Math Core

The Sun is about 150,000,000 kilometers from Earth. A single bacterium might be 0.000002 meters long. Numbers like these have so many zeros that they are easy to miscount. Scientific notation uses powers of 10 to write them in a short, clear form.

What scientific notation looks like

Multiplying by 10 moves the decimal point one place to the right. Multiplying by $10^3 = 1{,}000$ moves it three places. So you can write any number as a small number times a power of 10:

150{,}000{,}000 = 1.5 \times 100{,}000{,}000 = 1.5 \times 10^8

Definition

Scientific notation

A number is in scientific notation when it is written as

a \times 10^n

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

The first part must have exactly one nonzero digit before the decimal point. So $3.2 \times 10^5$ is in scientific notation, but $32 \times 10^4$ and $0.32 \times 10^6$ are not, even though all three equal 320,000.

Writing large numbers

To write a large number in scientific notation, move the decimal point left until just one nonzero digit is in front of it. The number of places you moved it is the exponent.

Worked example: A large number

Write 4,070,000 in scientific notation.

The decimal point is at the end: $4070000.$ Move it left until only the 4 is in front:

4\,070\,000. \;\longrightarrow\; 4.070000

That is 6 places, so $4{,}070{,}000 = 4.07 \times 10^6$ . Drop the trailing zeros after the 7.

Writing small numbers

For a number between 0 and 1, move the decimal point right until one nonzero digit is in front of it. Since the original number is small, the exponent is negative. Remember from the exponent rules that $10^{-1} = 0.1$ , $10^{-2} = 0.01$ , and so on.

Worked example: A small number

Write 0.00058 in scientific notation.

Move the decimal point right until the 5 is in front:

0.00058 \;\longrightarrow\; 5.8

That is 4 places to the right, so $0.00058 = 5.8 \times 10^{-4}$ .

Check: $5.8 \times 10^{-4} = 5.8 \times 0.0001 = 0.00058$ .

Which way is the exponent?

A number 10 or greater has a positive exponent.
A number between 1 and 10 has exponent 0, since $10^0 = 1$ .
A number between 0 and 1 has a negative exponent.

Converting back to standard form

To go from scientific notation back to an ordinary number (standard form), move the decimal point the number of places the exponent tells you. Positive exponent: move right. Negative exponent: move left. Fill empty places with zeros.

$2.9 \times 10^5$ : move right 5 places to get $290{,}000$ .
$7.1 \times 10^{-3}$ : move left 3 places to get $0.0071$ .

Common mistake

A negative exponent does not make the number negative. $6 \times 10^{-2} = 0.06$ , which is a small positive number. A negative number in scientific notation has the negative sign in front, like $-6 \times 10^{2} = -600$ .

Comparing and multiplying

To compare two positive numbers in scientific notation, look at the exponents first. The bigger exponent means the bigger number. Only if the exponents are equal do you compare the first parts.

To multiply, group the first parts together and the powers of 10 together, then use the exponent rule for multiplying powers.

Worked example: Multiplying in scientific notation

Find $(3 \times 10^4)(5 \times 10^6)$ . Write the answer in scientific notation.

\begin{aligned} (3 \times 10^4)(5 \times 10^6) &= (3 \times 5) \times (10^4 \times 10^6) \\ &= 15 \times 10^{10} \end{aligned}

But 15 is not less than 10, so this isn't in scientific notation yet. Write $15 = 1.5 \times 10^1$ :

15 \times 10^{10} = 1.5 \times 10^1 \times 10^{10} = 1.5 \times 10^{11}

Tip

A quick check: in the final answer, count the digits. $1.5 \times 10^{11}$ should be a 12-digit number, 150,000,000,000, because the exponent 11 counts the places after the first digit.