Scientific Notation

Sometimes the numbers that we work with are very large or very small, and several zeros are used to hold the decimal place. With such numbers, we may use a different way of writing them. It is called scientific notation. If you already know how to write and interpret numbers in scientific notation, feel free to move on to the next topic. Or, test your knowledge with the quick quiz at the end of this section, then decide how to continue.

Scientific notation makes use of the powers of ten to indicate the magnitude or size of very large numbers, or very small numbers. The use of scientific notation eliminates the need for zeros to hold the decimal place, so zeros shown in scientific notation are significant digits. (Significant digits come up later in this lesson.) In proper scientific notation the decimal place is adjusted to the position just right of the first non-zero digit (the first significant digit).

This section deals first with exponents and powers of 10, then with writing and adjusting numbers in scientific notation, and a few tips to keep in mind when using calculators.

Exponents

Look at these examples. The first says 10 to the second power, or 10 squared, is 10 times 10 or 100. Then 10 cubed is 10 times 10 times 10 or 1,000. Next, 10 to the fourth is 10 times 10 times 10 times 10 or 10,000. Also, 10 to the first power is simply 10. Less obviously, 10 to the zero power equals one. Note that 10 to any positive power is just one followed by that number of zeros.

10² = 10 x 10 = 100

10³ = 10 x 10 x 10 = 1,000

10⁴ = 10 x 10 x 10 x 10 = 10,000

also 10¹ = 10

and 10⁰ = 1

These examples deal with negative exponents. A negative sign in the exponent means the inverse or reciprocal of the same number without a negative sign. Ten to the minus one means one over ten (one over ten to the first power) which is point one (one tenth). Ten to the minus two means one over ten squared or point zero one (one hundredth). Ten to the minus three means one thousandth, and so forth. When negative exponents are used as powers of ten, the exponent indicates the number of places following the decimal point up to and including the one.

10^-1 = 1/10 = 0.1

10^-2 = 1/10² = 0.01

10^-5 = 1/10⁵ = 0.00001

In scientific notation, exponents are used to show the size of numbers so that place-holding zeros are not needed.

A number written in proper scientific notation has one non-zero digit, followed by a decimal point, followed by the rest of the number, followed by *10 with an exponent. The entire number before the "*10" is called the coefficient. The general form is shown below, followed by an example.

X.YZ*10ⁿ

2.46*10⁵ = 246000

Large Numbers

A simple example is that 10,000 can be written as 1 x 10 to the fourth power.

10,000 = 1*10⁴

Next, you can see how 30,000 can be written as 3 times 10,000; and since 10,000 is 10 to the fourth, then 30,000 can be written as 3 times 10 to the fourth.

30,000 = 3*10,000 = 3*10⁴

Here is a different way of figuring this change. If you start with 30,000 and you were to move the decimal place to the left until it was next to the 3, which is the first non-zero digit in the number, you would have moved the decimal four places. The number of places the decimal has to move becomes the exponent for the 10. So 30,000 is equal to 3 times 10 to the fourth power, the same as before.

30,000 = 30,000 = 3*10⁴

Here is another example. Thirty-four thousand can be written as 3.4 times 10,000, which in turn can be written as 3.4 times 10 to the fourth.

34,000 = 3.4*10,000 = 3.4*10⁴

Looking at it another way, you must move the decimal point four places to get it in position just to the right of the first non-zero digit in the number (the 3). So 34,000, with the decimal point moved over four places, becomes 3.4 times 10 to the fourth.

34,000 = 34,000 = 3.4*10⁴

The same process is shown for changing 34,567 into scientific notation. In scientific notation the decimal always goes just to the right of the first non-zero digit.

34,567 = 3.4567*10,000 = 3.45567*10⁴

Small Numbers

Next, consider what is involved in changing small numbers into scientific notation.

In this first example, 0.01 is the same as 1/100 which is the same as one over ten to the second, which, as we saw before, equals ten to the minus two.

0.01 = 1/100 = 1/10² = 10^-2

A number like .04 is the same as 4/100 or 4 over ten squared which equals 4 times ten to the minus 2.

0.04 = 4/100 = 4/10² = 4*10^-2

Another way of doing this is to move the decimal from where it is to just right of the first non-zero number, the 4 in this case. Since the decimal must be moved 2 places to the right, the exponent is minus 2. So .04 is the same as 4 times 10 to the minus 2.

0.04 = 0.04= 4*10^-2

When working with the number .0045, the decimal must be moved 3 places to the right so the exponent is minus 3. So .0045 is the same as 4.5 times 10 to the minus 3. Remember, in proper scientific notation, the correct position for the decimal place is just to the right of the first non-zero digit.

0.0045 = 0.0045 = 4.5*10^-3

Zeroes

A comment about zeros: zeros will come up when we deal with significant digits later in this lesson. When changing a number to scientific notation, drop any zeros which are used just to hold the decimal place, but keep any zeros which are known values. All zeros shown in scientific notation should be significant digits because the exponent, rather than the decimal place, does the job of letting you know how large or small the number is.

Here are some examples of zeros that are lost and kept.

In the number 0.04500 the first zero is lost when changed to scientific notation but the two zeros at the end are kept.

0.04500 = 4.5*10^-2

In this example, we have no indication that any of the zeros have been measured as significant digits. Therefore, we presume that they were all serving as decimal placeholders and write the number as 3.7 times 10 to the fifth.

370,000 = 3.7*10⁵

In this example, the plus or minus 1,000 tells you how many of those digits were actually measured. The last three zeros are just place holders, but the zero in the 1,000 place must have been measured (because we know the number to the nearest 1,000). That zero is kept; but the other zeros, which are only holding the decimal place, are lost when you move the decimal place to put the number into scientific notation.

370,000 (+/- 1000) = 3.70*10⁵

Adjusting the Decimal Place

Sometimes you have to adjust a number that already has an exponent to get it into proper scientific notation.

For example let's start with the number 64 times 10 to the third. It is not in proper scientific notation because the decimal place is not immediately to the right of the first non-zero number. It should be right after the 6. To get it there, it must be moved one place to the left. As the decimal moves one place to the left, the exponent must become one digit larger. Consequently, the number becomes 6.4 times 10 to the fourth.

64*10³ = 64*10³ = 6.4*10⁴

You could break the process down into steps. Change the 64 into 6.4 times 10. Then 10 times 10 to the third becomes 10 to the fourth.

64 x 10³ = (6.4 x 10¹)*10³ = 6.4*10⁴

Next, we have .0032 times ten to the fifth. In this case, in order to get the decimal point in the proper position, right behind the 3, the decimal point has to be moved to the right three places. Then the exponent is reduced by 3, from 5 down to 2. So the number becomes 3.2 times 10 to the second.

0.0032*10⁵ = 0.0032*10⁵ = 3.2*10²

Notice that when you move the decimal point to the left, the exponent gets larger; and when you move the decimal point to the right, the exponent gets smaller. The previous examples all had positive exponents.

When you are dealing with negative exponents, you must remember that the more negative the exponent, the smaller the number. For example negative 4 is less than negative 3. This is pointed out in the next examples.

Start with 0.015 times 10 to the negative 3. When you move the decimal two places to the right to get it behind the one, the value of the exponent has to go down by two. Since it is already at negative 3, going down makes it negative 5. So the number becomes 1.5 times 10 to the negative 5.

0.015*10^-3 = 0.015*10^-3 = 1.5*10^-5

Here is another example, 750 times 10 to the negative 8. Here we have to move the decimal two places to the left and make the exponent larger by 2. Starting with the exponent at negative 8, getting larger takes it up to negative 6. The value becomes 7.5 times 10 to the negative 6.

750*10^-8 = 750*10^-8 = 7.5*10^-6

Scientific Notation on Calculators

Probably, you will use a calculator to do your calculations. I recommend that you use one that handles scientific notation and learn how to use it properly. The reason for this is that a calculator can treat a number like 1.23*10⁴ in two different ways. First, it can treat the number as a single numerical value written in a particular format, scientific notation. Second, it can treat the number as two separate numbers multiplied together, (1.23) times (10⁴). For our purposes the first approach is better.

To tell if your calculator handles scientific notation, look for an "EE" button or an "EXP" button. When using scientific notation on a calculator, do not type in "x10", the "EE" or "EXP" button replaces that part of the number. Instead, enter 1.23"EE"4 or 1.23"EXP"4. Also, make note of how that value is displayed in the screen of the calculator.

As part of Practice Problem 3 in your workbook, you will be asked to bring your calculator to the lab instructor and have them show you how to perform calculations like these.

Quick Quiz - Scientfic Notation Calculations

Test your skill at working with calculations using scientific notation by carrying out the following calculations and expressing your answer in scientific notation. If you know how to do so, also round your answers off to the appropriate number of significant digits. (If you don't know how to determine significant digits, don't worry -- it's covered a little bit later in Lesson 1.)

Question 1: (2.1*10³) x (3.0*10⁴) =

Question 2: (1.2*10^-3) x (2.0*10⁷) x (1.5*10²) =

Question 3: (2.1*10³) / (3.0*10⁴) =

Answers

Question 1: (2.1*10³) x (3.0*10⁴) = 6.3*10⁷

Question 2: (1.2*10^-3) x (2.0*10⁷) x (1.5*10²) = 3.6*10⁶

Question 3: (2.1*10³) / (3.0*10⁴) = 7.0*10^-2

In addition to these questions, you can complete Practice Problems 3 and 4 in your workbook to become more familiar with scientific notation and calculations involving it.