Number Types | Purplemath Number Types Purplemath Numbers are classified according to type. The first type of number is the first type you ever learned about: the counting, or "natural" numbers: 1, 2, 3, 4, 5, 6, ... The next type is the "whole" numbers, which are the natural numbers together with zero: 0, 1, 2, 3, 4, 5, 6, ... (The number zero, spread from India by north-African scholars, was originally viewed by European authorities as being demonic.) Content Continues Below Then come the "integers", which are zero, the natural numbers, and the negatives of the naturals: ..., –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, ... The next type of number is the "rational", or fractional, numbers, which are technically regarded as ratios (divisions) of integers. In other words, a fraction is formed by dividing one integer by another integer. Note that each new type of number contains the previous type within it.

The wholes are just the naturals with zero thrown in. The integers are just the wholes with the negatives thrown in. And the fractions are just the integers with all their divisions thrown in. (Remember that you can turn any integer into a fraction by putting it over the number 1. For example, the integer 4 is also the fraction .) Affiliate Once you've learned about fractions, there is another major classification of numbers: the ones that can't be written as fractions. Remember that fractions (also known as rational numbers) can be written as terminating (ending) or repeating decimals; for example, 0.5 = and 0.76 = , are terminating decimals, while 0.333333.... = and 0.538461538461... = are repeating decimals. On the other hand, we have all those other numbers that can be written as non-repeating, non-terminating decimals; these number are non-rational (that is, they cannot be written as fractions), so they are called the "irrationals". Examples would be ("the square root of two") or the number ("3.14159...", from geometry).

The rationals and the irrationals are two totally separate number types; there is no overlap. Advertisement Putting these two major classifications, the rationals and the irrationals, together in one set gives you the "real" numbers. Unless you have dealt with complex numbers (the numbers with an "i" in them, such as 4 – 3i), then every number you have ever seen has been a "real" number. "But why", you ask, "are they called 'real' numbers? Are there 'pretend' numbers?" Well, yes, actually there are, though they're actually called "imaginary" numbers; they are what is used to make the complex numbers, and "imaginary" is what the "i" stands for. Affiliate WyzAnt Tutoring The commonest question I hear regarding number types is something along the lines of "Is a real number irrational, or is an irrational number real, or neither... or both?" Unless you know about complexes, everything you've ever done has used real numbers. Unless the number has an "i" in it, it's a real.

Here are some typical number-type questions (assuming that you haven't yet learned about imaginaries and complexes): True or False: An integer is also a rational number. Since any integer can be formatted as a fraction by putting it over 1, then this statement is true. True or False: A rational number is also an integer. Not necessarily; the integer 4 is also the rational number but, for instance, the rational number is not also a integer. So this statement is false. True or False: A number is either a rational number or an irrational number, but not both. True! In decimal form, a number is either non-terminating and non-repeating (so it's an irrational) or else it's not (so it's a rational); there is no overlap between these two number types! Content Continues Below Classify according to number type; some numbers may be of more than one type. 0.45 This is a terminating decimal, so it can be written as a fraction: . Since this fraction does not reduce to a whole number, then it's not an integer or a natural.

And everything is a real, so the answer is: rational, real 3.14159265358979323846264338327950288419716939937510... You probably recognize this as being π, though this may be more decimal places than you customarily use. The point, however, is that the decimal does not repeat, so π is an irrational. And everything (that you know about so far) is a real, so the answer is: irrational, real 3.14159 Don't let this fool you! Yes, you often use something like this as an approximation of π, but it isn't π! This is a rounded decimal approximation, and, since this approximation terminates, this is actually a rational, unlike π itself, which is irrational! The answer is: rational, real Affiliate 10 Obviously, this is a counting number. That means it is also a whole number and an integer. Depending on the text and teacher (there is some inconsistency), this may also be counted as a rational, which technically-speaking it is.

And of course it's also a real. The answer is: natural, whole, integer, rational (possibly), real This is a fraction, so it's a rational. It's also a real, so the answer is: rational, real This can also be written as , which is the same as the previous problem. The answer is: rational, real Your first impulse may be to say that this is irrational, because it's a square root, but notice that this square root simplifies: , which is just an integer. The answer is: integer, rational, real This number is stated a fraction, but notice that it reduces to –3, so this may also count as an integer. The answer is: integer (possibly), rational, real Except for the section in your book where you have to classify numbers according to type, you really won't need to be terribly familiar with this hierarchy. It's more important to know what the terms mean when you hear them. For instance, if your teacher talks about "integers", you should know that the term refers to the counting numbers, their negatives, and zero.

URL: Types of Numbers As you can see, the location of the ones digit in the binary representation indicates the numbers of the binary sequence that are to be added together to yield the base 10 number of interest. CARDINAL NUMBERS A cardinal number is a number that defines how many items there are in a group or collection of items. Typically, an entire group of items is referred to as the "set" of items and the items within the set are referred to as the "elements" of the set. For example, the number, group, or set, of players on a baseball team is defined by the cardinal number 9. The set of 200 high school students in the graduating class is defined by the cardinal number 200. (See ordinal numbers and tag numbers.) CATALAN NUMBERS Catalan numbers are one of many special sequences of numbers that derive from combinatorics problems in recreational mathematics. Combinatorics deals with the selection of elements from a set of elements typically encountered under the topics of probability, combinations, permutations, and sampling.

The specific Catalan numbers are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16,796 and so on deriving from This particular set of numbers derive from several combinatoric problems, one of which is the following. Given "2n" people gathered at a round table. How many person to person, non-crossing, handshakes can be made, i.e., no pairs of arms crossing one another across the table? A few quick sketches of circles with even sets of dots and lines will lead you to the first three answers easily. Two people, one handshake. Four people, two handshakes. Six people, 5 handshakes. With a little patience and perseverance, eight people will lead you to 14 handshakes. Beyond that, it is probably best to rely on the given expression. CHOICE NUMBERS Choice numbers, more commonly called combination numbers, or simply combinations, are the number of ways that a number of things can be selected, chosen, or grouped. Combinations concern only the grouping of items and not the arrangement of those items.

They typically evolve from the question how many combinations of "n" objects are possible using all "n" objects or "r" objects at a time? To find the number of combinations of "n" dissimilar things taken "r" at a time, the formula is: which can be stated as "n" factorial divided by the product of "r" factorial times (n - r) factorial. Example: How many different ways can you combine the letters A, B, C, and D in sets of three? Clearly, namely ABC, ABD, ACD, and BCD. (Note that ACB, BAC, BCA, CAB and CBA are all the same combination just arranged differently. In how many ways can a committee of three people be selected from a group of 12 people? We have: How many handshakes will take place between six people in a room when they each shakes hands with all the other people in the room one time? Here, Notice that no consideration is given to the order or arrangement of the items but simply the combinations. Another way of viewing combinations is as follows. Consider the number of combinations of 5 letters taken 3 at a time.

This produces: Now assume you permute (arrange) the r = 3 letters in each of the 10 combinations in all possible ways. Each group would produce r! permutations. Letting x = 5C3 for the moment, we would therefore have a total of x(r!) different permutations. This total, however, represents all the possible permutations (arrangements) of n things taken r at a time, which is shown under arrangement numbers and defined as nPr. Therefore, Using the committee of 3 out of 12 people example from above, Consider the following: How many different ways can you enter a 4 door car? It is clear that there are 4 different ways of entering the car. Another way of expressing this is: If we ignore the presence of the front seats for the purpose of this example, how many different ways can you exit the car assuming that you do not exit through the door you entered? Clearly you have 3 choices. This too can be expressed as: Carrying this one step further, how many different ways can you enter the car by one door and exit through another? Entering through door #1 leaves you with 3 other doors to exit through.

The same result exists if you enter through either of the other 3 doors. Therefore, the total number of ways of entering and exiting under the specified conditions is: Another example of this type of situation is how many ways can a committee of 4 girls and 3 boys be selected from a class of 10 girls and 8 boys? This results in: See ARRANGEMENT NUMBERS for determining the number of possible arrangements between items. CIRCULAR PRIMES A circular prime number is one that remains a prime number after repeatedly relocating the first digit of the number to the end of the number. For example, 197, 971 and 719 are all prime numbers. Similarly, 1193, 1931, 9311 and 3119 are all prime numbers. Other numbers that satisfy the definition are 11, 13, 37, 79, 113, 199 and 337.

Primes of two or more digits can only contain the digits 1, 3, 7 because, If 0, 2, 4, 5,6, or 8 were part of the number, in the units place, the number would be divisible by 2 or 5. It is thought that there are an infinite number of circular primes but has not yet been proven. COMPLEX NUMBERS Complex numbers are formed by the addition of a real number and an imaginary number, the general form of which is a + bi where i = = the imaginary number and a and b are real numbers. The "a" is said to be the real part of the complex number and b the imaginary part. Probably the easiest number to define after prime numbers. The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 is either a prime number or a composite number. As we know, a prime number "p" is any positive number the only divisors of which are 1 and p (or -1 and -p). Thus, by definition, any number that is not a prime number must be a composite number.

A composite number is any number having 3 or more factors/divisors and is the result of multiplying prime numbers together. Most of the positive integers are the product of smaller prime numbers. Examples: 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, etc., are all composite numbers, each being divisible by lower prime numbers. Every number divisible by 2, the only even prime, is composite. Every composite number can be broken down to a single unique set of prime factors and their exponents.

Examples: 210 = 2 x 3 x 5 x 7; 495 = 32 x 51 x 111 or 4500 = 2 x 2 x 3 x 3 x 5 x 5 x 5 = 22 x 32 x 53. This is the one and only possible factorization of the number 210. If a positive number N is evenly divisible by any prime number less than , the number N is composite. Wilson's Theorem states that for every prime number "p", [(p + 1)! + 1] is evenly divisible by "p". The converse was shown to also be true in that every integer "N" that evenly divides [(N + 1)! + 1] is prime. Combining these leads to the famous general theorem that a necessary and sufficient condition that an integer "N" be prime is that "N" evenly divide [(n + 1)! + 1]. Conversely, if "N" does not divide [(N + 1)! + 1], "N" is composite.

Unfortunately, the practical use of this method is minimal due to the large numbers encountered with high N's. CONGRUENT NUMBERS A number N is said to be congruent if there are two integers, x and y, that result in the expressions x2 + Ny2 and x2 - Ny2 being perfect squares. The smallest known congruent number is 5 which satisfies 412 + 5(122) = 492 and 412 - 5(122) = 312. The use of the square of a negative number results in another solution of 22 + 5(12) = 32 and 22 - 5(12) = (-1)2. While there are many congruent numbers, finding them is an arduous task. The expressions x2 + Ny2 and x2 - Ny2 are often useful in solving many problems in recreational mathematics.

COUNTING NUMBERS The counting numbers are the familiar set of whole numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,....., that we see and use every day. (The 0 is sometimes included.) The set of counting numbers is often referred to as the natural numbers. CUBIC NUMBERS or CUBES A cubic number is the third power of a number as in a x a x a = a3. Those familiar with the evolution of the squares from adding successive odd numbers might not be too surprised to discover how the cubes evolve from summing odd numbers also. Clearly, the nth cube is simply n3. Cubes can be derived in other ways also: The cube of any integer, "n", is the sum of the series of odd numbers beginning with (n2 - n + 1) and ending with (n2 + n - 1).

Example: For n = 6, (n2 - n + 1) = 31 and 31 + 33 + 35 + 37 + 39 + 41 = 216 = 63. The last line of numbers are the perfect cubes. The first ten cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000. The sum of the first n cubes starting with 1, is 13 + 23 + 33 +........+ n3. which, quite surprisingly, is the square of the nth triangular number, defined by Tn = n(n+1)/2. The sum of the cubes of the first n odd numbers is 2n4 - n2 = n2(2n2 - 1). The sum of the cubes of the first n even numbers is 2n4 + 4n3 + 2n2 = 2n2(n + 1)2.

The sum of the first n cubes, 13 + 23 + 33 + 43 +........+ n3 is equal to the square of the sum of the first n integers. Thus, 13 + 23 + 33 + 43 +........+ n3 = (1 + 2 + 3 + 4 +........+ n)2. The cube of any integer is the difference of the squares of two other integers. Every cube is either a multiple of 9 or right next to one. A perfect cube can end in any of the digits 0 through 9. Three digit numbers that are the sum of the cubes of their digits: 153, 370, 371, 407. The smallest number that is the sum of 2 cubes in two different ways. 1729 = 13 + 123 = 103 + 93. What are the dimensions of two cubes with integral sides that have their combined volume equal to the combined length of their edges. What are the dimensions of the cubes? x = 2 and y = 4.

The sum of any two cubes can never be a cube. The sum of a series of three or more cubes can equal a cube. Examples: 113 + 123 + 133 + 143 = 203 11343 + 11353 + .........21333 = 16,8303 The cube root of a number N is the number "a" which, when multiplied by itself twice, results in the number N or N = axaxa. There is no formula for extracting the cube root of a number. It can be obtained by means of a long division method or a simple estimation method. The sum of "n" terms of an arithmetic progression with the first term equal to the sum of the first "n" natural numbers and a common difference of "n" is n3. First, a method for approximating the cube root of a number to several decimal places which is usually sufficient for everyday use. Estimation Method 1--Make an estimate of the cube root of N = n lying between successive integers a and b.

2--Compute A = N - a3 and B = b3 - N 2--n = a + (bA)/[bA + aB] Example: Find the cube root of 146 1--With 146 lying between 125 and 216, let a = 5 and b = 6. 2--A = 21 and B = 70 3--n = 5 + (6 x 21)/[6 x 21 + 5 x 70] = 5 + 126/476 = 5 + .264 = 5.2647 4--The cube root of 146 is 5.2656 A more exact method for determining integer cube roots.

Note that each cube of the numbers 1 through 10 ends in a different digit: n3....1......8....27....64...125...216...343...512...729...1000 Also note that the last digit is the cube root for all cases except 2, 3, 7 and 8. A quick review of these exceptions leads to the fact that these four digits are the difference between 10 and the cube root, i.e., 8 = 10 - 2, 7 = 10 - 3, 3 = 10 - 7 and 2 = 10 - 8. How can this information be used to determine the cube root of a number? Given the cube of a number between 1 and 100, say 300,763. The last digit tells us that the last digit of the cube root is 10 - 3 = 7. Eliminating the last 3 digits of the cube leaves the number 300. The number 300 lies between the cubes of 6 and 7 in our listing above.

The first digit of our cube root will be the lowest of these two numbers, in this case 6. Therefore, the cube root of 30,763 becomes 67. The last digit of 4 is the last digit of the cube root. The number 592 lies between 512 and 729, the cubes of 8 and 9. Therefore, the cube root of 592,704 becomes 84. Of course, this is only useful if you know ahead of time that the cube is a perfect cube, i.e., having an integral cube root. CYCLIC NUMBERS A cyclic number is a number of "n" digits that when multiplied by 1, 2, 3,...n, results in the same digits but in a different order.

For example, the number 142,857 is a cyclic number since 142,857 x 2 = 285,714, 142,857 x 3 = 428,571, 142,857 x 4 = 571,428, and so on. It is not known just how many cyclic numbers exist. DECIMAL NUMBERS Decimal numbers are numbers expressed through the decimal, or base 10, number system where each digit represents a multiple of some power of 10. The term applies primarily to numbers that have fractional parts so indicated by a decimal point. A number less than 1 is called a decimal fraction, e.g., .673. A mixed decimal is one consisting of an integer and a decimal fraction, e.g., 37.937. . Rational numbers can be expressed in the form of a fraction, 1/2, or as a decimal, .50, 1/8, or as a decimal, .125.

From experience, we know that a fraction expressed in decimal form will either terminate without a remainder such as 3/8 = 0.375 or 7/8 = 0.875, repeat the same digit endlessly such as 1/3 = .3333333..... or 2/3 = .6666666....., repeat a series of different digits repeatedly such as 1/27 = .037037037... or 1/7 = .142857142857...., or repeat a series of digits after some non repeating digits such as 1/12 = .0833333..... All prime denominators produce repeating decimals. Fractions with the same denominator often produce decimals with the same period and period length but with the digits starting with a different number in the period.

For instance, 1/7 = .142857142857142857...., 2/7 = .285714285714285714...., 3/7 = .428571428571428571...., 4/7 = .571428571428571428...., 5/7 = .714285714285714285...., and 6/7 = .857142857142857142.... Other denominators produce two or more repeating periods in different orders. Investigate those with prime denominators of 11, 13, 17, 19, etc. and then 9, 12, 14, 16, 18, etc.and see what results. What do you notice? Another interesting property of repeating decimals of even period length is illustrated by the following.

Take the decimal equivalent of 2/7 = .285714285714...... The repeating period is 285714. Split the period into two groups of three digits and add them together. The result is 999. Do the same with any other repeating decimal period and the result will always be a series of nines. Take 1/17 = .0588235294117647........ Adding 05882352 and 94117647 gives you 99999999. All repeating decimals, regardless of the period and length, are rational numbers. This simply means that it can be expressed as the quotient of two integers. A question that frequently arises is how to convert a repeating decimal, which we know to be rational, back to a fraction. Rational decimal fractions may be converted to fractions as follows: Given the decimal number N = 0.078078078... Multiply N by 1000 or 1000N = 78.078078078...

Subtracting N = .078078078... .................................999N = 78 making N = 78/999 = 26/333. Given the decimal number N = .076923076923... Muliply N by 1,000,000 or 1,000,000N = 76,923.0769230769230... Subtracting N = .0769231769230... .........................................999,999N = 76,923 making N = 76,923/999,999 = 1/13 An easier way to derive the fraction is to simply place the repeating digits over the same number of 9's. For example, the repeating decimal of .729729729729 converts to the fraction of 729/999 = 27/37.

Another trick where zeros are involved is to place the repeating digits over the same number of 9's with as many zeros following the 9's as there are zeros in the repeating decimal. For example, .00757575 leads to 075/9900 = 1/132. DEFICIENT NUMBERS Deficient numbers are part of the family of numbers that are either deficient, perfect, or abundant. Deficient numbers, dN, are numbers where the sum of its aliquot parts (proper divisors), sa(N), is less than the number itself sa(N) < N. (In the language of the Greek mathematicians, the divisors of a number N were defined as any whole numbers smaller than N that, when divided into N, produced whole numbers. The factors/divisors of a number N, less the number itself, are referred to as the aliquot parts, or aliquot divisors, of the number.) Equivalently, N is also deficient if the sum, s(N), of all its divisors is less than 2N.

It can be readily seen that using the aliquot parts summation, sa(16) = 1+2+4+8 = 14 < N = 16 while using all of the divisors, s(16) = 1+2+4+8+16 = 31 < 2N = 32, making the number 16 deficient under either definition. 1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,19,21,22,23 are all deficient. A prime number or any power of a prime number is deficient. The divisors of a pefect or deficient number is deficient. DIGITAL ROOT The digital root of a number is the single digit that results from the continuous summation of the digits of the number and the numbers resulting from each summation. For example, consider the number 7935. The summation of its digits is 24. The summation of 2 and 4 is 6, the digital root of 7935.

Digital roots are used to check addition and multiplication by means of a method called casting out nines. For example, check the summation of 378 and 942. The DR of 378 is 9, 3+7+8=18, 1+8=9.. The DR of 942 is 6, 9+4+2=15, 1+5=6. Adding 9 and 6 produces 15, The DR of 15 is 6, 1+5=6. The summation of 378 and 942 is 1320. The DR of 1320 is 6. With the two final DR's are equal, the addition is correct. EGYPTIAN FRACTIONS Egyptian fractions are the reciprocals of the positive integers where the numerator is always one. They are often referred to as unit fractions. They were used exclusively by the Egyptians to represent all forms of fractions. The two fractions that they used that did not have a unit fraction was 2/3 and 3/4.

The only other fractions that they seemed to have a strong interest in were those of the form 2/n where n was any positive odd number. The Rhind papyrus contains a list of unit fractions representing a series of 2/n for odd n's from 5 to 501. It is unclear as to why they found these 2/n fractions so important. These and other 2/n fractions may be derived from 2/n = 1/[n(n+1)/2] + 1/[n+1)/2]. In the year 1202, Leonardo Fibonacci proved that any ordinary fraction could be expressed as the sum of a series of unit fractions in an infinite number of ways. He used the then named greedy method for deriving basic unit fraction expansions. He described the greedy method in his Liber Abaci as simply subtracting the largest unit fraction less than the given non unit fraction and repeating the process until only unit fractions remained. It was later shown that the greedy method, when applied to any fraction m/n, results in a series of no more than "m" unit fractions. An example will best illustrate the process. Reduce the fraction 13/17 to a sum of unit fractions.

Dividing the fraction yields .7647. Of the unit fractions 1/2, 1/3, 1/4, 1/5, etc., 1/2 is the largest that is smaller than 13/17 so we compute 13/17 - 1/2 = 9/34 making 13/17 = 1/2 + 9/34. Repeating with 9/34, we have 9/34 - 1/4 = 2/136 = 1/68 making 13/17 = 1/2 + 1/4 + 1/68. Alternatively, divide the numerator into the denominator and use the next highest integer as the new denominator. 17/13 = 1.307 making 2 the denominator of the fraction to be subtracted from 13/17. Reduce the fraction 23/37 to a sum of unit fractions.

23/37 - 1/2 = 9/74 - 1/16 = 70/1184 - 1/17 = 6/20,128 - 1/3355 = 1/33,764,720 making 23/37 = 1/2 + 1/16 + 1/17 + 1/3355 + 1/33,764,720. There are many methods or algorithms that derive the unit fractions for any fraction m/n. Much more information regarding unit fractions can be found at Unit Fractions Search at Egyptian Fractions at ~eppstein/numth/egypt/ Unit Fraction from Math World at Algorithms for Egyptian Fractions at ~eppstein/numth/egypt/force.html Creating Unit Fractions at The unit fractions derived by means of the method shown, or any other method, can be further broken down into other unit fractions by means of the identity 1/a = 1/(a+1) + 1/a(a+1), also known to Fibonacci.

For example, 1/2 = 1/(2+1) + 1/2(2+1) = 1/3 + 1/6. Further still, 1/3 = 1/(3+1) + 1/3(3+1) = 1/4 + 1/12 and 1/6 = 1/(6+1) + 1/6(6+1) = 1/7 + 1/42 yielding 1/2 = 1/4 + 1/7 + 1/12 + 1/42. While the number of unit fractions derivable for any given fraction is therefore infinite, there is apparently no known procedure for deriving a series with the least number of unit fractions or the smallest largest denominator. . EQUABLE TRIANGLES A shape is called equable if its area equals its perimeter. There are exactly five equable Heronian triangles: the ones with side lengths (5,12,13), (6,8,10), (6,25,29), (7,15,20), and (9,10,17).

EQUIVALENT NUMBERS Equivalent numbers are numbers where the aliquot parts (proper divisors other than the number itself) are identical. For instance, 159, 559 and 703 are equivalent numbers since their aliquot parts sum to 57. f(159) = 1, 3, 53 and 159 where 1 + 3 + 53 = 57. f(559) = 1, 13, 43 and 559 where 1 + 13 + 43 = 57. f(703) = 1, 19, 37 and 703 where 1 + 19 + 37 = 57. EVEN NUMBERS Probably the easiest number to define, an even number is any number that is evenly divisible by 2. The nth even number is given by Ne = 2n. The sum of the set of "n" consecutive even numbers beginning with 2 is given by Se = n(n + 1).

The sum of the set of m consecutive even numbers starting with n1 and ending with n2 is given by Se(n1-n2) = n2^2 - n1^2 + (n1 + n2) or (n1 + n2)(1 + n1 - n2). The sum of the squares of the even numbers starting with 2^2 is given by Se^2 = (4n^3 + 6n^2 + 2n)/3. An even number multiplied by any number, or raised to any power, results in another even number. The number 2 is the only even prime number. 2.1 - The Types of Numbers 2.1 - The Types of Numbers Let's survey the types of numbers. We will look at natural and whole numbers , integers , rational numbers , irrational numbers , real numbers , imaginary numbers and complex numbers . The Natural and Whole Numbers We start with the natural numbers. These are the numbers 1, 2, 3, … (The … symbol means that the sequence goes on forever.) They are used for counting. If we include zero then we get the whole numbers, 0, 1, 2, 3, ….

The natural and whole numbers are usually considered to be exact (e.g. there are 4 tires on a car, 8 legs on a spider). But sometimes they are approximate (e.g. there were 1000 people in the crowd). Here they are, shown on the number line: The Integers Now for each of the numbers 1, 2, 3, … let’s create its opposite or negative and put it on the opposite side of the number line, like this: We say that 1 and −1 are opposites, 2 and −2 are opposites, etc. and we also say that −1 is the opposite of 1 and that 1 is the opposite of −1. Negative numbers are used to describe debts as opposed to assets, temperatures below zero as opposed to temperatures above zero, heights below sea level as opposed to heights above sea level, and so on. The set of numbers …, −3, −2, −1, 0, 1, 2, 3, … (the whole numbers and their opposites) is called the integers. The Rational Numbers Next are the rational numbers. A rational number is any number that can be expressed as the quotient of two integers.

We can use any of these fraction notations to express rational numbers: a is called the numerator and b is called the denominator. The denominator cannot be zero. The notation means that we break something into b equal pieces and we have a of those pieces. For example if we break a pie into 4 pieces and take 1 piece then we have 1/4 of the pie: Rational numbers can also be written in decimal notation instead of fraction notation. For example: 1/4 = 0.25 The decimal notation 0.25 means literally “25/100” and 25/100 and 1/4 are equivalent fractions. Notice that some rational numbers have no exact decimal equivalent. For example 1/3 is approximately equal to 33/100 but not exactly: 1/3 ≈ 0.33 (The symbol ≈ means “is approximately equal to”.) Rational numbers are usually considered to be exact. For this reason the Algebra Coach program will not convert fractions to decimals when it is running in exact mode. Notice that since 3/1 = 3 and −5/1 = −5, the rational numbers include all the integers.

The Irrational Numbers The irrational numbers are those that cannot be expressed as a ratio of two integers. Examples are as well as the square roots of many other numbers, and special numbers like e and π. It turns out that there are as many irrational numbers as rational. Irrational numbers have no exact decimal equivalents. To write any irrational number in decimal notation would require an infinite number of decimal digits. Thus these are only approximations: ≈ 1.732, e ≈ 2.718 and π ≈ 3.14, For this reason the Algebra Coach program will not convert irrational numbers to decimals when it is operating in exact mode. The Real Numbers The rational numbers and the irrational numbers together make up the real numbers. The real numbers are said to be dense. They include every single number that is on the number line. The number line is useful for understanding the order of numbers. Smaller numbers are farther to the left and larger numbers are farther to the right. We use the symbol < to mean “is less than” and the symbol > to mean “is greater than”. Here are some examples of the use of these symbols: 5 < 8 because 5 is to the left of 8.

−5 < 2 because −5 is to the left of 2. In fact any negative number is less than any positive number. 5 < 8 and 8 > 5 are two ways of stating the same fact. 3.14 < π < 3.15 I call this a “less than sandwich”. It means that 3.14 < π and also π < 3.15. In other words the number π is somewhere between 3.14 and 3.15. Real numbers often result from making measurements and measurements are always approximate. For example, with one piece of equipment the length of an object might be measured to be 5.28 (in some units). This doesn’t mean the length is exactly 5.28. It just means that it is closer to 5.28 than it is to 5.27 or 5.29. With a more accurate (and usually much more expensive piece of equipment) the length might be measured to be 5.283, which just means it is closer to 5.283 than it is to 5.282 or 5.284, and so on.

If an expression contains an approximate number then that whole expression is also approximate. Click here for more information on accuracy and significant figures. The Imaginary Numbers and the Complex Numbers If real numbers include every single number on the number line, then what other numbers could there be? To answer that question, consider how we built up the number system so far: We started with the whole numbers (numbers like 3) We wanted opposites for these numbers so we created the integers (numbers like −3) We wanted to divide these numbers but needed the rationals to describe some of the results (numbers like 3/4) We wanted to take the square root of these numbers but needed the irrationals to describe some of the results (e.g. ) We now want to take the square root of negative numbers but we will need a new type of number to describe the result. We define the square root of a negative number to be an imaginary number. How much further can this process of creating new types of numbers go? The answer is one more step. We can add a real number to an imaginary number. The result is called a complex number.

This is the end of the line because it turns out that every possible operation with every possible complex number results only in other complex numbers. We say that complex numbers make the numbers complete. Where do imaginary and complex numbers go on the number line? The answer is they don’t. This picture shows the complex plane. It contains the number line (which is now called the real axis) and a new axis called the imaginary axis, perpendicular to it. Real numbers lie on the real axis, imaginary numbers lie on the imaginary axis and complex numbers generally lie off the real axis, either above it or below it. The Algebra Coach program can run in real mode or complex mode. In real mode it will not carry out any operation that leads to a non-real number (such as taking the square root of a negative number). Click here for more information on complex numbers. Whole Numbers and Integers Whole Numbers and Integers Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, ... (and so on) No Fractions!

Examples: 0, 7, 212 and 1023 are all whole numbers (But numbers like ½, 1.1 and 3.5 are not whole numbers.) Counting Numbers Counting Numbers are Whole Numbers, but without the zero. Because you can't "count" zero . So they are 1, 2, 3, 4, 5, ... (and so on). Natural Numbers "Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject. Integers Integers are like whole numbers, but they also include negative numbers ... but still no fractions allowed! So, integers can be negative {-1, -2,-3, -4, -5, ... }, positive {1, 2, 3, 4, 5, ... }, or zero {0} We can put that all together like this: Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...

} Examples: −16, −3, 0, 1 and 198 are all integers. (But numbers like ½, 1.1 and 3.5 are not integers) These are all integers (click to mark), and they continue left and right infinitely: Some People Have Different Definitions! Some people (not me) say that whole numbers can also be negative, which makes them exactly the same as integers. And some people say that zero is NOT a whole number. So there you go, not everyone agrees on a simple thing! My Standard I usually stick to this: Name { ... -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... } -15, 0, 27, 1102 And everyone agrees on the definition of an integer, so when in doubt say "integer". And when you only want positive integers, say "positive integers". It is not only accurate, it makes you sound intelligent.

Like this (note: zero isn't positive or negative): Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... } Negative Integers = { ..., -5, -4, -3, -2, -1 } Positive Integers = { 1, 2, 3, 4, 5, ... } Non-Negative Integers = { 0, 1, 2, 3, 4, 5, ... } (includes zero, see?) Other Numbers For an interesting look at other types of numbers read The Evolution of Numbers

Question: An integer is what sort of number?
Answer: Whole