Numbers - Decimal Numbers - In Depth © 2000-2005 Math.com. All rights reserved . Please read our Privacy Policy . Our decimal system of numbers lets us write numbers as large or as small as we want, using a secret weapon called the decimal point. In our number system, digits can be placed to the left and right of a decimal point, to indicate numbers greater than one or less than one. The decimal point helps us to keep track of where the "ones" place is. It's placed just to the right of the ones place. As we move right from the decimal point, each number place is divided by 10. We can read the decimal number 127.578 as "one hundred twenty seven and five hundred seventy-eight thousandths". But in daily life, we'd usually read it as "one hundred twenty seven point five seven eight." Here is another way we could write this number: Notice that the part to the right of the decimal point, five hundred seventy-eight thousandths, can be written as a fraction: 578 over 1000. However, you will hardly ever see a decimal number written like this. Why do you think this is?

You can see that our decimal code is a very handy and quick way to write a number of any size! Examples Here's how to write these numbers in decimal form: Three hundred twenty-one and seven tenths 321.7 (6 x 10) + (3 x 1) + (1 x 1/10) + (5 x 1/100) 63.15 Decimal Notation Decimal Notation Given decimal notation, write a word name. Convert between fraction notation and decimal notation. Given a pair of numbers in decimal notation, tell which is larger. Round decimal notation to the nearest thousandth, hundredth, tenth, ones, ten, hundred, or thousand. 4.2 Add using decimal notation. Subtract using decimal notation. Solve equations of the type x + a = b and a + x = b, where a and b may be in decimal notation. 4.3 Convert from notation like 45.7 million to standard notation, and between dollars and cents. 4.4 Solve equations of the type , where a and b may be in decimal notation. Simplify expressions using the rules for order of operations. 4.5 Convert from fraction notation to decimal notation.

Round numbers named by repeating decimals in problem solving. Calculate using fraction and decimal notation together. 4.6 Estimate sums, differences, products, and quotients. 4.7 Solve applied problems involving decimals. REHABILITATION ACT - SECTION 504: If you have a documented disability and need special accommondations, please contact the Director of Counseling, Michelle Kettler, at (409)938-1211, ext. 124. Ms. Kettler's office is located on the second floor of the Student Center Building in room 212. 4.1 DECIMAL NOTATION, ORDER, AND ROUNDING In Chapter 4, we will study the use of decimal notation. The word decimal comes from the Latin word decima, meaning a tenth part. Using decimal notation, we can write 0.875 for 7/8 or 48.97 for 48 97/100.

Decimal Notation and Word Names 1 Let's look at the meaning of the decimal 26.3385 min, the men's 10,000-meter run record held by Kenenisa Bekele from Ethiopia. The decimal notation 26.3385 means We read both of the meanings as: "Twenty-six and three thousand three hundred eighty-five ten-thousandths." To write a word name from decimal notation: 397.085 1. Write a word name for the whole number (the number named to the left of the decimal point). 2. Write the word "and" for the decimal point. 3. Write a word name for the number named to the right of the decimal point, followed by the place value of the last digit. 397.085 CONVERTING BETWEEN DECIMAL NOTATION AND FRACTION NOTATION Given decimal notation, we can convert to fraction notation as follows: To convert from decimal to fraction notation Four and eight hundredths 1. Count the number of decimal places 4.08 2. Move the decimal point that many places to the right to make a whole number. 408. 3.

Write the answer over a denominator with a 1 followed by that number of zeros. 2 zeros To convert from fraction notation to decimal notation when the denominator is 10, 100, 1000, 10,000 and so on. 1. Count the number of zeros in the denominator 3 zeros 2. Move the decimal point that number of places to the left. Leave off the denominator. 80.079 COMPARING NUMBERS IN DECIMAL NOTATION To compare two numbers in decimal notation, start at the left and compare corresponding digits moving from left to right. If two digits differ, the number with the larger digit is the larger of the two numbers. To ease the comparison, extra zeros can be written to the right of the last decimal place. Which of the two numbers is larger: 2.109 or 2.1 Number 2.1 2.100 There is a tie in the ones place; a tie in the tenths; a tie in the hundreths; however 9 > 0.

Thus, 2.109 > 2.1 Which of the two numbers is larger: 2.04 or 2.039 Number 2.039 2.039 There is a tie in the ones place; a tie in the tenths; however, 4 > 3. Thus, 2.04 > 2.039 Now we will look at ROUNDING To round to a certain place: a) Locate the digit in that place. b) Consider the next digit to the right. c) If the digit to the right is 5 or higher, round up; if the digit to the right is 4 or lower, keep the number the same. Round 14.8973 to the nearest hundredth. a) Locate the digit in the hundredths place. 14.8973 b) Consider the next digit to the right. 14.8973 c) Since the digit, 7, is 5 or higher, round up. When we add 1 to the 9 to round up, we carry 1 to the tenths place since 9 + 1 = 10.

d) Note that the 0 in 14.90 indicates that the answer is correct to the nearest hundredth. 14.90 Round 3872.2459 to the nearest: Round to: 4.2 ADDITION AND SUBTRACTION Adding and subtracting with decimal notation is similar to adding and subtracting whole numbers. First we line up the decimal points so that we can add or subtract corresponding place-value digits. Then we add or subtract the digits from the right. ADDITION: When you see a whole number like 789, think of money and write it as $789.00 (no dimes or pennies). If more zeros are needed, then add the zeros. Zeros are place holders showing that you have no tenths, hundredths, thousandths, and so on. When you subtract you must add your zeros for place holders in order to borrow. DO NOT ADD OR SUBTRACT THIS WAY ADDITION NO-NOS SOLVING EQUATIONS Now let's solve equations x + a = b and a + x = b, where a and b may be in decimal notation. Subtract 17.78 on both sides.

Check your answer in the original equation. Subtract 9.906 on both sides. Check your answer in the orginal equation. Subtract 241 on both sides. Check your answer in the original equation. 4.3 MULTIPLICATION Let's find the product of 2.3 x 1.12. To understand how we find such a product, we first convert each factor to fraction notation. Next, we multiply the whole numbers 23 and 112, and then we divide by 1000. 2.3 x 1.12 = ? 2 decimal places in 1.12 (100ths) 1 decimal place in 2.3 (10ths) 3 decimal places all together.

(1000ths) MULTIPLYING BY 0.1, 0.01, 0.001, AND SO ON 0.1 x 38 MULTIPLYING BY 10, 100, 1000, AND SO ON 10 X 97.34 4.3 Application Using Multiplication with Decimal Notation NAMING LARGE NUMBERS We often see notation like the following in newspapers, magazines and on television. The largest building in the world is the Pentagon, which has 3.7 million square feet of floor space. In 2003, consumers spent $51.7 billion on online retail products. To understand such notation, consider the information in the following table. Pay attention to the number of zeros in each number.

1 hundred 10 x 10 x 10 103 10 x 10 x 10 x 10 x 10 x 10 106 1 billion 1,000,000,000 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 109 1 trillion 1,000,000,000,000 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 1012 EXAMPLE: Convert the number in this sentence to standard notation: In 1999, the U.S. Mint produced 11.6 billion pennies.

11.6 billion 11.6 x 1,000,000,000 (9 zeros) 11,600,000,000 (move the decimal point 9 places to the right.) MONEY CONVERSION Converting from dollars to cents is multiplying by 100. To see why, consider $1.00. If you went to the bank to exchange $1.00 for pennies, how many pennies would you receive for your $1.00? Answer. Dollars to Cents To convert from dollars to cents, move the decimal point two places to the right and change the $ sign in the front to a cent sign at the end. Converting from cents to dollars is multiplying by 0.01. To see why, consider 65 cents. 65 cents Cents to Dollars To convert from cents to dollars, move the decimal point two places to the left and change the cent sign at the end to a $ sign in the front. LET'S REVIEW (click here for PowerPoint) First let's go over some terms that we will be using with division.

We use the following method when we divide a decimal quantity by a whole number. To divide by a whole number, a) place the decimal point directly above the decimal point in the dividend, and b) divide as though dividing whole numbers. DIVISORS THAT ARE NOT WHOLE NUMBERS Consider the division: Write your division problem as a fraction. Then multiply by 1 to change to a whole number divisor: To divide when the divisor is not a whole number, a) move the decimal point (multiply by 10, 100, and so on) to make the divisor a whole number; b) move the decimal point (multiply the same way) in the dividend the same number of places; and c) place the decimal point directly above the new decimal point in the dividend and divide as though dividing whole numbers. DIVIDING BY 10, 100, 1000 and SO ON If is often helpful to be able to divide quickly by a ten, hundred, or thousand, or by a tenth, hundredth, or thousandths.

Consider the following example: To divide by 10, 100, 1000, and so on, a) count the number of zeros in the divisor, and b) write the equation by moving the decimal point in the dividend that number of places to the left. We are dividing by a number greater than 1 so the result is smaller than 713.49. DIVIDING BY 0.1, 0.01, 0.001, AND SO ON Now consider the following example: To divide by 0.1, 0.01, 0.001 and so on, a) count the number of decimal places in the divisor. b) write the quotient by moving the decimal point in the dividend that number of places to the right. We are dividing by a number less than 1 so the result is larger than 23.789. 4.4 ORDER OF OPERATIONS: DECIMAL NOTATION The same rules for order of operations used with whole numbers and fraction notation apply when simplifying expressions with decimal notation. RULES FOR ORDER OF OPERATIONS 1. Do all calculations within grouping symbols before operations outside. 2.

Evaluate all exponential expressions 3. Do all multiplications and divisions in order from left to right. 4. Do all additions and subtractions in order from left to right. 4.5 CONVERTING FROM FRACTION NOTATION TO DECIMAL When a denominator has no prime factors other than 2's and 5's, we can find decimal notation by multiplying by 1. We multiply to get a denominator that is a power of ten, like 10, 100, or 1000. Example 1: Find decimal notation for 3/5. Example 2: Find decimal notation for 7/20. Example 3: Find decimal notation for 87/25. Example 4: Find decimal notation for 9/40. We can always divide to find decimal notation. Example 5: Find decimal notation for 3/5. Example 6: Find decimal notation for 7/8. In examples 5 and 6, the division terminated, meaning that eventually we got a remainder of 0.

A terminating decimal occurs when the denominator has only 2's or 5's, or both, as factors, as in 17/25, 5/8, or 83/100. This assumes that the fraction notation has been simplified. Let's consider a different situation: Since 6 has a 3 as a factor, the division will not terminate. Although we can still use division to get decimal notation, the answer will be a repeating decimal, as follows. Example 7: Find decimal notation for 1/6. How many jelly beans are in the jar? How large was your graduating class? How much is this vacation going to cost? In all of these situations you use estimation to make a guess of what the actual number might be. We use estimation to get a quick answer instead of doing the long calculation, or you use estimation after your long calculation to make sure your answer makes sense in your addition, subtraction, multiplication and division problems. Estimate by rounding to the nearest ten the total cost of one refrigerator priced at $474, and one computer priced at $595. Which of the following is an appropriate estimate?

a) $1000 4.7 APPLICATIONS AND PROBLEM SOLVING FIVE STEPS FOR PROBLEM SOLVING 1. Familiarize yourself with the situation. Carefully read and reread until you understand what you are being asked to find. Draw a picture or see if there is a formula that applies. Assign a letter, or variable, to the unknown. 2. Translate the problem to an equation using the letter or variable. 3. Solve the equation. 4. Check the answer in the original wording of the problem. (estimate to make sure your answer makes sense) 5. State the answer to the problem clearly with the appropriate units. Example 1: At a printing company, the cost of copying is 11 cents per page. How much, in dollars, would it cost to make 466 copies? 1. Familiarize yourself with the situation. We have 466 pages to copy at 11 cents per page. How much will it costs? We have 11 cents for one page, 11 cents for another page, 11 cents for another page, and so on. 2. Translate the problem to an equation using a letter of variable.

We are repeating 11 cents 466 times. The short cut for repeated addition is multiplication. 466 x 0.11 = C 3. Solve the equation. 4. Check the answer in the original wording of the problem. (estimate to make sure your answer makes sense) 466 x 0.10 = 46.60, we are multiply 466 by 0.11 so the answer will be just a little bit bigger than $46.60. 5. State the answer to the problem clearly with the appropriate units. The cost of copying 466 pages at 11 per copy is $51.26. Example 2: A loan of $4425 is to be paid off in 12 monthly payments. How much is each payment? 1. Familiarize yourself with the situation. We have a total loan of $4425 which needs to be paid off in 12 monthly payments. We have our total and so much of that total will be paid each month until we have a balance of $0.00. 2. Translate the problem to an equation using a letter of variable.

Each month we are making a payment and that payment will be subtracted from our total. We repeat the subtraction each month until we reach a zero balance. Repeated subtraction calls for division. Total/months = equal payments 4425 ÷ 12 = P 3. Solve the equation. 4. Check the answer in the original wording of the problem. (estimate to make sure your answer makes sense) 4425 ÷ 12 = 368.75, Let's check it with multiplication. (an estimate would be to multiply 386 x 10) 5. State the answer to the problem clearly with the appropriate units. Each monthly payment will be $368.75. Reading and Writing Decimals Write the following decimal using words. Roll your mouse over each digit for help. Answer: The decimal 1,729,405.008365 is written as: one million, seven hundred twenty-nine thousand, four hundred five and eight thousand, three hundred sixty-five millionths Summary: You learned how to read and write decimals in this lesson. When writing a mixed number as a decimal, the fractional part must be converted to decimal digits.

Decimals are named by the place of the last digit. The hyphen is an important indicator when reading and writing decimals. When writing a phrase as a decimal, some of the words indicate the place-value positions, and other words indicate the digits to be used. Exercises In Exercises 1 and 2, click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. ANSWER BOX: ANSWER BOX: RESULTS BOX: In Exercises 3 through 5, read each question below. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, choose a different button. 3. Comparing Decimals Comparing Decimals Unit 12 > Lesson 3 of 12 Example 1: Polygon Pizza Place caters children's parties with square-shaped pizza. Each pizza is exactly the same size and is divided into equal parts called slices. At Sam's party, each child had 2 out of 10 slices from a single pizza.

At Elena's party, each child had 15 out of 100 slices from a single pizza. At which party did each child have more pizza? Analysis: We can write a fraction to represent each party: Party Using our knowledge of decimals, we get: Party 0.15 Answer: Each child got more pizza at Sam's Party. In Example 1, we compared two decimal numbers and found that 0.2 is greater than 0.15. Some students would argue that 0.15 is a longer decimal with more digits, and is therefore greater than 0.2. However, if we think about money, we know that 20 cents is greater than 15 cents. Thus, our answer in Example 1 is correct. Decimal numbers are compared in the same way as other numbers: by comparing the different place values from left to right. We use the symbols <, > and = to compare decimals as shown below.

Example 1: 0.2 is greater than 0.15 0.15 < 0.2 0.15 is less than 0.2 0.2 = 0.2 0.2 is equal to 0.2 0.15 = 0.15 0.15 is equal to 0.15 When comparing two decimals, it is helpful to write one below the other. This is shown in the next example. Example 2: Which is greater, 0.57 or 0.549? Analysis: Let's compare these decimals using a place-value chart. 9 Answer: 0.57 is greater than 0.549. Notation: 0.57 > 0.549 As you can see in the example above, 0.57 has fewer decimal digits than 0.549. It is easier to compare two decimals when you have the same number of decimal digits, so an extra zero was written to the right of the digit 7 in the decimal 0.57.

We are able to do this because 0.57 and 0.570 are equivalent decimals . Use Caution With Writing Extra Zeros It is easier to compare decimals when you have the same number of decimal digits. Thus, we often write extra zeros to the right of the last digit of one of the decimals being compared. These extra zeros are place holders and do not change the value of the decimal. However, if you inserted a zero between the decimal point and a decimal digit, that would change the value of the decimal. This is shown in the table below: 0.57 The meaning of decimals -- A complete course in arithmetic First, second, third, fourth, and so on. An ordinal number answers the question Which one? We will now see that the ordinal numbers express division into equal parts. An ordinal number names which part -- the third part, the fourth, the fifth, and so on. Division into equal parts If we divide 15, for example, into three equal parts -- into three 5's -- then we say that 5 is the third part of 15. We say that because 15 is the third multiple of 5: 5, 10, 15.

We use that same ordinal number to name the part. If we divide a quantity into four equal parts, then we call each part a fourth; if into five equal parts, a fifth; if into one hundred equal parts, a hundredth; but if into two equal parts, we say half. And so with the exception of half, an ordinal number indicates into how many parts a quantity has been divided. A third, a fourth, a fifth, and so on, are the names of parts. Those names belong to language itself, not just to mathematics. Those ordinals are prior to the names of numbers we call fractions, which, we are about to see, are numbers we need for measuring, and are the parts of number 1. We will go into this more in Lesson 15 . The decimal system Since our numbering system is based on the powers of 10 , it is called a decimal system. Decem in Latin means ten. In the previous Lessons we learned about whole numbers, which are the repeated additions of 1: 1, 2, 3, 4, and so on. Here we will learn about numbers that are less than 1.

They are the numbers we create when we divide 1 into equal parts -- where 1 is now a continuous unit. And since this is a decimal system, those parts of 1 will have the ordinal names of the powers of 10: tenths, hundredths, thousandths, and so on. First, we will divide One into ten equal parts, and so each part is called a Tenth. If we divide each Tenth into ten equal parts, then One will be in one hundred equal parts -- count them And so we have divided One into Hundredths. If we divide each Hundredth into ten equal parts, then each tiny piece will be a Thousandth part of One. And so on. Those Tenths, Hundredths, Thousandths are called the decimal units . 3. Which numbers are the decimal units? They are the units less than 1 -- they are the parts of 1 -- named as tenths, hundredths, thousandths, ten-thousandths, and so on. The sequence of names of the decimal units follows the same sequence as the powers of 10. They are those powers in their ordinal form.

Whole units 412.387056 The decimal point signifies "and," in the sense of "plus." That is the whole number 412 plus 3 tenths of 1, plus 8 hundredths of 1, plus 7 thousandths; and so on. Any number written with a decimal point is loosely called a decimal. A number less than 1 -- a proper fraction -- written as a decimal, we call a decimal fraction. A number written with a numerator and denominator we call a common fraction (Lesson 20). The positions to the right of the decimal point are called the decimal 412.387056 places; the 1st, the 2nd, the 3rd, and so on. Each decimal digit occupies a decimal place. 5. Which decimal unit is at each decimal place? The digit at the 1st decimal place shows the number of Tenths. In the number above, there are 3 tenths. The 2nd decimal place shows the number of Hundredths. In that number, there are 8 hundredths. And so on.

The first digit to the left of the decimal point -- the 2 -- shows the number of Ones. As with whole numbers, each digit has its place value : 2.3875. . . Ones, tenths, hundredths, thousandths, ten-thousandths, . . . As with the whole numbers, each place is ten times the place to its right. One is ten tenths. Example 1. In this number 534.267 a) there are how many ones? Answer. 4. The ones place is the first digit to the left of the decimal point. b) How many hundredths? Answer. 6. Hundredths (ordinal) is a decimal unit. It falls to the right of the decimal point. c) How many hundreds? Answer. 5. Hundred (cardinal) is a whole unit. It falls to the left of the decimal point. Example 2. Expanded form. Just as the numeral for every whole number stands for a sum (Lesson 2), so does the numeral for every decimal. Here is the expanded form of 534.267.

534.267 = 5 Hundreds + 3 Tens + 4 Ones + 2 Tenths + 6 Hundredths + 7 Thousandths. At this point, please "turn" the page and do some Problems . or Big Numbers and Scientific Notation radioactive decay and geologic age, geologic time , metric system Big Numbers and Scientific Notation by Jennifer M. Wenner, Geology Department, University of Wisconsin-Oshkosh Jump down to: Teaching strategies | Materials & Exercises | Student Resources What is scientific notation? The concept of very large or very small numbers is something that is difficult for many students to comprehend. In general, students have difficulty with two things when dealing with numbers that have more zeros (either before OR after the decimal point) than they are used to. They often do not understand: that "big" and "small" are relative terms, the concept of "order of magnitude" Scientific notation is a way to assess the order of magnitude and to visually decrease the zeros that the student sees. It also may help students to compare very large (or very small numbers) But students still have little intuition about scientific notation.

Teaching them to recognize that scientific notation is a short hand way to better understand big and small numbers can be useful to them in all aspects of their academic career. In science, often we work with very large or very small numbers. For example, in geology, the age of the Earth = 4,600,000,000 years old, or, in chemistry, one a.m.u. = 0.00000000000000000000000000166 kilograms. It seems like a lot of work to keep track of all those zeros. Fortunately, we can easily keep track of zeros and compare the size of numbers with scientific notation. Scientific notation allows us to reduce the number of zeros that we see while still keeping track of them for us. For example the age of the Earth (see above) can be written as 4.6 X 109 years. This means that this number has 9 places after the decimal place - filled with zeros unless a number comes after the decimal when writing scientific notation.

So 4.6 X 109 years represents 4600000000 years. Very small numbers use the same type of notation only the exponent on the 10 is usually a negative number. For example, 0.00000000000000000000000000166 kg (the weight of one atomic mass unit (a.m.u.)) would be written 1.66 x 10-27 using scientific notation. A negative number after the 10 means that we count places before the decimal point in the scientific notation. You can count how many numbers are between the decimal point in the first number and the second number and it should equal 27. The nice thing about scientific notation is that it also tells us something about significant figures . Teaching Strategies: Ideas from Math Education Put quantitative concepts in context There are a number of geologic contexts in which big (and small) numbers and scientific notation come up.

A few of them include: Radioactive decay/Radiometric dating Use multiple representations Because everyone has different ways of learning, mathematicians have defined a number of ways that quantitative concepts can be represented to individuals. Here are some ways that big numbers and scientific notation can be represented. Verbal representation We represent large (and small) numbers with modifiers that tell us how many zeros are attached to the number. One million (1. x 106) tells us that the decimal (after the 1) moves 6 places to the right (1,000,000.). One thousandth (1. x 10-3) tells us that the decimal moves 3 spaces to the left (0.001). 4.5 billion years looks like this: 4500000000 years or 4,500,000,000 years That's a lot of zeros!

We can also write that number in scientific notation: 4.5 X 109 yrsor (on a calculator) 4.5E9 yrs (1 billion in scientific notation means 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10!) What does that number actually mean? Can you actually imagine 4.5 billion of anything? Graphical representation Although many topics in geoscience use scientific notation, graphical representations are often plotted on logarithmic scales as in the electromagnetic spectrum below. Log scale of electromagnetic spectrum. Details There are many reasons for this - most importantly, it takes up less space - but a log scale does not give a sense of true scale: Bar illustrating true scale of powers of ten. Details Geologic time is a great context in which to talk about big numbers and to give a sense of scale. We have the geologic time scale to give students a sense of what parts of "time" we know the most about. And how big "time" really is. Here is a geologic time scale that gives a sense of scale Geologic time scale.

Details It is interesting to note that the time periods that we know the most about (the Cenozoic, Mesozoic and Paleozoic) only make up a small fraction of the whole of geologic time! Symbolic representation There are a number of analogies for thinking about big numbers. Students (particularly in entry-level classes) often relate well to analogies with money: Money analogies: A local radio station is tring to collect a million pennies for a worthy cause. How many dollars is this? Answer: $10,000. You have just won a contest. The prize is that you will get $1 bills from a bank teller at the rate of 1 per second for as long as you can stand there taking them (no bathroom breaks, no napping, etc.). How long will you have to stand there to become a millionaire? Answer: about 12 days. A very rich man decides that he will help to pay off the national debt by paying $1 into the coffers of the US at the rate of one per second. How long will it take him to pay off the debt (provided no more debt accumulates).

(You can check the debt (to the penny) at the Treasury Dept. - over 7.6 trillion dollars in Feb 2005)? Answer: about 32,000 years per trillion dollars Use technology appropriately Students have a number of tools that can be used to help them become more familiar with both scientific notation and very large and very small numbers. Calculators can be set up to display scientific notation. Spreadsheet programs can tabulate data, expressing it in a number of forms. Calculators: Scientific notation provides an excellent way for students to become familiar with their (probably new) calculators. Many students do not know what scientific notation is and have little intuition about it. Even fewer really know how to use their calculators to express it. Often I see students just ignoring "the little number in the upper right" and so their answer is off by orders of magnitude. Have students figure out how to make their calculators express scientific notation. A quick exercise to do in class is have them multiply some numbers in scientific notation on their calculator and then write the answer in numerical notation. Computers: Many spreadsheets can be formatted to display numbers numerically and in scientific notation.

Showing students a table of these data expressed both ways may help them to get a sense of what is happening "in their calculators" when numbers are displayed in scientific notation. These numbers can also be plotted so that students can get a sense of scale. Work in groups to do multiple day, in-depth problems Mathematicians also indicate that students learn quantitative concepts better when they work in groups and revisit a concept on more than one day. Therefore, when discussing quantitative concepts in entry-level geoscience courses, have students discuss or practice the concepts together. Also, make sure that you either include problems that may be extended over more than one class period or revisit the concept on numerous occasions. Very big and very small numbers come up over and over in introductory geoscience: Geologic time, radiometric dating, unit conversions (especially metric), etc. When each new topic is introduced, make sure to point out that this is not new material, it's just a matter of remembering. Teaching Materials and Exercises Jerry Johnson of University of Nevada at Reno has an excellent example of Understanding Big Numbers that applies to population growth and government spending. This file includes student exercises and instructor notes.

In the Quantitative Skills site, Barb Tewksbury discusses Back of the Envelope Calculations and has a number of exercises that give students practice in quantitative skills and in estimating numbers. A View from the Back of the Envelope has a number of links to interesting ways to explain (and estimate) big numbers and orders of magnitude. Number Formats in Microsoft Excel Number Formats in Microsoft Excel The original version of this article appeared in TechTrax Ezine . Number Formats are little codes that help you control the appearance of numbers in Excel. They are not difficult, and can also be used to help with a few tricks. I'm using Excel charts in many of these examples, but these number formats also apply to values in cells. You can view or edit a number format by double clicking on a text element, then clicking on the Number tab in the dialog, shown below. There are a number of built in categories, and you will have to experiment with different ones to learn all the options you have. Most likely you will find an appropriate number format within the choices provided, but it is also very easy to add your own. If your desired number format is similar to one you can find in the dialog, select it, then click on the Custom category.

The chosen number format will be displayed in an edit box, where you can adjust it as needed. Number Format Dialog A number format consists of up to 4 items, separated by semicolons. Each of the items is an individual number format. The first by default applies to positive numbers, the second to negative numbers, the third to zeros, and the fourth to text. If you don't apply any special formatting to text, Excel uses the General number format, which basically means anything that will fit. That means Excel will show as many decimal places that the text item has room for, it won't use a thousands separator, and if the number can't fit, perhaps due to a narrow column width, Excel uses a scientific number format. If only one format is specified, Excel uses it for any contents of the cell. The Y axis tick labels above and the column of numbers to the right use the General number format. 1.23E+08 1.23E-05 1.23E-06 You can make long numbers easier to read by inserting a thousands separator (a comma for those working in the US).

The #,##0 number format is used below; the comma in the format tells Excel to use commas, the octothorpe (pound sign) is merely a placeholder for the comma, and tells Excel not to put digits there if none are needed, and there are no decimal digits. This works well for the axis labels below left, but the decimal fractions have all been truncated to their integer values, zero. Also, this format doesn't allow Excel to use the shorter scientific format, so the longest numbers merely show #######, signifying that they don't fit in their cells. There is only one format specified, so negative numbers use the same format with the comma thousands separator. Without anything different specified, the negative numbers are signified with a minus sign. Make the labels easier to read: Insert a thousands separator with the #,##0 number format. ####### 0.00000123 Defining Colors in Number Formats The chart below shows how to make negative numbers appear different, something accountants need to know at a glance. The number format used for the Y axis labels is $#,##0;[Red]$#,##0.

The first part of this two part format tells Excel to use a dollar sign and a comma thousands separator for positive numbers. The second part tells Excel to make negative numbers red, then use the dollar sign and thousands separator (but no minus sign, since it isn't explicitly stated). Since there is no third number format, Excel uses the first for zero values. Negative numbers stand out with a number format like $#,##0;[Red]$#,##0. Let's use the three part number format [Blue]$#,##0;[Red]$#,##0;$#,##0, so we can color positive numbers blue, negatives red, and keep zeros in their default black. The colors that are available in number formats are [Black], [Blue], [Cyan], [Green], [Magenta], [Red], [White], and [Yellow]. The [Blue]$#,##0;[Red]$#,##0;$#,##0 number format produces tricolored axis labels. Changing the Default Number Format Conditions By default, the first three items in the number format are applied to positive, negative, and zero values. But you can assign your own simple conditions.

For example, in this chart, numbers equal to or greater than 3000 are colored blue, with the [Blue][>=3000]$#,##0;[Red][<0]$#,##0;$#,##0 number format. The [Blue][>=3000]$#,##0;[Red][<0]$#,##0;$#,##0 number format provides a variation on the tricolored axis labels. You can format just a single value in blue by replacing ">=" with "=" in the [Blue][=3000]$#,##0;[Red][<0]$#,##0;$#,##0 number format. The [Blue][=3000]$#,##0;[Red][<0]$#,##0;$#,##0 number format shows yet another variation on the theme. With this level of control over the display of numbers, number formats are like a poor man's conditional formatting. Varying Decimal Digits in Number Formats The following chart uses a logarithmic X axis to show the variation in its data more uniformly. By default (General) the data labels would have many digits after their decimal points, but I only wanted to show two significant digits to reduce clutter.

I used the graduated number format [Blue][<0.1]0.000;[<1]0.00;[Red]0.0, which has three, two, or one decimal digits, depending on the value being displayed. I also colored the data labels for the three ranges differently to help you spot the difference; the axis labels are all black ([<0.1]0.000;[<1]0.00;0.0). Use [Blue][<0.1]0.000;[<1]0.00;[Red]0.0 for a staggered number of decimal digits. Hide Zeros in Data Labels One of the commonly asked questions in the on-line forums is, "How do I hide data labels that show a zero value?" One easy way is to use a custom number format that provides no format for zeros. The two pie charts below have General and 0% number formats for the values and percentages of each pie slice. Notice that each has an entry for item C in the legend.

Item C was zero, so it doesn't appear, but its data label (0 or 0%) shows up in between slices B and D. The General number format shows all labels, including C's big fat zero. The 0% number format shows all five labels, including C's zero percent. You can hide the zero values with a number format that has a place for a zero number format, but no actual format. I've used General;;; and 0%;;; for the same two charts to hide the zeros. Since negative numbers make no sense in a pie chart, either, I have left them out. This chart's General;;; number format only allows display of positive numbers. The 0%;;; number format in this chart only shows positive percentages. Other Number Formats Scientific Notation Scientific Notation is a shorthand used to keep short the number of characters in a value no matter how large or small the number is. It looks like 1.23x103, or 1.23E+3, where the number 1.23 is multiplied by 10 raised to the power 3. 1.23x103 or 1.23E+3 equals 1230.

We didn't save any characters with this number, but when you consider that it's 9.3x107 miles from here to the sun, well, now we're saving some digits. Excel has no native way to write a number with superscripted notation and retain its numerical value, but neither did FORTRAN back in the good old days, so we'll use the E notation. Essentially, the E notation consists of a single digit, a decimal point, any number of decimal digits, the letter E, a plus or minus sign and an exponent. This table shows a few examples: #,##0 1.235E-06 1.235E-06 I show a technique to apply scientific notation to a fake chart axis elsewhere on this web site . Dates and Times Internally, Excel stores dates as a number representing the elapsed days since January 1, 1900 (actually, since December 31, 1899, because the Lotus programmers thought 1900 was a leap year, and Microsoft kept the error either because they also thought it was a leap year, or for compatibility with Lotus 1-2-3, then the spreadsheet market leader).

Excel stores times as the fraction of a day since midnight. Any given date-time is a number whose integer part represents the date and whose fractional part represents the time. I am using 1:30 in the afternoon of July 4, 2004 for this illustration. Just as there are several ways to represent numbers, there are many ways to format dates and times. You construct a date-time format by stringing together smaller building blocks. Fortunately these are pretty easy to remember, and they are not case sensitive. Any punctuation you want in the value is included in the number format. YYYY is the four-digit year (2004) and YY is the two-digit year (04). MMMM is the month spelled out (July), MMM is the abbreviated month (Jul), MM is the two-digit month, with a leading zero if required (07), and M is the one or two digit month (7). Since M is already reserved for a single numerical digit, so Excel uses MMMMM for a one-letter abbreviation (in the sequence J-F-M-A-M-J-J-A-S-O-N-D).

DDDD is the day spelled out (Sunday), DDD is the abbreviated day (Sun), DD is the two-digit day, with a leading zero if required (04), and D is the one or two digit day (4). Unlike MMMMM above for one character month abbreviations, there is no DDDDD format that gives just the first character of the day's name. HH is the two-digit hour (01 or 13, depending on whether AM/FM is specified), H is the one-digit hour (1 or 13), MM is the two-digit minute (30), M is the one-digit minute, SS is the two-digit second, and S is the one-digit second. AM/PM is specified for a 12-hour clock, or omitted for a 24-hour clock. If you are dealing with times only, you can use [H] to increment hours beyond 24, [M] to increment minutes beyond 60, or [S] to increment seconds beyond 60. You wouldn't use all three in a format, you would use a format like [H]:MM:SS.

The table below illustrates how Sunday July 4, 2004 01:30 PM is displayed using various number formats: General 2006 Leading Zeros A frequently asked question is "Why does Excel remove the zero in my zip code?" The reason is that the zero is not required to display the value of the number. However, since number formats are not about the value, they are about the appearance, you can use a special format to retain the leading zeros, 00000 (use as many zeros in the number format as you want digits in your number). There is even a Zip Code number format listed in the Special category. What if you don't want to show leading zeros? For example, you want a number to appear as .123 instead of 0.123. We saw above how a number format of #,##0 uses the pound sign to hold places that indicate the comma thousands separator should be used, but if the places only contain leading zeros, they are not used and the comma is not needed.

In the same way, the pound sign in the number format #.000 holds the place for the decimal point, but if it is a leading zero, the place contains no zero digit. You can use the pound sign after the decimal point, to indicate how many digits to show if they are not trailing zeros. The format #.### shows up to three decimal digits, except for trailing zeros. The following table shows how #.### treats various numbers. Notice that the decimal point is displayed for the first two examples even if there are no nonzero decimal digits. Also notice how zero is displayed: since the zero is a leading zero, all that is displayed is a lone decimal point. Number 12.25 Labels If you type a number with a label, for example, "15 feet", Excel interprets this entry as text, and cannot use the value in a calculation. However, if you incorporate this label into the number format (0" feet"), then the number is stored in the cell as a number, but displayed with the label.

You should enclose the label in double quotes, but if the label cannot be confused with one of the formatting characters, Excel will treat it as a label (for example, M cannot be used as is for a label, since "M" represents Month or Minute in a custom format). Number Format Using Numbers, Writing Lists Decimals, percentages, and fractions: 235.485, 55%, 14 1/4 Scores: The Bulls won the final game by a score of 114 to 106. Addresses: 1032 Maple Avenue. Sometimes, though, an address is part of a building's name, and then you'll want to spell it out: One Corporate Plaza. Unless space is at a premium, write out numerical street names (of one hundred or less): 1032 Fifth Avenue. For proper envelope addressing form (U.S. Post Office recommendations), click HERE . Political and military units (for numbers of one hundred or less): Seventh Precinct, Fourteenth Congressional District, Fifty-third Regiment, Third Batallion, 112d Artillery Finances: Tickets cost $35.50 apiece.

The city spent $1.1 million for snow removal last year. (Or use $1,100,000.) You can leave the comma out of figures in the thousands: They spent $7500 on that car before junking it. Also, leave the comma out of addresses and year-dates: In 1998, they moved to NE 12887 53rd Avenue. Ranges: Between 18 and 25 bald eagles have been counted near the Connecticut River this spring. Time: 9:15 a.m. If you use the word o'clock, however, for rounded off times, spell out the number in words: We left at seven o'clock. Use a.m. and p.m., not AM and PM. Numbered, Vertical ("Display"), and Bulleted Lists Writing and reference manuals offer different advice for creating lists. It seems that as long as you're consistent within your document, you can devise just about any means you want for creating your lists, whether you want them as run-in lists (built into the flow of your text) or as vertical lists (indented and stacked up).

Technical writing may have its own requirements in this regard, and you should consult a technical writing manual for specific rules. Use parentheses around the numbers (no periods after the number, though) when using a run-in list: I have three items to discuss: (1) the first item; (2) the second item; and (3) the third item. Use semicolons to separate the items, whether they're expressed as fragments or full sentences. For a vertical list (sometimes called a display list), you may choose to capitalize the items or not, and you may choose to put a comma after each item or not. (If you use commas, put a period after the last item.) We will now review the following three principles: fairness in recruiting academic eligibility scholarly integrity Your choice to capitalize or not may depend on how elaborate your lists are and how many of them you have in your text. If a vertical list contains complete sentences or lengthy and complex items, you may prefer to end each element in the list with a semicolon, except for the last element, which you will end with a period.

Most coaches conform to three basic principles in recruiting new players: Look for players first who can fill those positions you will need the subsequent year; Look for players who are "court smart" as opposed to being merely athletic; Look for players who are academically eligible and who have an academic purpose in going to college. Although the elements in the list above begin with capital letters, that is not absolutely necessary. Notice that there is no "and" at the end of the next-to-last element (although some reference manuals allow for or recommend its use). Although we have used numbers for this list, bullets would work equally well if numbering seems inappropriate or irrelevant. The list below is based on a format suggested by the New York Public Library's Writer's Guide to Style and Usage: Most coaches conform to three basic principles in recruiting new players Look for players first who can fill those positions you will need the subsequent year Look for players who are "court-smart" as opposed to being merely athletic Look for players who are academically eligible and who have an academic purpose in going to college Note that this format does not include a period even at the end of the last element. Most writers, however, want to use some kind of punctuation in their listed items.

When the introductory statement is a complete sentence, you can end it with either a period or a colon. Use a colon if the sentence is clearly anticipatory of the list, especially if it contains phrasing such as the following or as follows. A colon is also appropriate if the list that follows will be numbered or will establish a priority order. If the introductory statement is not a complete statement, however, neither a period nor a colon would be appropriate since that would interrupt the grammatical structure of the statement; use either no punctuation or try the dash technique noted above. Listing Names in Alphabetical Order Putting people's names in alphabetical order is done on a letter-by-letter basis, taking into consideration all the letters before the comma that separates the last from the first name. Omit titles (such as Lady, Sir, Sister), degrees (M.D., Ph.D.), etc., that precede or follow names. A suffix that is an essential part of the name  such as Jr., Sr., or a roman numeral  appears after the given name, preceded by a comma.

(Ford, Henry J., III or Pepin, Theophilus, Jr.) Beethoven, Ludwig van (The van or von in Dutch or German names, if not capitalized by family usage,appears after the first name; if capitalized, it appears before the last name and determines the alphabetical order.) D'Annunzio, Gabriele Deere-Brown, Juan (Ignore the hyphen.) Deere-Brown, Juan-Poivre Dante Alighieri (Some Italian names of the 15th century or before are alphabetized by first name) D'Arcy, Pierre de Gaulle, Charles (With French names, the de goes before the last name when the last name contains only one syllable. See de Maupassant, below.) Descartes, René Garcia Lorca, Federico (Use full surnames for Spanish names.) López y Quintana, María O'Keeffe, Georgia (Ignore the apostrophe.) Pepin, R. E.

Rueda, Lope de (For Spanish names, de comes after the first name) Saint-Exupéry, Antoine de Von Braun, Werner (See Beethoven, above.) Significant Digits and Rounding / MATH200 / Brown, TC3 What’s New What are Significant Digits? What strikes you about the difference between a measurement of 1.615 in and 1.6 in? The first measurement is more precise than the second. Specifically, the first measurement is accurate to the nearest thousandth (0.001) of an inch, while the second measurement is accurate to only the nearest tenth (0.1) of an inch. No measurement in the real world is exact; all have some degree of “slop”. When we present a numerical result of 1.615 in, it is understood to be accurate to the nearest thousandth: we know that the true measurement is between 1.6145 and 1.6155 in.

By contrast, if we present a value of 1.6 in, we are really saying that it is 1.6 to the nearest tenth: the true measurement could be anywhere between 1.55 and 1.65 in. You can see that every real-world number carries information both about its magnitude and about its precision. 1.615 in and 1.6 in have about the same magnitude, but the first one is more precise. We talk about that level of precision as how many significant digits the number has. The significant digits in a number start at the first non-zero digit and end at the last digit. Examples: 1417 has four significant digits, and so do 1.417 and 0.00001417. What about 14.1700? It has six significant digits, not four, because only zeroes at the start of a number are non-significant. Finally, 14.07 has four significant digits. There can be some ambiguity with trailing zeroes in a large whole number. For instance, we quote the average distance from earth to sun as 93 million miles.

In that form, the number has two significant digits. (Remember what significant digits mean: they mark the non-“slop” part of the measurement. All we’re saying is that the average distance is 92½ to 93½ million. But suppose we write the number as 93,000,000? Does it now have eight significant digits? Are we saying the average distance is between 92,999,999.5 and 93,000,000.5 miles? Surely not! When you see a large round whole number, the zeroes may represent “slop”. To get around this problem, numbers are often expressed in scientific notation. For instance, the figure of 93 million miles is 9.3×107 miles (“nine point three times ten to the seventh”). On your calculator it appears as 9.3E7. You’ll find some more about scientific notation later on this page . Practice How many significant digits are in each of these numbers?

4800, 4800.0, 4.8, 0.0000067, .0000067. Answers: 4800 has two to four significant digits. The 4 and 8 are definitely significant, but just by looking at the number we can’t tell whether it’s accurate to the nearest whole number (4800, four significant digits), to the nearest ten (480x, three significant digits), or to the nearest hundred (48xx, two significant digits). 4800.0 has five significant digits. In the previous example (plain 4800), the two zeroes might indicate precision of measurement or be there simply as place holders. But with 4800.0 the last zero is obviously not needed as a place holder and therefore it must be significant. 4.8 has two significant digits. 0.0000067 and .0000067 are the same number. Leading zeroes are never significant, and therefore this number has two significant digits.

Rounding the Results of Calculations Now you see what’s wrong with “25 feet divided by 6.0 equals 4.166666667 feet”, which is what your calculator will tell you. A measurement of 25 feet is accurate only to the nearest foot; you can’t get an answer that is accurate to a billionth of a foot! The rule for multiplying and dividing is this: find the number of significant digits in each factor. The answer will have the smaller number of significant digits. Example: 25 and 6.0 each have two significant digits. The smaller of 2 and 2 is 2; therefore the answer will have two significant digits. 25÷6.0 = 4.2 is correct. Example: Multiply 81.7×2.405. Your calculator says 196.4805. But 81.7 has three significant digits and 2.405 has four; therefore the answer must have three significant digits (smaller of 3 and 4). 81.7×2.405 = 196.

That last illustration makes an important point: when multiplying or dividing, your guide to rounding is not the decimal places but the number of significant digits. Even though each factor had one or more decimal places, the answer has no decimal places because it must have only three significant digits. For powers and roots, the answer should have the same number of significant digits as the original. For example, taking the square root of 167.21 (five significant digits), your calculator gives 12.93097057, which you round to 12.931 (five significant digits). Why is that? Because the square root is defined as the number such that, when you multiply itself, you get back to the original. If the original has five significant digits, then the number that you multiply by itself to get back to the original must have five significant digits. Practice Compute and round your answer properly: 34.78×11.7÷0.17. Answer: the least significant number has two significant digits, and therefore the answer must have two significant digits.

You take the 2393.682353 from your calculator and round it to 2400 (or 2.4×102 in scientific notation). Compute and round your answer properly: 16.2². Answer: the original has three significant digits, and the answer must have the same. Your calculator gives 262.44, which you round to 262. Three people will share a lottery prize of $24.8 million equally. How much will each one receive, before taxes? Answer: 24.8÷3 = 8.2666666667 on the calculator. 24.8 has three significant digits. The 3 looks like it has only one significant digit, but it is an exact number: not between 2½ and 3½ people but 3 people exactly. Think of it as being 3.000000000... people. Therefore the answer will have three significant digits: each person gets $8.27 million.

Quick Rules of Thumb for Statistics In your statistics work, rather than worry about significant figures, you can use the following rules of thumb: mean and standard deviation: round to one more decimal place than your original data. If your original data were mixed, round to one decimal place more than the least precise. For example, consider the set 10, 8.7, 9.2, 11.4, 3.5, 6, 4.5. Some are whole numbers, some have one decimal place. The least precise are the whole numbers, so you will report your mean and standard deviation to one decimal place. This suggestion follows Sullivan, Michael, Fundamentals of Statistics 3/e (Pearson Prentice Hall, 2011) page 118. It won’t always be strictly correct according to the rules of significant figures, but it’s usually right for data sets of size roughly 10–30. probabilities: round to four decimal places. If the probability is less than 0.0001 (1E−4 or 1×10-4), report it as “<0.0001”.

This suggestion and the following one match the conventions used in tables in a number of statistics textbooks. test statistics z, t, χ² in hypothesis tests: round to two decimal places How to Round an Answer Once you’ve computed an unrounded answer, how do you round it correctly? Decide how many significant digits (or decimal places) you’ll need, and then round all at once. Example: if the TI-83 computes a mean of numbers with one decimal place as 3.876, that needs to be rounded to one decimal place. Draw a line at the spot where the rounding must happen: 3.8|76. Because what’s to the right of the line is bigger than 5, you round up to 3.9. If the first digit after the line is 5 to 9, round up; if the first digit after the line is 0 to 4, round down. Example: to round 2.884 and 2.885 to the nearest hundredth, write 2.88|4 → 2.88 and 2.88|5 → 2.89.

It’s important to round all at once, not digit by digit. Example: Round 30.4746 to the nearest hundredth. Incorrect solution: 30.4746 → 30.475 → 30.48. Correct solution: 30.47|46 → 30.47. What’s to the right of the line is less than 5, and you round down to 30.47. Practice Round to the nearest whole number: 17.514, 24.500, 24.501, 27.499. Answers: 17.514 → 18 (5 to 9, round up), 24.500 → 25 (5 to 9, round up), 24.501 → 25 (5 to 9, round up), 27.499 → 27 (0 to 4, round down). Round to one decimal place: 17.545, 24.451, 38.989.

Answers: 17.545 → 17.5 (0 to 4, round down), 24.451 → 24.5 (5 to 9, round up), 38.987 → 39.0 (38.9|87 — the 8 of 87 means that the 9 of 38.9 must increase; 38.9 rolls over to 39.0). The Big No-no Never round in the middle of a calculation; always round only the final answer. Example: 1.2×1.2×1.5. Since the factors have two significant digits, the answer will have two significant digits. But you must carry along your intermediate results without rounding. 1.2×1.2 = 1.44, 1.44×1.5 = 2.16 → 2.2. But if you had rounded 1.44 to 1.4 in mid-stream, you would have 1.4×1.5 = 2.1, which is off. Always wait till the end of a calculation to round.

When you’re working with your calculator, never re-enter an intermediate result you see on your screen. Instead, chain your later calculation to the earlier one. Example: (√2.00)×6.000. The factors have three and four significant digits, and therefore the answer will have three significant digits. Find √2 on your calculator; you should get something like 1.414213562. Do not re-enter this number. Instead, press the [×] key and then 6 and [ENTER]. Your calculator will display Ans*6 and then an answer of 8.485281374, which you round to 8.49. Ans tells you that the previous answer is being carried along in its original unrounded form. Suppose instead you had rounded, entering 1.41×6? You would get 8.46, pretty far off from the correct answer. Always let the calculator carry full precision along for you. What if you have a big decimal, like 11.25055509, and you’re supposed to square it and round the answer to one decimal place?

While it is wrong to round to one decimal place before squaring, you certainly don’t need to type in eight decimals. You are usually safe if you compute with two more significant figures than your ultimate answer will have. In this case you’re squaring 11, and the ultimate answer will be 120-odd; with one decimal place that’s four significant figures. You enter 11.2506 and square it: 126.576 → 126.6. Practice A goat is tethered to a post in your back yard with a chain 8.4 feet long. Assuming the goat doesn’t eat the chain, how much area of your yard will the goat be able to munch on? (Recall that the area of a circle is πr².) Answer: 8.4 has two significant digits, and therefore the answer will also have two significant digits. You need 8.4²×π. Enter 8.4, then press the [x²] key and [ENTER] to obtain 70.56. Now press the [×] key.

To enter π, find the symbol in gold just over the [^] key (which is at the right of the keyboard, just above the [÷] key). Press [ 2nd ^ makes π] and [ENTER] to obtain 221.6707776; round to two significant digits for an answer of 220 square feet. (By the way, it’s possible to do quite complicated calculations in one step on your calculator. That’s a fine approach, as long as you think about the order of operations and put in parentheses where they are needed. Both methods will give the same answer.) Scientific Notation Scientific notation was developed to express very large and very small numbers. To write a large number in scientific notation, move the decimal point to the left until it is between the first and second significant digits; the number of places moved is the exponent. For example, 167 becomes 1.67, but the decimal point moved two places left. Therefore 167 = 1.67×102 or 1.67E2. Scientific notation removes the guesswork about how significant a large number is.

9.3E7 has two significant digits; it is accurate to the nearest million miles. 9.30000E7 has five significant digits, and it is accurate to the nearest hundred miles (0.00001×107 = 100 miles). To write a small number in scientific notation, move the decimal point right until it has just passed the first non-zero digit. Write the number of places moved as a negative number in the exponent. Example: 0.0000894 must move the decimal point five places right to become 8.94E–5 or 8.94×10–5. Notice that big numbers end up with a positive exponent after the E and small numbers end up with a negative exponent after the E. To convert a number from scientific notation to ordinary decimals, reverse the process. A positive exponent indicates a big number: move the decimal point to the right. A negative exponent indicates a small number: move the decimal point to the left.

Example: if the probability of an event is 6.014E–4 or 6.014×10–4, you must move the decimal point four places left to convert it to 0.0006014. If the population of the earth is about 6.1×109 people, you move the decimal point right nine places to convert to 6,100,000,000. To enter scientific notation in your calculator, find the EE symbol in gold just over the comma key (which is above the 7). Example: to enter 6.1×109, press [6] [.] [1] [2nd] [EE] 9. If you then press the [ENTER] key, you will probably see 6100000000. In normal mode, the calculator tries to present results in ordinary numbers if they fit on the screen.

Practice Write in scientific notation: 6,370,000 m (radius of the earth, assuming accuracy to the nearest 1000 m); 0.00000110 in (diameter of a polio virus). Answers: 6,370,000 accurate to the nearest 1000 m must be rounded as 6,370,|000: it has four significant digits. Since the decimal point must move six places left, the exponent is 6: 6.370×106 m. In 0.00000110 in, there are three significant digits including that last zero. The decimal point moves six places to the right: 1.10×10–6. What’s New 20 Dec 2010: update page reference for third edition of Sullivan (intervening changes suppressed)

Question: When writing 'a millionth' in decimals (using numbers and a decimal point/dot), how many zeros are between the decimal point and the figure 1?
Answer: Five