they have a large number of digits after the decimal. But using those representations of the names of the sizes makes comparing and computing with these numbers problematic. That’s where scientific notation comes in.

### Writing Numbers in Standard or Scientific Notation Form

When we say that a number is in scientific notation, we are specifying the form in which that number is written. That form begins with an integer with an absolute value between 1 and 9, then perhaps followed the decimal point and then some more digits. This is then multiplied by 10 raised to some power. When the number only has one non-zero digit, the scientific notation form is the digit multiplied by 10 raised to an exponent. When the number has more than one non-zero digit, the scientific notation form is a single digit, followed by a decimal, which is then followed by the remaining digits, which is then multiplied by 10 to a power.
The following numbers are written in scientific notation:

The following numbers are not written in scientific notation:
 because it isn't multiplied by 10 raised to a power
 because the absolute value of −50.053 is not at least 1 and less than 10
 because 41.7 is not at least 1 and less than 10
 because 0.036 is not at least one and less than 10
Some numbers are so large or so small that it is impractical to write them out fully. Avogadro’s number is important in chemistry. It represents the number of units in 1 mole of any substance. The substance many be electrons, atoms, molecules, or something else. Written out, the number is: 602,214,076,000,000,000,000,000. Another example of a number that is impractical to write out fully is the length of a light wave. The wavelength of the color blue is about 0.000000450 to 0.000000495 meters. Such numbers are awkward to work with, and so scientific notation is often used. We need to discuss how to convert numbers into scientific notation, and also out of scientific notation.
Recall that multiplying a number by 10 adds a 0 to the end of the number or moves the decimal one place to the right, as in  or . And if you multiply by 100, it adds two zeros to the end of the number or moves the decimal two places to the right, and so on. For example,  and . Multiplying a number by 1 followed by some number of zeros just adds that many zeros to the end of the number or moves the decimal place that many places to the right. Numbers written as 1 followed by some zeros are just powers of 10, as in , , , etc. Generally, .
We can use this to write very large numbers. For instance, Avogadro’s number is 602,214,076,000,000,000,000,000, which can be written as . The multiplication moves the decimal 23 places to the right.
Similarly, when we divide by 10, we move the decimal one place to the left, as in . If we divide by 100, we move the decimal two places to the left, as in . In general, when you divide a number by a 1 followed by  zeros, you move the decimal  places to the left, as in . This denominator could be written as . If we use that in the expression and allow for negative exponents, rewrite the number as . With this, we can write division by a 1 followed by  zeros as multiplication by 10 raised to .
Using that information, we can demonstrate how to convert from a number in standard form into scientific notation form.
Case 1: The number is a single-digit integer.
In this case, the scientific notation form of the number is .
Case 2: The absolute value of the number is less than 1.
Follow the process below.
1. Step 1: Count the number of zeros between the decimal and the first non-zero digit. Label this .
2. Step 2: Starting with the first non-zero digit of the number, write the digits. If the number was negative, include the negative sign.
3. Step 3: If there is more than one digit, place the decimal after the first digit from Step 2.
4. Step 4: Multiply the number from Step 3 by .
Case 3: The absolute value of the number is 10 or larger.
Follow the process below.
1. Step 1: Count the number of digits that are to the left of the decimal point. Label this .
2. Step 2: Write the digits of the number without the decimal place, if one was present. If the number was negative, include the negative sign.
3. Step 3: If there is more than one digit, place the decimal point after the first digit.
4. Step 4: Multiply the number from Step 3 by .
When we write numbers in scientific notation form, we can manipulate the representation of the number by moving the decimal around, and making an appropriate change to the exponent of the 10. For instance, let’s look at . If we wanted to move the decimal one place to the left, we’d have to increase the power of 10, as shown here: . Since we moved the decimal one to the left, we balance that with moving the exponent up by one. Similarly, if we move the decimal one place to the right, we have to balance that by moving the exponent one to the left, or subtracting one from the exponent, as shown here: . Generally, for a number in the form :
1. If you move the decimal to the left by  digits, you increase the exponent by .
2. If you move the decimal to the right by  digits, you decrease the exponent by  digits.

### Converting Numbers from Scientific Notation to Standard Form

In the previous section, converting a number from standard form to scientific notation was explored. Now, we explore converting from scientific notation back into standard form. Doing so involves moving the decimal according to the power of the 10. The decimal is moved a number of steps equal to the exponent of the 10. As demonstrated previously, when the exponent of the 10 is negative, the decimal is moved to the left and when the exponent of the 10 is positive, the decimal is moved to the right.

### Adding and Subtracting Numbers in Scientific Notation

To add or subtract numbers in scientific notation, the numbers first need to have the same exponent for the 10s. It is possible to add the following since the powers of 10 match:

Notice that the number parts were added, but the exponent part remained the same. This is due to the distributive property of the real numbers. The  is factored from the two terms, as shown:

Numbers in scientific notation can be added or subtracted directly using a calculator. Simply enter the values in scientific form and set your calculator to display scientific notation.
Adding and subtracting in scientific notation is straightforward when the exponents are the same. There are two issues that can arise. The first issue is what to do if after adding or subtracting the result is not in scientific notation.
The second issue that might be encountered when adding or subtracting is that the powers of 10 do not match. In that case, one of the numbers must be changed so that the powers of 10 match. It is easiest to make the smaller power of 10 larger to match the other power of 10.
For example, to perform the following, , we’d change the  so that the power of 10 is 5. To do so, we need to increase the power of 10 and move the decimal in the number part two places to the left. That would alter  into . We would use  in the addition problem, so that the exponents match, allowing the addition to occur.

The steps to take when the exponents of the 10s are not equal are:
Step 1: Increase the smaller exponent to equal the larger exponent. Label the amount increased as .
Step 2: For the number with the smaller power of 10, move the decimal point of the number part to the left  places.
Step 3: Perform the addition or subtraction.
Step 4: If the result is not in scientific notation, adjust the number to be in scientific notation.

### Multiplying and Dividing Numbers in Scientific Notation

Multiplying and dividing numbers in scientific notation is somewhat easier than adding or subtracting, because the exponents of the 10s do not have to match. However, it is much more likely that the result will not be in scientific notation, and so that will have to be adjusted at the end. Generally, we multiply or divide the number parts of the two values, and then apply exponent rules to the 10 raised to the powers.
To multiply two numbers in scientific notation:
Step 1: Multiply the number parts.
Step 2: Add the exponents of the 10s.
Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2.
Step 4: If the number is not in scientific notation, adjust it appropriately.

### Dividing Numbers in Scientific Notation

To divide two numbers that are in scientific notation:
Step 1: Divide the number parts.
Step 2: Subtract the exponent of the denominator from the exponent of the numerator.
Step 3: The answer is the result from Step 1 times 10 raised to the result from Step 2.
Step 4: If the number is not in scientific notation, adjust it appropriately.

### Using Scientific Notation in Computing Real-World Applications

As noted at the start of this section, scientific notation is useful when the standard representation of a number is awkward or impractical, which occurs when the numbers being used are extremely large or extremely small. For example, Venus is 67,667,000 miles from the sun. In scientific notation, this is . Planetary and galaxy distances is one set of numbers that is easier to express using scientific notation.

### What Numbers Could Be Considered “Too Big” or “Too Small”?

One wonders when the numbers we represent become too large or small for consideration. Perhaps the following examples put limits on what is meaningful. The number of particles in the known universe has been estimated at  particles. The smallest distance that has been measured is , though the theoretical smallest measurable value is . The distance across the universe is . Considering what those numbers represent, the extreme largest and extreme smallest, they might be numbers that constrain what we should reasonably be expected to deal with.

### Check Your Understanding

### Key Terms

1. scientific notation
2. standard notation

### Key Concepts

1. Some numbers are so large or so small that writing the number out is clumsy and make it difficult to determine the true size of the number. Scientific notation makes the number more readable and make the relative size of the number immediately apparent.
2. A number written in scientific notation is a number at least 1 and smaller than 10 multiplied by 10 raised to an exponent. Converting between scientific notation and standard notation involves correctly applying multiplication and division by powers of 10, which in practice equates to understanding how moving the decimal point of a number impacts the exponent of 10.
3. Adding and subtracting numbers in base 10 requires the exponent of 10 in each number be the same. Once the numbers are converted to have the same exponent with the ten, then the numbers are added or subtracted as indicated, with the power of 10 remaining the same. If the result is not in scientific notation (for instance, the number has exceeded 10), then then number must be converted into scientific notation.
4. Multiplying and dividing numbers in scientific notation is done by multiplying or dividing the number parts, then multiplying or dividing the 10 raised to the power parts, then multiplying those two results. If the new number is not in scientific notation, then the result must be converted into scientific notation.

### Videos

1. Converting from Standard Form to Scientific Notation Form
2. Converting from Scientific Notation Form to Standard Form
3. Multiplying Numbers in Scientific Notation
4. Dividing Numbers in Scientific Notation
5. Application of Scientific Notation


# Real Number Systems and Number Theory

## Arithmetic Sequences

### Learning Objectives

After completing this section, you should be able to:
1. Identify arithmetic sequences.
2. Find a given term in an arithmetic sequence.
3. Find the th term of an arithmetic sequence.
4. Find the sum of a finite arithmetic sequence.
5. Use arithmetic sequences to solve real- world applications
As we saw in the previous section, we are adding about 2.5 quintillion bytes of data per day to the Internet. If there are 550 quintillion bytes of data today, then there will be 552.5 quintillion bytes tomorrow, and 555 quintillion bytes in 2 days. This is an example of an arithmetic sequence. There are many situations where this concept of fixed increases comes into play, such as raises or table arrangements.

### Identifying Arithmetic Sequences

A sequence of numbers is just that, a list of numbers in order. It can be a short list, such as the number of points earned on each assignment in a class, such as {10, 10, 8, 9, 10, 6, 10}. Or it can be a longer list, even infinitely long, such as the list of prime numbers. For example, here’s a sequence of numbers, specifically, the squares of the first 12 natural numbers.
{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144}
Each value in the sequence is called a term. Terms in the list are often referred to by their location in the sequence, as in the th term. For the sequence {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144}, the first term of the sequence is 1, the fourth term is 16, and so on. In the sequence of assignment scores {10, 10, 8, 9, 10, 6, 10}, the first term is 10 and the third term is 8 ().
The notation we use with sequences is a letter, which represents a term in the sequence, and a subscript, which indicates what place the term is in the sequence. For the sequence {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144}, we will use the letter  as a value in the sequence, and so  would be the term in the sequence at the fifth position. That number is 25, so we can write .
In this section, we focus on a special kind of sequence, one referred to as an arithmetic sequence. Arithmetic sequences have terms that increase by a fixed number or decrease by a fixed number, called the  constant difference (denoted by ), provided that value is not 0. This means the next term is always the previous term plus or minus a specified, constant value. Another way to say this is that the difference between any consecutive terms of the sequence is always the same value.
To see a constant difference, look at the following sequence: {7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87}.  illustrates that each term of the sequence is the previous term plus 8. Eight is the constant difference here.
Arithmetic sequences can be expressed with a formula. When we know the first term of an arithmetic sequence, which we label , and we know the constant difference, which is denoted , we can find any other term of the arithmetic sequence. The formula for the  term of an arithmetic sequence is .
Let’s examine the formula with this arithmetic sequence: . In this sequence  and . The table below shows the values calculated.
We can see how the  term can be directly calculated. In this sequence, the formula is  where the first term, , is 4 and the constant difference  is 3. We can then determine the  term of this sequence: .
If we know two terms of the sequence, it is possible to determine the general form of an arithmetic sequence, .

### Finding the Sum of a Finite Arithmetic Sequence

Sometimes we want to determine the sum of the numbers of a finite arithmetic sequence. The formula for this is fairly straightforward.

### Using Arithmetic Sequences to Solve Real-World Applications

Applications of arithmetic sequences occur any time some quantity increases by a fixed amount at each step. For instance, suppose someone practices chess each week and increases the amount of time they study each week. The first week the person practices for 3 hours, and vows to practice 30 more minutes each week. Since the amount of time practicing increases by a fixed number each week, this would qualify as an arithmetic sequence.

### Check Your Understanding

### Key Terms

1. sequence
2. term of a sequence
3. arithmetic sequence
4. first term
5. constant difference

### Key Concepts

1. A sequence is a list of numbers. Any individual number in that list, or sequence, is a term of the sequence. A specific term of a sequence is denoted by the sequence symbol with a subscript indicating where the term in the sequence is.
2. A special form of a sequence is an arithmetic sequence. Each arithmetic sequence is determined by its first term and its constant difference. Any term in an arithmetic sequence is determined by adding the constant difference to the preceding term.
3. If the first term and the constant difference of an arithmetic sequence are known, then any term of the sequence can be found directly.
4. Because arithmetic sequences follow such a strict pattern, the sum of the first  terms of an arithmetic sequence can be determined with the formula .

### Formulas

1.
2.
3.
4.

### Videos

1. Arithmetic Sequences
2. Finding the First Term and Constant Difference for an Arithmetic Sequence
3. Finding the Sum of a Finite Arithmetic Sequence
4. Fibonacci Sequence and “Lateralus”


# Real Number Systems and Number Theory

## Geometric Sequences

### Learning Objectives

After completing this section, you should be able to:
1. Identify geometric sequences.
2. Find a given term in a geometric sequence.
3. Find the th term of a geometric sequence.
4. Find the sum of a finite geometric sequence.
5. Use geometric sequences to solve real-world applications.
One of the concerns when investing is the doubling time, which is length of time it takes for the value of the investment to be twice, or double, that of its starting value. A shorter doubling times means the investment gets bigger, sooner. For example, if you invest $200 in an account with an 8-year doubling time, then in 8 years the value of the account will be double the starting amount, or . After another 8 years (for a total of 16 years) the investment would be twice its value after the first 8 years, or . Every 8 years, the investment would double again, so after the third 8-year period, the investment would be worth . This process exhibits exponential growth, an application of geometric sequences, which is explored in this section.

### Identifying Geometric Sequences

We know what a sequence is, but what makes a sequence a geometric sequence? In an arithmetic sequence, each term is the previous term plus the constant difference. So, you add a (possibly negative) number at each step. In a geometric sequence, though, each term is the previous term multiplied by the same specified value, called the common ratio. In the sequence  the common ratio is 2. To see the difference between an arithmetic sequence and geometric sequence, examine these two sequences (Figures 3.52 and 3.53).
Each term in this arithmetic sequence is the previous term plus 5.
Each term in this geometric sequence is the previous term times 2.
In the sequence , the numbers get big fairly quickly, and stay positive. However, that’s not always the case with geometric sequences. Depending on the value of the common ratio, the terms could increase each time (like in the one shown in ), or the terms can get smaller each time, or the terms can alternate between positive and negative values. It all depends on the value of the common ratio, .
Consider this geometric sequence:

Each term is the previous term times 5, which means the common ratio is 5. This common ratio is larger than 1, and so the terms increase each time. Now, look at this geometric sequence:

Each term is the previous term times −3, and the sign of the terms alternate from positive to negative. Then, there’s this geometric sequence:

Each term is the previous term times , and the terms decrease each time. What we should take away from these three examples is if the common ratio is a positive number larger than 1, then the sequence increases. If the common ratio is a negative number, then the sign of the terms alternates between positive and negative. If the common ratio is between 0 and 1, then the terms decrease.
Two special cases of geometric sequences are when the constant ratio is 1 and when the common ratio is 0. When the constant ratio is 1, every term of the sequence is the same, as in . This is referred to as a constant sequence. When the constant ratio is 0, the first term can be any number, but every term after the first term is 0, as in .
As with arithmetic sequences, the first term of a geometric sequence is labeled . The number that is multiplied by each term is called the common ratio and is denoted . So, if the first term is known, , and the common ratio is known, , then the  term, , can be calculated with the formula .
Return to the sequence . We observe that the first term is 3, so . We also found that the common ratio is 2, so . The table below shows how any term can be calculated using just  and .

### Finding the Sum of a Finite Geometric Sequence

As with arithmetic sequences, it is possible to add the terms of the geometric sequence. Like arithmetic sequences, the formula for the finite sum of the terms of a geometric sequence has a straightforward formula.

### Using Geometric Sequences to Solve Real-World Applications

Geometric sequences have a multitude of applications, one of which is compound interest. Compound interest is something that happens to money deposited into an account, be it savings or an individual retirement account, or IRA. The interest on the account is calculated and added to the account at regular intervals. This means the interest that was earned later gains its own interest. This allows the money to grow faster. If that interest is added every month, we say it is compounded monthly. If the interest is added daily, then we say it is compounded daily. The amount of money that is deposited into the account is called the principal and is denoted . The account earns money on that principal. The amount it earns is a percentage of the money in the account. The interest rate, expressed as a decimal, is denoted .
Another application of geometric sequences is exponential growth. This arises in biology quite frequently, especially in relation to bacterial cultures, but also with other organism population models. In bacterial cultures, the time it takes the population to double is often recorded. This time to double is the same, regardless of how big the population gets. So, if the population doubles after 3 hours, it doubles again after another 3 hours, and again after another 3 hours, and so on. Put into geometric sequence language, it has a common ratio of 2.

### Check Your Understanding

### Key Terms

1. geometric sequence
2. common ratio

### Key Concepts

1. A special form of a sequence is a geometric sequence. Each geometric sequence is determined by its first term and its constant ratio. Any term in a geometric sequence is determined by multiplying the constant ratio to the preceding term.
2. If the first term and the constant ratio of a geometric sequence are known, then any term of the sequence can be found directly.
3.  Because geometric sequences follow such a strict pattern, the sum of the first  terms of a geometric sequence can be determined with the formula .
4. Finding the sum of a finite geometric sequence
5. Applying arithmetic sequences

### Formulas

1.
2.

### Videos

1. Geometric Sequences
2. Sum of a Finite Geometric Sequence

### Encryption Throughout History

Encryption began at least as far back as the Roman Empire. During the reign of Caesar, a particular cypher was used, fittingly named the Caesar Cypher. This encryption process granted the Romans a great tactical advantage. Even if a message was intercepted, it would not make sense to the person intercepting the message.
Find four instances when encryption was used and cracked over the course of history.

### Projects

### The Golden Ratio in Art and Architecture

The golden ratio has been used in art and architecture as far back as ancient Greece (possibly further). It also appears in South America (Incan architecture). Find five instances of the use of the golden ratio in art or architecture and describe its use in each of those instances.

### Your Budget

Budgeting either is, or will shortly be, an important aspect of your life. Managing money well reduces stress in your life, and provides space for planning for future expenses, such as vacations or home improvements.
Imagine your life 10 years from now. Estimate your monthly income. Identify expenses you will encounter monthly (mortgage or rent, car payment, insurance, entertainment, etc.). Decide on an amount you plan to save monthly (this is treated as an expense). Create a spreadsheet with those values. Record your monthly net income (your income minus your expenses). Determine how much money you will have saved over the course of 5 years (ignore interest). Write a reflection on your anticipated financial health.

### Estimating Pi

The value of pi is the ratio of the circumference of a circle to the diameter of the circle. It is also equal to the ratio of the area of the circle to the square of the radius of the square.
Research three ways to physically estimate pi.
Estimate pi using all three processes you found.
Present your process and solutions in class.

### Design Your Own Shift Cypher

A cypher is a message written in such a way as to mask its contents. Changing a message into its cypher form is called encryption. Decryption or deciphering is the process of changing a cyphertext message back into the original (legible) message. One process of encryption is to scramble the letters, symbols, and punctuation of a message according to a mathematical rule. One rule that could be used for such a cypher is addition in a chosen modulus. In this project, you will create such a cypher, encrypt a message, and then decrypt the message.
Step 1: Choose the letters, symbols, and punctuation marks you want to allow in your messages. This
    should include at least the uppercase letters and a space character. This is your character set.
Step 2: Count the number of characters you will use. Label this number .
Step 3: Pair each character of your character set an integer from 0 to . Do not assign more than one character to an integer.
Step 4: Choose an integer between 1 and . This will be the number used to create the cypher. Label this number .
Step 5: Write a message using your character set.
Step 6: Replace every character in your message by the integer with which it was paired in Step 3.
Step 7: For every number, , from Step 7, perform the addition  (mod ).
Step 8: Replace every number found in Step 7 with the character with which it was paired in Step 3. This is your cyphertext.
To decrypt your cyphertext, reverse the steps above.
Step 1: Replace the cyphertext characters with the paired values.
Step 2: For each value , perform the subtraction  (mod ).
Step 3: Replace the numbers from Step 2 with their paired characters from the character set.
The message is then deciphered.

### Design Your Own Cypher Using Multiplication

A cypher is a message written in such a way as to mask its contents. Changing a message into its cypher form is called encryption. Decryption or deciphering is the process of changing a cyphertext message back into the original (legible) message. One process of encryption is to scramble the letters, symbols, and punctuation of a message according to a mathematical rule. One rule that could be used for such a cypher is multiplication in a chosen modulus. In this project, you will create such a cypher, encrypt a message, then decrypt the message.
Step 1: Choose the letters, symbols, and punctuation marks you want to allow in your messages. This should include at least the uppercase letters and a space character. This is your character set.
Step 2: Count the number of characters you will use. Label this number .
Step 3: Pair each character of your character set an integer from 0 to . Do not assign more than one character to an integer.
Step 4: Choose an integer, labeled , between 1 and  so that . This will be the number used to create the cypher.
Step 5: Write a message using your character set.
Step 6: Replace every character in your message by the integer with which it was paired in Step 3.
Step 7: For every number, , from Step 6, perform the multiplication  (mod ).
Step 8: Replace every number found in Step 7 with the character with which it was paired in Step 3. This is your cyphertext.
Before beginning to decrypt in this cypher, you need to know the multiplicative inverse of the value you chose as s.
Step 1: The multiplicative inverse of  is the number that, when multiplied by  in your modulus, equals 1. To find this, you will have to multiply  and every number between 2 and () until the product is 1 (mod ). Once this number is found, the message can be decrypted. Call this number .
Step 2: To decrypt your cyphertext, replace the cyphertext characters with the paired values.
Step 3: For each of the value, , perform the multiplication .
Step 4: Replace the numbers from Step 3 with their paired characters from the character set.
The message is then deciphered.

### Key Terms

1. common ratio
2. geometric sequence

### Key Concepts

1. Geometric sequence.
2. Finding an arbitrary term in a geometric sequence.
3. Constant ratio.
4. Finding the sum of a finite geometric sequence.
5. Applying arithmetic sequences.

### Formulas

### Chapter Review

### Prime and Composite Numbers

### The Integers

### Order of Operations

### Rational Numbers

### Irrational Numbers

### Real Numbers

### Clock Arithmetic

### Exponents

### Scientific Notation

### Arithmetic Sequences

### Geometric Sequences

### Chapter Test


# Number Representation and Calculation

## Introduction

Right now, almost all cultures use the familiar Hindu-Arabic numbering system, which uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 along with place values based on powers of ten. This is a relatively recent development. The system didn’t develop until the 6th or 7th century C.E. and took some time to spread across the world, which means other cultures at other times had to develop their own methods of counting and recording quantity. Being different cultures and different times means there were significant differences in counting systems. Cultures needed to count and measure time for agriculture and for religious observations. It was needed for trade. Some languages had words only for one, two, and many. Other cultures developed more complex ways to represent quantity, with the Oksapmin people of New Guinea using an astonishing 27 words for their system.

Representing these quantities in a recorded form likely began with a simple marking system, where one scratch on a stick or bone represented one of whatever was being counted. We still see this today with tally marks. These systems use repeated symbols to represent more than one. We also have systems where different symbols represent different quantities but still use some repetition, such as in Roman numerals.
Other systems were devised that rely on place values, like the Hindu-Arabic system in use today. Place value systems needed a zero, though, and weren't immediately recognized and took time to develop. And within these positional systems there is variation. Some systems counted in twenties, others in tens, and some in a mix (adding another reason to visit Hawaii). Even now, though we all use and think using tens, computers are designed to work in groups of two, which requires a different perspective on numbering.
In this chapter, we explore different numbering systems and grouping systems, eventually discussing base 2, the language of computers.


# Number Representation and Calculation

## Hindu-Arabic Positional System

### Learning Objectives

After completing this section, you should be able to:
1. Evaluate an exponential expression.
2. Convert a Hindu-Arabic numeral to expanded form.
3. Convert a number in expanded form to a Hindu-Arabic numeral.
The modern system of counting and computing isn’t necessarily natural. That different symbols are used to indicate different quantities or amounts is a relatively new invention. Simple marking by scratches or dots, one for each item being counted, was the norm long into human history. The modern system doesn’t use repeated symbols to indicate more than one of a thing. It uses the place of a digit in a numeral to determine what that digit represents. A numeral is a symbol used to represent a number. A number is an abstract idea that represents quantity or amount.
Being clear about the difference between numeral and number is important. Just like a person can be called by various names, such as brother, father, husband, uncle, they are all representing the same person, John Smith. The person John Smith is the number, and the names brother, father, husband, and uncle are the numerals.
The system we use for counting and computing uses place values based on powers of 10. In this section, we review exponents and our positional system.

### Evaluating Exponential Expressions

Most modern numerical systems depend on place values, where the quantity represented depends not only on the digit, but also on where the digit is in the number. The place value is a power of some specific number, which means most numbering systems are actually exponential expressions. An exponential expression is any mathematical expression that includes exponents. So, evaluating such an expression means performing the calculation. In this chapter, we will be using exponents that are positive integer values. Before we do so, let’s remind ourselves about exponents and what they represent. Suppose you want to multiply a number. Let’s label that number , by itself some number of times. Let’s label the number of times . We denote that as . We say , or the base, raised to the th power, or the exponent. For example, if we are multiplying 13 by itself eight times, we write  and say 13 to the eighth power.
When computing exponential expressions, we should be careful to remember the order of operations. Using the order of operation rules, calculations inside the parentheses are done first, then exponents are calculated, then multiplication and division calculations are performed, and then addition and subtraction.

### Converting Hindu-Arabic Numerals to Expanded Form

When you see the number 738, and you speak the number out loud, what do you say? You probably said “seven hundred thirty-eight” while wondering what point could possibly be made by asking this. What you didn’t say was “seven, and three, and eight.” A pre-K student might say that. Which should make you wonder, why?
The reason is that you’ve been taught place values, or the positions of digits in a number that determine the values of those digits. You know that in a three-digit number, the first digit is hundreds, the second digit is tens, and the last digit is ones. These place values rely on powers of 10, which makes this system a base 10 system.
This sense of place value is what makes our system of numbers so useful. You’ve also been taught the Hindu-Arabic numeration system. This system, which uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, and also employs place value based on powers of 10, is in use today.
Writing a number using these place values is writing them in expanded form. For a number with  digits, the expanded form is the first digit times 10 raised to one less than , plus each following digit times 10 raised to one less than the previous power of 10. For example, the number 738 would be written as .
What about a four-digit number, like 5,825? Out loud, we’d say five thousand, seven hundred twenty-five. In expanded form, it would be . Notice that the largest exponent is one less than the number of digits, and that the exponents go down by one as we move through the number.

### Converting Numbers in Expanded Form to Hindu-Arabic Numerals

Converting from expanded form back into a Hindu-Arabic numeral is the reverse process of expanding a number, and is equivalent to evaluating the exponential expression.

### Check Your Understanding

### Key Terms

1. numeral
2. number
3. exponential expression
4. base
5. exponent
6. place value
7. base 10 system
8. Hindu-Arabic numeration system
9. expanded form

### Key Concepts

1. Exponents are used to represent repeated multiplication of a base.
2. In arithmetic, exponents are computed before multiplication, division, addition, and subtraction. Computing an exponent is done by multiplying the base by itself the number of times equal to the exponent.
3. The system of numbers currently used is the Hindu-Arabic system. Digits in this system take on values based on their place in the number. The place values are determined by multiplying the digit by 10 raised to the appropriate power.
4. The expanded form of a Hindu-Arabic number is the sum of each digit times 10 raised to the exponent for that place value.

### Videos

1. Exponential Notation


# Number Representation and Calculation

## Early Numeration Systems

### Learning Objectives

After completing this section, you should be able to:
1. Understand and convert Babylonian numerals to Hindu-Arabic numerals.
2. Understand and convert Mayan numerals to Hindu-Arabic numerals.
3. Understand and convert between Roman numerals and Hindu-Arabic numerals.
Each culture throughout history had to develop its own method of counting and recording quantity. The system used in Australia would necessarily differ from the system developed in Babylon that would, in turn, differ from the system developed in sub-Saharan Africa. These differences arose due to cultural differences. In nearly all societies, knowing the difference between one and two would be useful. But it might not be useful to know the difference between 145 and 167, as those quantities never had a practical use. For example, a shepherd likely didn't manage more than 100 sheep, so quantities larger than 100 might never have been encountered. This can even be seen in our use of the term few, which is an inexact quantity that most would agree means more than two. However, as societies became more complex, as commerce arose, as military bodies developed, so did the need for a system to handle large numbers. No matter the system, the issues of representing multiple values and how many symbols to use had to be addressed. In this section, we explore how the Babylonians, Mayans, and Romans addressed these issues.

### Understand and Convert Babylonian Numerals to Hindu-Arabic Numerals

The Babylonians used a mix of an additive system of numbers and a positional system of numbers. An additive system is a number system where the value of repeated instances of a symbol is added the number of times the symbol appears. A positional system is a system of numbers that multiplies a “digit” by a number raised to a power, based on the position of the “digit.”
The Babylonian place values didn’t use powers of 10, but instead powers of 60. They didn’t use 60 different symbols though. For the value 1, they used the following symbol:  For values up to 9, that symbol would be repeated, so three would be written as  To represent the quantity 10, they used  For 20, 30, 40, and 50, they repeated the symbol for 10 however many times it was needed, so 40 would be written  When they reached 60, they moved to the next place value. The complete list of the Babylonian numerals up to 59 is in .
You can see how Babylonians repeated the symbols to indicate multiples of a value. The number 6 is 6 of the symbol for 1 grouped together. The symbol for 30 is three of the symbols for 10 grouped together. However, their system doesn’t go past 59. To go past 59, they used place values. As opposed to the Hindu-Arabic system, which was based on powers of 10, the Babylonian positional system was based on powers of 60. You should also notice there is no symbol for 0, which has some impact on the number system. Since the Babylonian number system lacked a 0, they didn’t have a placeholder when a power of 60 was absent. Without a 0, 101, 110, and 11 all look the same. However, there is some evidence that the Babylonians left a small space between "digits" where we would use a 0, allowing them to represent the absence of that place value. To summarize, the Babylonian system of numbers used repeating a symbol to indicate more than one, used place values, and lacked a 0.
So how do we convert from Babylonian numbers to Hindu-Arabic numbers? To do so, we need to use the symbols from , and then place values based on powers of 60. If you have  digits in the Babylonian number, you multiply the first “digit” by 60 raised to one less than the number of “digits.” You then continue through the “digits,” multiplying each by 60 raised to a power that is one smaller. However, be careful of spaces, since they represent a zero in that place.

### Understand and Convert Mayan Numerals to Hindu-Arabic Numerals

The Mayans employed a positional system just as we do and the Babylonians did, but they based their position values on powers of 20 and they had a dedicated symbol for zero. Similar to the Babylonians, the Mayans would repeat symbols to indicate certain values. A single dot was a 1, two dots were a 2, up to four dots. Then a five was a horizontal bar. The horizontal bars could be used three times, since the fourth horizontal bar would make a 20, which was a new position in the number. The 0 was a special picture, which appears like a turtle lying on its back. The shell would then be "empty," so maybe that’s why the symbol was 0. The complete list is provided in . Another feature of Mayan numbers was that they were written vertically. The powers of 20 increased from bottom to top.
To summarize, the  Mayan system of numbers  used repeating symbol to indicate more than one, used place values, and employed a 0. So how do we convert from Mayan numbers to Hindu-Arabic numbers? To do so, we need to use the symbols from  and then place values based on powers of 20. If you have  digits in the Mayan number, you multiply the first “digit” by 20 raised to one less than the number of “digits.” You then continue through the “digits,” multiplying each by 20 raised to a power that is one smaller than the previous power. Fortunately, there is an explicit 0, so there is no ambiguity about numbers like 110, 101, and 11.

### Understand and Convert Between Roman Numerals and Hindu-Arabic Numerals

The Mayan and Babylonian systems shared two features, one of which we are familiar with (place value) and one that we don’t use (repeated symbols). The  Roman system of numbers  used repeated symbols, but does not employ a place value. It also lacks a 0. The Roman system is built on the following symbols in .
As in the Mayan and Babylonian systems, a symbol may be repeated to indicate a larger value. However, at 4, they did not use IIII. They instead used IV. Since the I came before the V, the number stands for “one before five.” A similar process was used for 9, which was written IX, or “one before ten.” The value 40 was written XL, or “ten before fifty,” while 49 was written XLIX, or “forty plus nine.”
The following are the rules for writing and reading Roman numerals.
1. The representations for bigger values precede those for smaller values.
2. Up to three symbols may be grouped together; for example, III for 3, or XXX for 30, or CC for 200.
3. A larger value followed by a smaller value indicated addition; for example, VII for 7, XIII for 13, LV for 55, and MCC for 1200.
4. I can be placed before V to indicate 4, or before X, to indicate 9. These are the only ways I is used as a subtraction.
5. X can be placed before L to indicate 40, and before C to indicate 90. These are the only ways X is used as a subtraction.
6. C can be placed before D to indicate 400, and before M to indicate 900. These are the only ways C is used as a subtraction.
7. If multiple symbols are used, and a subtraction involving that symbol, the subtraction part comes after the multiple symbols. For example, XXIX for 29 and CCXC for 290.
Of course, we can convert from Hindu-Arabic numerals, to Roman numerals, too.

### Check Your Understanding

### Key Terms

1. additive system of numbers
2. positional system of numbers
3. Babylonian system of numbers
4. Mayan system of numbers
5. Roman system of numbers

### Key Concepts

1. Historically, there have been many systems for numbering. One system is an additive system, in which symbols are repeated to express larger numbers. Another system is a positional system, in which the digits and their positions determine the quantity being represented.
2. The Babylonian system was a combination of a positional and additive system. It used 60 as its base. Using that in the positional system makes it possible to convert between Babylonian and Hindu-Arabic numbers.
3. The Mayan system was a combination of a positional and additive system. It used 20 as its base. Using that in the positional system makes it possible to convert between Mayan and Hindu-Arabic numbers.
4. The Roman system was an additive system. Knowing what each symbol represents makes it possible to convert between Roman and Hindu-Arabic numbers.

### Videos

1. Converting Between Babylonian and Hindu-Arabic numbers
2. Converting Mayan Numbers to Hindu-Arabic Numbers
3. Converting From Roman Numbers to Hindu-Arabic Numbers
4. Converting From Hindu-Arabic Numbers to Roman Numbers


# Number Representation and Calculation

## Converting with Base Systems

### Learning Objectives

After completing this section, you should be able to:
1. Convert another base to base 10.
2. Write numbers in different base systems.
3. Convert base 10 to other bases.
4. Determine errors in converting between bases.
In our system of numbers, we use base 10, but using base 10 was not a given within other systems. There were other systems that used bases other than 10, as we saw with the Mayans and the Babylonians. The base 10 system comes down to grouping objects in sets of 10, but grouping in sets of 10 only happens if the culture values grouping by that many. We feel 10 is natural because we have 10 fingers. There are other systems using other grouping values, such as 4 or 20.
One good reason for examining other bases is to remind ourselves how we had to learn arithmetic when we were young, memorizing rules for our base 10 system. We had to learn why those arithmetic rules made sense, such as why  and . Another good reason for learning other base systems is due to computers; their circuitry instead uses base 2.
In this section, we explore other base systems and how to convert between them.

### Conversion of Another Base into Base 10 and Other Bases

We saw in Hindu-Arabic Positional System that our Hindu-Arabic system uses base 10, which is a system using place values of digits that depend on powers of 10 (or, are based on powers of 10). We’ve already worked with bases other than base 10: The Babylonian system was base 60, while the Mayan system was base 20.
To explore how our base 10 system is used, answer the following question: What’s the following quantity: 4,572? You probably said four thousand five hundred seventy-two (no, there is no “and” between hundred and seventy). But why do you think that 4 means four thousand? A very young person when learning their numbers might say that’s a four five seven and two. But you added the context of thousands to the four. Why?
Place value, that’s why. You learned early on that where the numeral was gave it different meanings. Ten thousands, thousands, hundreds, tens, and ones. So, you translate that symbol string (4,572) into “four thousand five hundred seventy-two.” As we saw in Hindu-Arabic Positional System, expanding a Hindu-Arabic number involved writing the number using each digit times its appropriate power of 10. So, we could write 4,572 as .
One possible reason we use base 10 is that we have 10 fingers, and in the cultures where the Hindu-Arabic system developed, that became the standard. Other cultures may have used other ways of organizing numbers, perhaps using 20 by including toes, or using 60 because 60 has many divisors. Mathematically though, base 10 is an awkward base to work in since 10 has limited divisors. But we think it is easy and simple because that’s what we’ve been taught to use.
Using a base 10 system means we need 10 symbols to make our numbering system work: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Now imagine that we all only had 6 fingers instead of 10 and our counting system was based on those 6 fingers. We would be counting in groups of 6, not groups of 10. How would this change how we work with quantity?
First, we’d need only six symbols. Let’s use 0, 1, 2, 3, 4, 5. Second, our place values would be based on powers of 6, not powers of 10. For instance, the number 3,024 in base 6 would be .
    That is how you can translate a base 6 number into a base 10 number. When we calculate that expression we get .
This means the base 6 number 3,024 is equal to the base 10 number 664.
From now on, if we are using a base 6 number, we will follow it with the subscript 6, like the following: 3,0246 means the number is in base 6.
A base 10 number gets no subscript (it’s the standard). So, 3,024 is a base 10 number. A base 13 number would be 4,67213.
So, a base 6 system uses only the symbols 0, 1, 2, 3, 4, and 5. Also, the place values use powers of 6. However, we still don’t know how to count in base 6. In order to do so, we’d have to know how to represent the quantities larger than five in base 6. Let’s review how our base 10 system works by counting from 0 to 100, which shows how larger values are represented.
In writing the base 10 numbers, you start with these first 10 values:
But you’ve run out of symbols. So, we use two digits:
The 1 out front means you’ve run out of digits one time.
But now you’ve run out twice. Continuing with those numbers gives:
And so on,
Eventually, you hit the 90s,
And you’ve run out of the digits again! So, we say we’ve run out of digits in the tens place one time, hence:
That’s the pattern we use in base 10. We write out the symbols until we’ve used all the symbols, then add a digit in front that counts how many times we’ve used the digits. Knowing the numbers, or being able to count higher and higher, is necessary to understand how all the arithmetic works, as it all goes back to counting.
The counting pattern is the same for any other base, including base 6. So, let’s start:
But we’ve run out of symbols! Just like in base 10, we use a second digit, where the first digit will tell us we’ve run out of symbols one time.
And we use the same pattern:
But we’ve run out of symbols for that front digit. So, we indicate it the same way as in base 10…by adding a third digit in front, indicating we’ve run out of symbols once in the second place:
The symbol pattern is the same, but truncated. We only use the six symbols. So that is how we represent base 6. Being able to write out these numbers is important when working with addition in the base.
When using a base larger than 10, though, we need more symbols. Instead of creating new symbols, we use capital letters, with A representing the digit for "10," B representing the digit for "11," and so on.
Base 2 is important in the digital age, as it is the system used by computers. It is the simplest base to work with, but has the drawback that the numbers in base 2 may use many, many digits. In Addition and Subtraction in Base Systems and Multiplication and Division in Base Systems, we will look at base 2 in each situation.

### Conversion of Base 10 into Another Base

Converting from base 10 into another base uses repeated division, recording the remainder at each step. Then, the number in the new base is the remainder starting from the last remainder found. To be accurate in what we’re saying, we need to remind ourselves of some terminology associated with division. When integers are divided, the one being divided is the dividend, and the one that is dividing the dividend is the divisor. The quotient is the largest natural number that can be multiplied by the divisor where the product is less than the dividend.
When the integer  is divided by the integer ,  is called the dividend and  is the divisor.
To convert a base 10 number  into base , we divide  by , recording the remainder. Then we divide the quotient from that step by the base , and record the remainder again. We continue this process until the quotient is 0. Then, the base  number has digits that start with the last remainder and use each remainder in reverse order.

### Converting from Hindu-Arabic Numbers to Mayan Numbers

To convert from a Hindu-Arabic number to a Mayan number involves two distinct processes. First, the number must be converted to base 20, using the process described and demonstrated previously. Next, that base 20 number has to be written using Mayan numerals. For reference, the Mayan numerals and their values are below.

### Errors in Converting Between Bases

There are some common errors that are made when converting between bases. Often, it comes down to using an “illegal” symbol in the new base.
When converting from base 10 to another base, an illegal symbol will be used if a mistake was made in the division process used to find the number in the new base. Since the digits are based on the remainders, any remainder that is an illegal symbol would indicate an error.
Another possible way to detect an error in converting between bases is to count the number of digits. When converting from a higher base to a lower base, the number of digits cannot get smaller. Similarly, when converting from a lower base to a higher base, the number of digits cannot get bigger. So, if a base 10 number is converted to a base 3 number, the number of digits in the new base 3 numbers cannot be less than the number of digits in the base 10 number. Similarly, if a base 7 number is converted to base 10, the number of digits in the base 10 number cannot be more than the number of digits in the original base 7 number.

### Check Your Understanding

The Babylonian system used base 60. To convert from Hindu Arabic numbers into Babylonian numbers, the process for converting from base 10 to a different base would be done first. Then, the results found in the conversion process would be changed to Babylonian numerals. This process is similar to the one for Mayan numbers.

### Key Terms

1. base 10
2. remainder
3. dividend
4. divisor
5. quotient

### Key Concepts

1. The system we use is the base 10 system. Base 10 is not the only base that can be used. To use another base, one could start with a list of numbers in that base.
2. To indicate that a number is written in a base other than 10, a subscript is appended to the end of the number. That subscript indicates the base for the number.
3. Numbers written in a base smaller than 10 use the same symbols as base 10. However, when using bases larger than 10, the symbols A, B, C, … are used to represent digits larger than 9.
4. To convert from a number written in a base other than 10 into a base 10 number, the number is written in expanded form and then that expression is computed.
5. To convert a number from base 10 into another base, the base 10 number is repeatedly divided by the new base. The remainders when performing these divisions become the digits for the number in the new base.
6. Common errors can be detected when performing base conversions.

### Videos

1. Convert Base 7 to Base 10
2. Converting from Base 10 to Another Base


# Number Representation and Calculation

## Addition and Subtraction in Base Systems

### Learning Objectives

After completing this section, you should be able to:
1. Add and subtract in bases 2–9 and 12.
2. Identify errors in adding and subtracting in bases 2–9 and 12.
Once we decide on a system for counting, we need to establish rules for combining the numbers we’re using. This begins with the rules for addition and subtraction. We are familiar with base 10 arithmetic, such as  or . How does that change if we instead use a different base? A larger base? A smaller one? In particular, computers use base 2 for all number representation. When your calculator adds or subtracts, multiplies or divides, it uses base 2. This is because the circuitry recognizes only two things, high current and low current, which means the system is uses only has two symbols. Which is what base 2 is.
In this section, we use addition and subtraction in bases other than 10 by referencing the processes of base 10, but applied to a new base system.

### Addition in Bases Other Than Base 10

Now that we understand what it means for numbers to be expressed in a base other than 10, we can look at arithmetic using other bases, starting with addition. When you think back to when you first learned addition, it is very likely you learned the addition table. Once you knew the addition table, you moved on to addition of numbers with more than one digit. The same process holds for addition in other bases. We begin with an addition table, and then move on to adding numbers with two or more digits.
We worked with base 6 earlier, and have the numbers in base 6 up to 1006. Using that table of values, we can create the base 6 addition table.
Here’s the beginning of the base 6 addition table:
Many of the cells are not filled out. The ones filled in are values that never get past 5, which is the largest legal symbol in base 6, so they are acceptable symbols. But what do we do with 5 + 3 in base 6? We can’t represent the answer as “8” since “8” is not a symbol available to us. Let’s go back to the list of numbers we have for base 6.
So, what is 5 + 1 equal to in base 6? Well, start at the 5, and jump ahead one step. You land on 10.
This means that, in base 6, 5 + 1 = 10.
So, what is 5 + 2 in base 6? Well, 5 + 2 = 5 + 1 + 1, so 10 + 1…jump one more space and you land on 11. So, 5 + 2 = 11 in base 6.
And so it goes. Using that process, stepping one more along the list, we can fill in the remainder of the base 6 addition table ().
With this table, and with our understanding of “carrying the one,” we can then use the addition table to do addition in base 6 for numbers with two or more digits, using the same processes you learned for addition when you did it by hand.
To summarize the creation of the addition tables for a given base, do the following.
Step 1: Set up the table.
Step 2: Fill in all the additions that use the “legal” symbols for the base. The diagonal that goes from upper left to lower right that is immediately next to the filled boxes all get the value 10, regardless of base.
Step 3: Enter the values that are in the “teens.” This can all be done on one table without creating multiple copies of previously done work.
As seen previously, when performing addition in another base, set up the problem exactly as you would for addition in base 10. At each step, check the addition table for the base. As in base 10 addition, move right to left, adding down the columns using the rules in the addition table. When necessary and just as in base 10, be sure to carry the 1.

### Subtraction in Bases Other Than Base 10

Subtraction in bases other than base 10 follow the same processes as base 10 subtraction, but, as with addition, using the addition table for the base.

### Errors When Adding and Subtracting in Bases Other Than Base 10

Errors when computing in bases other than 10 often involve applying base 10 rules or symbols to an arithmetic problem in a base other than base 10. The first type of error is using a symbol that is not in the symbol set for the base. For instance, if a 9 shows up when working in base 7, you know an error has happened because 9 is not a legal symbol in base 7.
The second type of error is using a base 10 rule when the numbers are not in base 10. For instance, if you are working in base 13, then 913 + 913 is not 1813, even though 18 is the correct answer in base 10.

### Check Your Understanding

### Key Concepts

1. Addition tables for bases other than 10 can be built using the same processes that are used in base 10, including using a number line.
2. Addition in bases other than base 10 use the same processes as addition in base 10, but use the addition table for that base.
3. Subtraction in bases other than base 10 use the same processes as subtraction in base 10, but use the addition table for that base.


# Number Representation and Calculation

## Multiplication and Division in Base Systems

### Learning Objectives

After completing this section, you should be able to:
1. Multiply and divide in bases other than 10.
2. Identify errors in multiplying and dividing in bases other than 10.
Just as in Addition and Subtraction in Base Systems, once we decide on a system for counting, we need to establish rules for combining the numbers we’re using. This includes the rules for multiplication and division. We are familiar with those operations in base 10. How do they change if we instead use a different base? A larger base? A smaller one?
In this section, we use multiplication and division in bases other than 10 by referencing the processes of base 10, but applied to a new base system.

### Multiplication in Bases Other Than 10

Multiplication is a way of representing repeated additions, regardless of what base is being used. However, different bases have different addition rules. In order to create the multiplication tables for a base other than 10, we need to rely on addition and the addition table for the base. So let’s look at multiplication in base 6.
Multiplication still has the same meaning as it does in base 10, in that  is 4 added to itself six times, .
So, let’s apply that to base 6. It should be clear that 0 multiplied by anything, regardless of base, will give 0, and that 1 multiplied by anything, regardless of base, will be the value of “anything.”
Step 1: So, we start with the table below:
Step 2: Notice  is there. But we didn’t hit a problematic number there (4 works fine in both base 10 and base 6). But what is ? If we use the repeated addition concept, . According to the base 6 addition table (), . So, we add that to our table:
Step 3: Next, we need to fill in . Using repeated addition,  (if we use our base 6 addition rules). So, we add that to our table:
Step 4: Finally, . And so we add that to our table:
Step 5: A similar analysis will give us the remainder of the entries. Here is  demonstrated: .
This is done by using the addition rules from Addition and Subtraction in Base Systems, namely that , and then applying the addition processes we’ve always known, but with the base 6 table (). In the end, our multiplication table is as follows:
Notice anything about that bottom line? Is that similar to what happens in base 10?
To summarize the creation of a multiplication in a base other than base 10, you need the addition table of the base with which you are working. Create the table, and calculate the entries of the multiplication table by performing repeated addition in that base. The table needs to be drawn only the one time.
The multiplication table in base 2 below is as minimal as the addition table in the solution for . Since the product of 1 with anything is itself, the following multiplication table is found.
As with the addition table, we can use the multiplication tables and the addition tables to perform multiplication of two numbers in bases other than base 10. The process is the same, with the same carry rules and placeholder rules.
Summarizing the process of multiplying two numbers in different bases, the multiplication table is referenced. Using that table, the multiplication is carried out in the same manner as it is in base 10. The addition rules for the base will also be referenced when carrying a 1 or when adding the results for each digit’s multiplication line.

### Division in Bases Other Than 10

Just as with the other operations, division in a base other than 10, the process of division in a base other than 10 is the same as the process when working in base 10. For instance,  because, we know that . So, for many division problems, we are simply looking to the multiplication table to identify the appropriate multiplication rule.

### Errors in Multiplying and Dividing in Bases Other Than Base 10

The types of errors encountered when multiplying and dividing in bases other than base 10 are the same as when adding and subtracting. They often involve applying base 10 rules or symbols to an arithmetic problem in a base other than base 10. The first type of error is using a symbol that is not in the symbol set for the base.
The second type of error is using a base 10 rule when the numbers are not in base 10. For instance, in base 17,  would be incorrect, even though in base 10, . That rule doesn’t apply in base 17.

### Check Your Understanding

### Key Concepts

1. Multiplication tables for bases other than 10 can be built using the same processes that are used in base 10, including using repeated addition and the addition table for the base.
2. Multiplication in bases other than base 10 use the same processes as multiplication in base 10, but use the multiplication table for that base.
3. Basic division in bases other than base 10 use the same processes as basic division in base 10, where the missing factor process is used.

### Projects

### Additive Systems

Go online. Google “additive number systems.”What system comes up?
1. Describe the additive system you found.
Using Google, identify three more additive systems of numbers.
1. Compare and contrast the systems you found. For instance, how many times can a symbol be used before a new symbol is used.
2. Identify three situations where additive systems are still used.

### Computers and Bases

Use Google to determine what base computers use.
Were other bases attempted for use in computers?
Determine why the base used in computers is appropriate.
Determine how the base used in computers is related to the circuitry in computers.
Determine how Boolean logic and the base used in computers are related, and might be identical.
There is research into using quibits in computers. Find out what quibits are and how can they improve computing speed.

### Cultures Using Base Systems Other Than 10

Using Google, find three cultures, other than Babylonian or Mayan, that use base systems other than 10.
1. Tell what base is used for each system.
2. If possible, determine why the culture used that base system.
3. Choose one of those systems. Explain that base system. Be sure to address whether the system is additive, place-value based, a blend of the two, and if it employs a zero.

### History of Zero

Using any resources available to you, determine the history of 0 in at least three different numbering systems. Address at least when and why such a development occurred and why a 0 is vital to the use of a positional system.

### Numbering Systems from Other Global Regions

Using any resources available to you, find at least three numbering systems from sub-Saharan Africa, Australia, China, or the Pacific Islands. Explore if they are positional or additive systems (or combinations!), the terminology of the system, if they used a 0, and what base they employed (if positional).

### Chapter Review

### Hindu-Arabic Positional System

### Early Numeration Systems

### Converting with Base Systems

### Addition and Subtraction in Base Systems

### Multiplication and Division in Base Systems

### Chapter Test


# Algebra

## Introduction

The jump from arithmetic to algebra can be a difficult one for many students. Many students struggle with the idea that mathematics can include situations that aren’t static and do change. In elementary arithmetic, a situation such as:
is a static situation and will yield the answer of 8 every time. However, a situation such as:

can yield many different answers because the answer depends on what amount (number) that  represents. Since the value of  can vary (represent different values), it is known as a variable.
Algebra is useful to better model real life situations. In the first equation shown,  can only model situations where you add those two numbers together. For example, if your uncle gives you five dollars and your aunt gives you three dollars, then you will always receive eight dollars. The second equation  can model more complex situations. For example, you wish to buy a game that costs $38 but you only have three dollars. Your uncle will pay you five dollars an hour to work for him. If you’ve worked five hours, have you earned enough money? If not, how many hours will you have to work?
Algebra and algebraic thinking open up a world of possibilities that arithmetic alone cannot do.