Standard form

Light travels about $300000000$ metres in one second. A thin wire may be only $0.000003$ metres across. Numbers like these have so many zeros that they are hard to read and easy to miscount. Powers of $10$ give a neat way to write them, called standard form or scientific notation. In this lesson you will write very large and very small numbers in standard form, change them back, and multiply them.

What standard form is

A number is in standard form when it is written as

$k \times 10^n$

where $k$ is at least $1$ but less than $10$ , and $n$ is an integer (positive, negative or zero).

So $3 \times 10^8$ and $6.74 \times 10^9$ are in standard form. But $67.4 \times 10^8$ is not, because $67.4$ is bigger than $10$ .

Large numbers

Remember that $10^1 = 10$ , $10^2 = 100$ , $10^3 = 1000$ and so on: the exponent counts the zeros.

Put the point just after the first digit, so that the number is between $1$ and $10$ .
Count how many places the point moved to the left. That count is the exponent.

Example 1

Write $300000000$ in standard form.

The point sits at the end: $300000000.$ Moving it to just after the $3$ takes $8$ places to the left.

$300000000 = 3 \times 10^8$

Example 2

Write $6740000000$ in standard form.

Place the point after the $6$ : $6.740000000$ . It moved $9$ places to the left. Drop the zeros at the end.

$6740000000 = 6.74 \times 10^9$

Example 3

The distance from the Earth to the Sun is about $149600000$ km. Write it in standard form.

The point moves $8$ places to the left, to just after the $1$ .

$149600000 = 1.496 \times 10^8 \text{ km}$

Example 4

Write the number " $1$ followed by $52$ zeros" in standard form.

It is $10 \times 10 \times \cdots$ with $52$ tens, which is $10^{52}$ , or $1 \times 10^{52}$ .

Small numbers

For numbers less than $1$ , the point moves to the right, and the exponent is negative.

Example 5

Write $0.000003$ in standard form.

$0.000003$ has six digits after the point, so it is $\displaystyle \frac{3}{1000000}$ .

$\displaystyle 0.000003 = \frac{3}{10^6} = 3 \times 10^{-6}$

Here we used Law 7: $\displaystyle \frac{1}{10^6} = 10^{-6}$ . In short, the point moved $6$ places to the right, to just after the $3$ , so the exponent is $-6$ .

Example 6

Write $0.0000021$ in standard form.

Move the point to just after the $2$ : that is $6$ places to the right.

$0.0000021 = 2.1 \times 10^{-6}$

Example 7

Write $0.00045$ in standard form.

The point moves $4$ places to the right, to just after the $4$ .

$0.00045 = 4.5 \times 10^{-4}$

Remember. Big number: the point moves left and the exponent is positive. Small number: the point moves right and the exponent is negative.

Back to ordinary form

Reverse the move. A positive exponent moves the point right; a negative exponent moves it left. Fill gaps with zeros.

Example 8

Write $4.8 \times 10^4$ and $7.05 \times 10^{-4}$ in ordinary form.

$4.8 \times 10^4$ : move the point $4$ places right. $4.8000$ becomes $48000$ .

$7.05 \times 10^{-4}$ : move the point $4$ places left. $0007.05$ becomes $0.000705$ .

Multiplying and dividing in standard form

Multiply the numbers in front, and use Law 1 on the powers of $10$ . If the front number reaches $10$ or more, adjust it.

Example 9

Find $(3 \times 10^4) \times (2 \times 10^5)$ .

$\begin{aligned} (3 \times 10^4) \times (2 \times 10^5) &= (3 \times 2) \times 10^{4+5} \\ &= 6 \times 10^9 \end{aligned}$

Example 10

Find $0.000002 \times 0.0000003$ in standard form.

$\begin{aligned} 0.000002 \times 0.0000003 &= (2 \times 10^{-6}) \times (3 \times 10^{-7}) \\ &= 6 \times 10^{-6 + (-7)} \\ &= 6 \times 10^{-13} \end{aligned}$

Example 11

Find $(6.5 \times 10^3) \times (2 \times 10^4)$ .

$(6.5 \times 2) \times 10^{3+4} = 13 \times 10^7$

But $13$ is not less than $10$ . Write $13 = 1.3 \times 10$ .

$13 \times 10^7 = 1.3 \times 10^1 \times 10^7 = 1.3 \times 10^8$

Example 12

Find $\displaystyle \frac{2.4 \times 10^7}{1.2 \times 10^3}$ .

$\displaystyle \frac{2.4 \times 10^7}{1.2 \times 10^3} = \frac{2.4}{1.2} \times 10^{7-3} = 2 \times 10^4$

Comparing numbers in standard form

Compare the exponents first. The bigger exponent means the bigger number. Only if the exponents are the same do you compare the front numbers.

For example, $3.2 \times 10^5$ is bigger than $9.8 \times 10^4$ , because $10^5$ is ten times $10^4$ .

Common mistakes

Leaving the front number outside the range. $67.4 \times 10^8$ should be $6.74 \times 10^9$ .
Giving a positive exponent to a small number. $0.00045 = 4.5 \times 10^{-4}$ , not $4.5 \times 10^4$ .
Counting the zeros instead of the places the point moves. In $0.0000021$ the point moves $6$ places.
Comparing front numbers before exponents.

Practice

Write $35000000$ in standard form.
Write $5200000$ in standard form.
Write $0.0000021$ in standard form.
Write $0.00702$ in standard form.
Write $0.00000008$ in standard form.
Write $5.2 \times 10^6$ in ordinary form.
Write $7.05 \times 10^{-4}$ in ordinary form.
Find $(4 \times 10^3) \times (2 \times 10^{-5})$ in standard form.
Find $(1.2 \times 10^4) \times (3 \times 10^2)$ in standard form.
Which is bigger, $4.2 \times 10^5$ or $9.9 \times 10^4$ ?

Answers

Show answers

$3.5 \times 10^7$
$5.2 \times 10^6$
$2.1 \times 10^{-6}$
$7.02 \times 10^{-3}$ . The point moves $3$ places to the right.
$8 \times 10^{-8}$ . The point moves $8$ places to the right.
$5200000$
$0.000705$
$8 \times 10^{-2}$ . $(4 \times 2) \times 10^{3 + (-5)}$ .
$3.6 \times 10^6$ . $(1.2 \times 3) \times 10^{4+2}$ .
$4.2 \times 10^5$ , because its exponent $5$ is bigger than $4$ .

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What stan­dard form is#

Large num­bers#

Exam­ple 1#

Exam­ple 2#

Exam­ple 3#

Exam­ple 4#

Small num­bers#

Exam­ple 5#

Exam­ple 6#

Exam­ple 7#

Back to ordi­nary form#

Exam­ple 8#

Mul­ti­ply­ing and divid­ing in stan­dard form#

Exam­ple 9#

Exam­ple 10#

Exam­ple 11#

Exam­ple 12#

Com­par­ing num­bers in stan­dard form#

Com­mon mis­takes#

Prac­tice#

Answers#