# 3.01 Scientific notation

Lesson

## Review powers of ten

Recall that an exponent (or power) tells us the number of times to multiply a certain number by itself. Let's review the patterns in the powers of ten with the following exploration.

#### Exploration

Move the slider in the applet below and think about the answers to the following questions.

1. What patterns do you see between the power of ten and its expanded form?
2. What patterns do you see between the power of ten and the number of zeros?

We can see that the power of ten is the same as the number of zeros after it is evaluated.

1. What do you think happens if we evaluate to a power that is less than one?
2. Why do you think $10^0=1$100=1?

Slide the slider to the left in the applet below to find out.

Notice that the power of ten is the same as the number of zeros before or after the decimal point.  Negative powers of ten are decimal numbers less than $1$1, while positive powers of ten are whole numbers. That's because of the place value system.

The following table demonstrates another way to think of some of the powers of ten that we see in the applet.

Power of Ten Meaning Value (fraction) Value (decimal) In Words
$10^5$105 $10\times10\times10\times10\times10$10×10×10×10×10 $100000$100000 $100000$100000 One hundred thousand
$10^4$104 $10\times10\times10\times10$10×10×10×10 $10000$10000 $10000$10000 Ten thousand
$10^3$103 $10\times10\times10$10×10×10 $1000$1000 $1000$1000 One thousand
$10^2$102 $10\times10$10×10 $100$100 $100$100 One hundred
$10^1$101 $10$10 $10$10 $10$10 Ten
$10^0$100 $1$1 $1$1 $1$1 One
$10^{-1}$101 $\frac{1}{10^1}$1101 $\frac{1}{10}$110 $0.1$0.1 One tenth
$10^{-2}$102 $\frac{1}{10^2}=\frac{1}{10\times10}$1102=110×10 $\frac{1}{100}$1100 $0.01$0.01 One hundredth
$10^{-3}$103 $\frac{1}{10^3}=\frac{1}{10\times10\times10}$1103=110×10×10 $\frac{1}{1000}$11000 $0.001$0.001 One thousandth

## Scientific notation

Consider the mass of the sun, which is approximately $1988000000000000000000000000000$1988000000000000000000000000000 kg. That's a very large number! How do scientists deal with numbers so large?

One way to write this number is to use scientific notation, as shown below. Writing numbers in scientific notation will help shorten the amount of writing or typing when doing calculations.

Scientific notation

A number is written in scientific notation if it has the form $a\times10^n$a×10n where $a$a is greater than or equal to $1$1 and less than $10$10, and $n$n is an integer.

Standard form Product form Scientific notation
$2680000$2680000 $2.68\times1000000$2.68×1000000 $2.68\times10^6$2.68×106

We may also see standard form referred to as basic numeral form or be asked to write it as a decimal.

Using the definition above, we can rewrite the mass of the sun as $1.988\times10^{30}$1.988×1030 kg. That takes much less space to write!

#### Worked examples

##### Question 1

What value should go in the space?

$300=\editable{}\times10^2$300=×102

Think: Let's write this expression without scientific notation.

Do:

$10^2$102 is equivalent to $10\times10$10×10 or $100$100. So we can rewrite the question as:

$300=\editable{}\times100$300=×100

So the missing value is $3$3 because $3\times100=300$3×100=300.

Reflect:

This means that $300$300 in scientific notation is $3\times10^2$3×102.

##### Question 2

Evaluate: Scientist recorded the diameter of Mercury as $4.879\times10^3$4.879×103 km. Write the diameter of mercury in standard form (as a basic numeral).

Think: First, we can find the value of the power of ten. Then we can multiply it by the constant term to get the number in standard form.

Do:

 $4.879\times10^3$4.879×103 $=$= $4.879\times1000$4.879×1000 Because $10^3=1000$103=1000 $=$= $4879$4879 Multiply

Reflect: Check that $4879$4879 and $4.879\times10^3$4.879×103 are the same number.

##### Question 3

Evaluate: The diameter of Saturn is approximately $120000$120000 km. Write this number using scientific notation.

Think: We know that $100000=10^5$100000=105. That will give us the power of ten in scientific notation.

Do: Rewrite $120000$120000 as $1.2\times10^5$1.2×105.

Reflect: How can we compare two numbers in scientific notation? For example, we may want to compare the sizes of Mercury and Saturn. How can we very quickly identify which planet is larger from the scientific notation form?

#### Practice questions

##### Question 4

Express $0.07$0.07 in scientific notation.

##### Question 5

Express $3.66\times10^{-6}$3.66×106 as a decimal number.

##### Question 6

The world's oceans hold approximately $188000000000000000000$188000000000000000000 gallons of water. Express this volume of water in scientific notation.

### Outcomes

#### 8.C.2

Solve real-world and other mathematical problems involving numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Interpret scientific notation that has been generated by technology, such as a scientific calculator, graphing calculator, or excel spreadsheet.