# Appendix C – Scientific Notation: Working Effectively with Large and Tiny Numbers

**SCIENTIFIC NOTATION**

The universe contains material in very tiny and extremely large quantities that scientific notation allows us to express with ease. By convention, numbers given in scientific notation are always shown with the ‘coefficient’ followed by the ‘exponent term’. In the coefficient, as illustrated below, one figure is given before the decimal, which is then followed by any remaining figures. The coefficient is multiplied by the exponent term—a power of the base 10, or in other words, 10 raised to some exponent.

the coefficient – one figure before the decimal (3), then any remaining figures (527)

3.527 x 10^{3}

the exponent term – a base of 10 raised to the 3rd power, indicating that the number is multiplied by

10 x 10 x 10 (or 10^{3})

When we convert numbers out of scientific notation, we multiply them by 10 by the number of times expressed in the exponent. In the example given above, the exponent 3 indicates that the coefficient is multiplied by 10 three times. In other words, 3.527 x 10^{3 }= 3.527 x 10 x 10 x 10 = 3527. A rapid conversion can be made by moving the decimal three places to the right (showing that we are multiplying by 10 three times).

3.527 x 10^{3 }= 3527

_{ →}

_{decimal moved 3 places to the right}

At a quick glance, the value of the exponent tells us a lot about the value of the number. A large exponent indicates a BIG number. 2.3 x 10^{24 }= 2,300,000,000,000,000,000,000,000!

## Negative Powers, or Reciprocals

Sometimes we encounter numbers in scientific notation that contain a negative exponent. Any negative exponent represents a reciprocal, which indicates a number smaller than 1.

The reciprocal of 10^{-1 }is 1 /10^{1}

The reciprocal of 10^{-3 }is 1/10^{3} or 1 .

The reciprocal of 10^{-19 }is103 10x10x10

1 .1019

1

These reciprocals indicate that the coefficient is divided by 10 by the number of times expressed in the exponent. For instance, 3.9 x 10-3 is converted out of scientific notation by dividing 3.9 by 10 three times. In other words, 3.9 x 10-3 = 3.9 10 10 10 = 0.0039. A rapid conversion can be made by moving the decimal three places to the left (showing that we are dividing by 10 three times).

3.9 x 10-3 = 0.0039

3 2 1 decimal places to the left

Again, a quick glance at the exponent reveals details about the number. A negative exponent means that we have a value less than 1.0. The larger the negative exponent the smaller the

number.

5.9 x 10-15 = 0.0000000000000059

Performing Calculations with Scientific Notation

Adding and Subtracting Numbers in Scientific Notation

How do we add 2.5 x 102 and 1.0 x 103 when their exponents are different? 103 (1,000) is an order of magnitude larger than 102 (100). We first need to convert these numbers to the same power of ten. Then we can simply add the coefficients and reiterate the exponent term. In this example we can convert the first number to match the power of ten in the second number.

2.5 x 102 = 0.25 x 103. Now we can add the coefficients and write in the exponent.

0.25×103 +1.0×103 =1.25×103.

Do you need to prove to yourself that this is correct? If so, take the numbers out of scientific notation, add them, and then convert your answer to scientific notation to compare it to our response above.

Prove it to yourself!

0.25 x 103 = 250 and 1.0 x 103 = 1000 250 + 1000 = 1250, which is 1.25 x 103

Multiplying Numbers in Scientific Notation

When multiplying numbers in scientific notation we multiply the coefficients and add the exponents.

multiply the coefficients

For example, (2.5 x 102) (2.0 x 103) = 5.0 x 105.

add the exponents

Here’s another example. (6.0x 102) (2.0 x 103) = 12.0 x 105

2

But now we have two figures to the left of the decimal when, by convention, we only leave one. We need to change the location of the decimal in our answer so there is only one figure to the left of the decimal. 12.0 x 105 = 1.2 x 106

Dividing Numbers in Scientific Notation

When dividing numbers in scientific notation we divide the coefficients and subtract the exponents.

divide the coefficients

8.06 x10 7

For example, 2.0x104 = 8.06 / 2.01 x 107-4 = 4.03 x 103

subtract the exponents

Here’s another example. 3.4 x 109 / 4.0 x 103 = 3.4/4.0 x 109-3 = 0.85 x 106.

But now we have no figures (other than zero) to the left of the decimal when, by convention, we should leave one. We need to change the location of the decimal in our answer so there is one figure to the left of the decimal. 0.85 x 106 = 8.5 x 105

Dealing with Exponents Raised to an Exponent

Sometimes we will encounter numbers expressed in scientific notation that are being raised to a power. In these instances, we raise the coefficient to the indicated power, and multiply the exponent term by that power.

For example, (4.07 x 106)4 = (4.07)4 x 106×4 = 274.3 x 1024 = 2.743 x 1026

raise to 4th power multiply: 6 x 4

Recap:

100 = 1 Powers:

Reciprocals:

100 always equals 1

102 = 10 x 10, or symbolically, 10n = 10 times itself “n” times

101 1 , or symbolically, 10n 110 10n

When adding or subtracting, first convert to the same power of 10. |
When multiplying, add the exponents. 10n x 10m = 10n+m |

When dividing, subtract the exponents. 10n 10m = 10n-m |
When exponents are raised to a power, multiply the exponents. |

## Practice Questions:

Try these questions, first on your own, then on your calculator. No cheating! 😎 Check the answers provided below after you’ve formed your own responses!

### Transpose the following numbers into or out of scientific notation:

- 3,872,000 =
- 9.82×10
^{6}= - 9134 =
- 1.574 x 10
^{3 }= - 0.127 =
- 2.751 x 10
^{-4 }=

### Perform the following calculations:

- 3.5×10
^{4 }+ 2.2×10^{3 }= - 3.5×10
^{4}– 3.9 x 10^{3 }= - (1.9 x 10
^{3}) (3.2 x 10^{1}) = - (5.2 x 10
^{-5}) (1.1 x 10^{4}) = - (8.7 x 10
^{6}) / (4.3 x 10^{3}) = - (9.5 x 10
^{21}) / (3.2 x 10^{19}) = - (3.22 x 10
^{15})^{3 }= - (5.67 x 10
^{-8})^{4 }=

### The Answers

- 3.872 x 10
^{6} - 9,820,000
- 9.134 x 10
^{1} - 1,574
- 1.27 x 10
^{-1} - 0.0002751

- 3.5 x 10
^{4}+ 0.22 x 10^{4}= 3.72 x 10^{4} - 3.5 x 10
^{4}– 0.39 x 10^{4}= 3.11 x 10^{4} - (1.9)(3.2) x 10
^{3+1}= 6.08 x 10^{4} - (5.2)(1.1) x 10
^{-5+4}= 5.72 x 10^{-1} - (8.7) / (4.3) x 10
^{6-3}= 2.02326 x 10^{3} - (9.5) / (3.2) x 10
^{21-19}= 2.97 x 10^{2} - (3.22)
^{3}x 10^{15x}^{3 }= 3.339 x 10^{46}after adjusting to leave the decimal behind the first figure - (5.67)
^{4}x 10^{-8x}^{4 }= 1.03 x 10^{-29}after adjusting to leave the decimal behind the first figure

You might be wondering how many figures to leave in your answers for these practice questions. Where should they be rounded off? Read on! Your questions will be answered in the following section on significant figures.