Math Core

The Sun is about 150,000,000 kilometers from Earth. A red blood cell is about 0.000008 meters wide. Numbers like these are hard to read and easy to miscopy, because you have to count all those zeros. Scientific notation uses powers of 10 to write very large and very small numbers in a short, clear form.

Powers of 10

Scientific notation is built on powers of 10, including the negative ones from the last lesson:

$10^3$	$10^2$	$10^1$	$10^0$	$10^{-1}$	$10^{-2}$	$10^{-3}$
$1{,}000$	$100$	$10$	$1$	$0.1$	$0.01$	$0.001$

Multiplying by $10^n$ moves the decimal point $n$ places to the right. Multiplying by $10^{-n}$ moves it $n$ places to the left.

Definition

Scientific notation

A number is in scientific notation when it is written as

a \times 10^n

where the coefficient $a$ is at least 1 but less than 10 ( $1 \le a < 10$ ), and the exponent $n$ is an integer.

For example, $3.7 \times 10^5$ is in scientific notation. But $37 \times 10^4$ is not (37 is too big), and $0.37 \times 10^6$ is not (0.37 is too small), even though all three equal $370{,}000$ .

From scientific notation to standard form

To write $a \times 10^n$ as an ordinary number (standard form), move the decimal point in $a$ by $n$ places. Fill in zeros as needed.

Worked example: Converting to standard form

Write each number in standard form.

$6.4 \times 10^5$
$2.05 \times 10^{-3}$

Solutions.

The exponent is positive, so move the decimal 5 places right: $6.4 \to 640{,}000$ . So $6.4 \times 10^5 = 640{,}000$ .
The exponent is negative, so move the decimal 3 places left: $2.05 \to 0.00205$ . So $2.05 \times 10^{-3} = 0.00205$ .

From standard form to scientific notation

Go the other way in three steps:

Place the decimal point right after the first nonzero digit. That gives the coefficient.
Count how many places the decimal point moved.
A big number (10 or more) gets a positive exponent. A small number (less than 1) gets a negative exponent.

Worked example: Converting to scientific notation

Write each number in scientific notation.

$150{,}000{,}000$
$0.000008$

Solutions.

The first nonzero digit is 1, so the coefficient is $1.5$ . The decimal point moves from the end of $150{,}000{,}000$ to just after the 1: that is 8 places. The number is big, so the exponent is positive: $1.5 \times 10^8$ .
The first nonzero digit is 8, so the coefficient is $8$ . The decimal point moves from $0.000008$ to just after the 8: that is 6 places. The number is small, so the exponent is negative: $8 \times 10^{-6}$ .

Common mistake

Check that the exponent's sign makes sense. A small number like $0.0042$ must have a negative exponent: $4.2 \times 10^{-3}$ . If you wrote $4.2 \times 10^3$ , you would have $4{,}200$ , which is a million times too big.

Tip

To check a conversion, ask: "Is my number bigger than 10 or smaller than 1?" Positive exponent means bigger than 10, negative exponent means less than 1, and $10^0$ means between 1 and 10.

Comparing and estimating

To compare two numbers in scientific notation, look at the exponents first. The larger exponent gives the larger number (for positive numbers). If the exponents match, compare the coefficients.

$2 \times 10^{7}$ is greater than $9 \times 10^{6}$ , because $10^7$ is ten times $10^6$ .
$3 \times 10^{-4}$ is greater than $5 \times 10^{-5}$ , because $-4 > -5$ .

You can also use scientific notation to estimate how many times bigger one quantity is than another. Round each number to a single digit times a power of 10, then divide.

Worked example: How many times as large?

A large city has about 8,000,000 people. A small town has about 2,000 people. About how many times as many people live in the city?

Round and write each number with a power of 10: the city has $8 \times 10^6$ people and the town has $2 \times 10^3$ .

\frac{8 \times 10^6}{2 \times 10^3} = \frac{8}{2} \times \frac{10^6}{10^3} = 4 \times 10^3 = 4{,}000

The city has about 4,000 times as many people as the town.

Practice

Practice 1

Write $4.5 \times 10^6$ in standard form.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Write $3.08 \times 10^{-4}$ in standard form.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Which shows $72{,}000{,}000$ in scientific notation?

Practice 4

The number $38{,}400{,}000{,}000$ is written as $3.84 \times 10^n$ . What is $n$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

The number $0.0000591$ is written as $5.91 \times 10^n$ . What is $n$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Which number is written in scientific notation?

Practice 7

Which number is the greatest?

Practice 8

A country has about $2 \times 10^8$ people. One of its cities has about $4 \times 10^5$ people. About how many times as many people live in the whole country as in the city?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.