# 2.3 Scientific Notation

## Learning Objectives

By the end of this section, you will be able to:

• Recall the Laws of Exponents
• Convert from decimal notation to scientific notation
• Convert scientific notation to decimal form
• Multiply and divide using scientific notation

### Try It

Before you get started, take this readiness quiz:

1) What is the place value of the 6 in the number $64,891$?
2) Name the decimal: $0.0012$.
3) Subtract: $5-(-3)$.

## Review of the Exponent Properties

If $a$ and $b$ are real numbers, and $m$ and $n$ are integers, then:

 Product Property $\frac{a^m}{a^n}=a^{m+n}$ Power Property $({a}^{m})^{n} = {a}^{mn}$ Product to a Power $(ab)^{m} = {a}^{m}{b}^{m}$ Quotient Property $\frac{a^m}{a^n}=a^{m-n}$ if $m>n$ and $\frac{a^m}{a^n}=\frac1{a^{n-m}}$, if $n>m$, $a\neq 0$ Zero Exponent Property $a^0=1, a\neq 0$ Quotient to a Power Property $a^{-n}=\frac{1}{a^n}$ and $\frac{1}{a^{-n}}=a^n$, $a\neq 0$ Quotient to a Negative Exponent $\left(\frac{a}{b}\right)^{-n}=\left(\frac{b}{a}\right)^n$

## Scientific Notation

Sometimes, in real-life scenarios, we may need to deal with numbers that are very large, or numbers that are very small. Thus, when dealing with numbers like $1,280,000,000$ or $0.00000000274$, it may be beneficial to represent these numbers in a different way. We will use scientific notation to help us with this.

### Scientific Notation and Significant Figures

Before we work on converting between forms of numbers, let’s consider the concept of significant figures and scientific notation. We can determine the number of significant figures in a number in scientific notation in the same way we would a number in decimal notation. Let’s remind ourselves of the rules for significant figures that were covered in the previous section:

The following conventions dictate which numbers in a reported measurement are significant and which are not significant:

1. Any nonzero digit is significant.
2. Any zeros between nonzero digits (i.e., embedded zeros) are significant.
3. Zeros at the end of a number without a decimal point (i.e., trailing zeros) are not significant; they serve only to put the significant digits in the correct positions. However, zeros at the end of any number with a decimal point are significant.
4. Zeros at the beginning of a decimal number (i.e., leading zeros) are not significant; again, they serve only to put the significant digits in the correct positions.

Of course, in scientific notation, we no longer have to worry about leading zeros. This is one of the benefits of using scientific notation, as the digits present are always significant.

### Example 2.3.1

Determine the number of significant figures in the following numbers:

a. $1.41 \times 10^3$
b. $1.034 \times 10^{-5}$
c. $4.0000 \times 10^{-2}$

Solution

a. All non-zero digits are significant so there are 3 significant figures.
b. All confined zeros are significant, as well as all non-zero digits, so there are 4 significant figures.
c. All zeros at the end of the number after a decimal point are significant, so there are 5 significant figures.

### Try It

4) Determine the number of significant figures in the following numbers:

a. $1.9 \times\ 10^3$
b. $4.00 \times\ 10^{-5}$
c. $5.001 \times\ 10^{-2}$

Solution

a. 2 significant figures
b. 3 significant figures
c. 4 significant figures

### Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 4,000 and 0.004. We know that 4,000 means $4\;\times\;1000$ and $0.004$ means $4\times \frac{1}{1,000}$

If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way:

 $4000$ ${4}\times{1000}$ $4\;\times\;10^3$ $0.004$ $4\;\times\;\frac1{1000}$ $4\;\times\;\frac1{10^3}=4\times10^{-3}$

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.

### Scientific Notation

A number is in scientific notation if it is in the form $M\times 10^n$ where $1\le M < 10$.

It is customary in scientific notation to use as the $\times$ multiplication sign, even though we avoid using this sign elsewhere in algebra.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.

${4\;\times\;10^3\;=\;\text{Decimal moved 3 to the right.}\;\;4}{\color{red}{\overrightarrow.}}{0_{\color{red}{1}}0_{\color{red}{2}}0_{\color{red}{3}}\;=\;4000}{\color{red}{.}}$

${4\;\times\;10^{-3}\;=\;\text{Decimal moved 3 to the left.}\;\;0_{\color{red}{3}}}{0_{\color{red}{2}}0_{\color{red}{1}}4{\color{red}{\overleftarrow.}}\;=\;0{\color{red}{.}}}{\color{black}{004}}$

In both cases, the decimal was moved 3 places to get the first factor between 1 and 10.

The power of 10 is positive when the number is larger than 1:                  ${4,000}=4\times 10^3$

The power of 10 is negative when the number is between 0 and 1:         $0.004=4\times 10^{-3}$

### Example 2.3.2

Write in scientific notation: 37,000.

Solution

Step 1: Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.

Remember, there is a decimal at the end of 37,000.

Move the decimal after the 3. 3.700 is between 1 and 10.

Step 2: Count the number of decimal places, $n$, that the decimal point was moved.

The decimal place was moved 4 places to the left.

${\text{Decimal moved 4 to the left. }3_{\color{red}{4}}7_{\color{red}{3}}0_{\color{red}{2}}0_{\color{red}{1}}0{\color{red}{\overleftarrow.}}{\color{red}{\;}}{\color{black}{=}}{\color{red}{\;}}{\color{black}{3}}{\color{red}{.}}{\color{black}{7}}}$

Step 3: Write the number as a product with a power of 10.

If the original number is:

Greater than 1, the power of 10 will be $10^a$.
Between 0 and 1, the power of 10 will be $10^{-a}$.

37,000 is greater than 1 so the power of 10 will have exponent 4.

$3.7\times10^4$

$10^4$ is 10,000 and 10,000 times 3.7 will be 37,000.

$3.7\times10^4=37,000$

### Try It

5) Write in scientific notation: $96,000$.

Solution

$9.6\times \ {10^4}$

6) Write in scientific notation: $48,300$.

Solution

$4.83\times \ {10^4}$

7) Write in scientific notation: $54,\tilde{0}00$

Solution

$5.40\times \ {10^4}$

### HOW TO

#### Convert from decimal notation to scientific notation

1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
2. Count the number of decimal places, $n$, that the decimal point was moved.
3. Write the number as a product with a power of 10.
If the original number is:

• greater than 1, the power of 10 will be $10^n$.
• between 0 and 1, the power of 10 will be $10^{-n}$.
4. Check.

### Example 2.3.3

Write in scientific notation: 0.0052

Solution

The original number, 0.0052, is between 0 and 1 so we will have a negative power of 10.

Step 1: Move the decimal point to get 5.2, a number between 1 and 10.

${0}{\color{red}{.}}{0_{\color{red}{\uparrow}}0_{\color{red}{\uparrow}}5_{\color{red}{\uparrow}}2\;=\;5}{\color{red}{.}}{2}$

Step 2: Count the number of decimal places the point was moved.

${0}{\color{red}{.}}{0_{\color{red}{1}}0_{\color{red}{2}}5_{\color{red}{3}}2\;=\;\text{The decimal was moved }}{\color{red}{\text{3}}}{\text{ places.}}$

Step 3: Write as a product with a power of 10.

$5.2\times \ {10}^{-3}$

Step 4: Check.

$\begin{eqnarray*}5.2&\times&\;10^{-3}\\5.2&\times&\frac1{10^3}\\5.2&\times&\frac1{1000}\\5.2\times0.001\;&=&\;0.0052\\0.0052&=&5.2\times\;10^{-3}\\\end{eqnarray*}$

### Try It

8) Write in scientific notation: $0.0078$

Solution

$7.8\times \ {10}^{-3}$

9)Write in scientific notation: $0.0129$

Solution

$1.29\times \ {10}^{-2}$

### Preserving Significant Zeros Using Scientific Notation

As we saw in the previous section, when considering a number with trailing zeros, it is not easy to determine whether or not they are significant. For example, the number 10,000 could have one significant figure only, but is is possible that some of the zeros are significant as well. One way to communicate a significant zero, when there is no decimal present, is to use a tilde to indicate its significance. For instance, $10,0\tilde{0}0$ indicates that the third zero is significant, which makes the zeros in between significant as well. In this sense, this number $10,0\tilde{0}0$ has 4 significant figures. If we wanted to put this number into scientific notation, we would need to include those significant zeros and it would be represented by $1.000\times 10^4$.

### Try It

10) Convert the following numbers to scientific notation. Be sure to preserve the number of significant figures.

a. $0.002840$
b. $129.00$
c. $1800\tilde{0}$

Solution

a. $2.840 \times 10^{-3}$
b. $1.2900 \times 10^2$
c. $1.8000 \times 10^4$

### Convert Scientific Notation to Decimal Form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.

$\begin{eqnarray*}9.12\;&\times&\;10^4\\9.12\;&\times&\;10^{-4}\\9.12\;&\times&\;10,000\\9.12\;&\times&\;0.0001\\91,200\;&\times&\;0.000912\end{eqnarray*}$

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

${9.12\times\;10^4=\text{Move the decimal 4 to the right.}\;9}{\color{red}{\overrightarrow.}}{1_{\color{red}{1}}2_{\color{red}{2}}0_{\color{red}{3}}0_{\color{red}{4}}\;=\;91,200}{\color{red}{.}}$

${9.12\times\;10^{-4}=\text{Move the decimal 4 to the left.}\;=\;0_{\color{red}{4}}}{0_{\color{red}{3}}0_{\color{red}{2}}0_{\color{red}{1}}9{\color{red}{\overleftarrow.}}12\;=\;0}{\color{red}{.}}{000912}$

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

### Example 2.3.4

Convert to decimal form: $6.2\times\ {{10}^{3}}$

Solution

Step 1: Determine the exponent, $n$, on the factor 10.

The exponent is $3$.

$6.2\times10^3$

Step 2: Move the decimal n spaces, adding zeros if needed.

If the exponent is positive, move the decimal point $n$ places to the right.
If the exponent is negative, move the decimal point $|n|$ places to the left.

$\begin{array}{c}\text{Exponent is positive so we move the decimal 3 spaces to the right, adding 2 zeros as placeholders.}\\6{\color{red}{\overrightarrow.}}{2_{\color{red}{1}}0_{\color{red}{2}}0_{\color{red}{3}}\;=\;6,200}{\color{red}{.}}{}\end{array}$

$10^3$ is $1000$ and $1000$ times $6.2$ is $6,200$.

### Try It

11) Convert to decimal form: $1.3 \times 10^3$

Solution

${1,300}$

12) Convert to decimal form: $9.25 \times 10^4$

Solution

${92,500}$

13) Convert to decimal form: $3.900 \times 10^5$

Solution

$390,\tilde{0}00$

### HOW TO

#### Convert scientific notation to decimal form.

The steps are summarized below.

To convert scientific notation to decimal form:

1. Determine the exponent, $n$, on the factor 10.
2. Move the decimal $n$ places, adding zeros if needed.
• If the exponent is positive, move the decimal point $n$ places to the right.
• If the exponent is negative, move the decimal point $\left|n\right|$ places to the left.
3. Check.

### Example 2.3.5

Convert to decimal form: $8.9\times 10^{-2}$

Solution

Step 1: Determine the exponent, n, on the factor 10.

The exponent is $-2$.

Step 2: Since the exponent is negative, move the decimal point 2 places to the left.

${8.9\;\times\;10^{-2}\;=\;}{0_{\color{red}{2}}{0_{\color{red}{1}}8}}{\color{red}{\overleftarrow.}}{9}$

Step 3: Add zeros as needed for placeholders.

$8.9\;\times\;10^{-2}\;=\;0.089$

### Try It

14) Convert to decimal form: $1.2\times 10^{-4}$

Solution

$0.00012$

15) Convert to decimal form: $7.5\times 10^{-2}$

Solution

$0.075$

### Multiply and Divide Using Scientific Notation

Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

### Example 2.3.6

Multiply. Write answers in decimal form: $(4\times 10^{5})(2\times 10^{-7})$

Solution

Step 1: Use the Commutative Property to rearrange the factors.

$4 \times \ {2} \times \ 10^5 \times \ 10^{-7}$

Step 2: Multiply.

$8 \times \ 10^{-2}$

Step 3: Change to decimal form by moving the decimal two places left.

$0.08$

### Try It

16) Multiply $(3\times {10}^{6})(2\times 10^{-8})$. Write answers in decimal form.

Solution

$0.06$

17) Multiply $(3\times 10^{-2})(3\times 10^{-1})$. Write answers in decimal form.

Solution

$0.009$

### Example 2.3.7

Divide. Write answers in decimal form: $\frac{9\times{10^3}}{3\times{10^{-2}}}$

Solution

Step 1: Separate the factors, rewriting as the product of two fractions.

$\frac{9\times{10^3}}{3\times{10^{-2}}}$

Step 2: Divide.

$3\times\ 10^5$

Step 3: Change to decimal form by moving the decimal five places right.

$300,000$

### Try It

18) Divide $\frac{{8}\times 10^{4}}{2\times 10^{-1}}$ Write answers in decimal form.

Solution

$400,000$

19) Divide $\frac{8\times 10^{2}}{4\times 10^{-2}}$. Write answers in decimal form.

Solution

$20,000$

Access these online resources for additional instruction and practice with integer exponents and scientific notation:

### Key Concepts

• To convert a decimal to scientific notation:
1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
2. Count the number of decimal places, $n$, that the decimal point was moved.
3. Write the number as a product with a power of 10. If the original number is:
• greater than 1, the power of 10 will be ${10}^{n}$
• between 0 and 1, the power of 10 will be ${10}^{−n}$
4. Check.
• To convert scientific notation to decimal form:
1. Determine the exponent, $n$, on the factor 10.
2. Move the decimal $n$ places, adding zeros if needed.
• If the exponent is positive, move the decimal point $n$ places to the right.
• If the exponent is negative, move the decimal point $\left|n\right|$ places to the left.
3. Check.
• Use a tilde to indicate a significant zero when necessary.

### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

## Glossary

### scientific notation

A number is expressed in scientific notation when it is of the form $M\times 10^n$ where $1\le M < 10$ and $n$ is an integer.
definition