Standard Form

One digit before the point

Very large and very small numbers are long to write and easy to misread: 150,000,000 has so many zeros that it is hard to count them. Standard form writes any number as a number from 1 up to (but not including) 10, multiplied by a power of ten.

Take 4700. The number in front must have exactly one digit before the decimal point, so it is 4.7. Now find what 4.7 must be multiplied by to make 4700. Multiplying by 1000 moves every digit three places to the left, into columns worth 1000 times as much, and 4.7 becomes 4700. The decimal point stays where it is.

Choose × 1000 and watch each digit of 4.7 move three places to the left, while the decimal point stays still: 4.7 becomes 4700.

The power counts the places

So 4700 = 4.7 × 1000. And 1000 is three tens multiplied together, 10 × 10 × 10, which is $10^3$10³. So $4700=4.7\times 10^3$4700 = 4.7 × 10³, and that is 4700 in standard form.

The index of the power of ten is the number of places every digit moves. 4,700,000 is 4.7 with every digit moved six places to the left, so 4,700,$000=4.7\times 10^6$000 = 4.7 × 10⁶. And 150,000,$000=1.5\times 10^8$000 = 1.5 × 10⁸, because 1.5 needs its digits moved eight places to make 150,000,000.

Standard form also makes sizes easy to compare: a number times $10^8$10⁸ is bigger than any number in standard form times $10^6$10⁶, so compare the powers first.

Small numbers move the other way

A number smaller than 1 works the same way, with the digits moving to the right. Take 0.0047. The number in front is again 4.7, and dividing 4.7 by 1000 moves every digit three places to the right: 4.7 ÷ 1000 = 0.0047.

Dividing by $10^3$10³ is the same as multiplying by $10^{-3}$10⁻³, because a negative index means one over the power: $10^{-3}=\frac{1}{10^3}=\frac{1}{1000}$10⁻³ = 1/10³ = 1/1000. So $0.0047=4.7\times 10^{-3}$0.0047 = 4.7 × 10⁻³. A negative power of ten means the number is less than 1.

In the same way, $0.00032=3.2\times 10^{-4}$0.00032 = 3.2 × 10⁻⁴, because the digits of 3.2 move four places to the right to make 0.00032.

Choose ÷ 1000 and watch each digit of 4.7 move three places to the right: 4.7 becomes 0.0047, which is $4.7\times 10^{-3}$4.7 × 10⁻³.

Two common mistakes

Count the places from the number in front, not from the number of digits. 36,000 is $3.6\times 10^4$3.6 × 10⁴, because the digits of 3.6 move four places to make 36,000. Writing $3.6\times 10^5$3.6 × 10⁵ moves them one place too far, which makes 360,000, ten times too big.

$36\times 10^3$36 × 10³ is also equal to 36,000, but it is not standard form, because 36 has two digits before the decimal point. The number in front must be at least 1 and less than 10.