Standard form — writing very big and small numbers — Mathematics

Before this

Index notation and the basic laws · coming soon

This leads to

Calculating with standard form
Rounding and significant figures
Negative and fractional indices · coming soon

Numbers too long to write

The mass of the Earth is about 5 970 000 000 000 000 000 000 000 kg. The mass of a proton, one of the particles inside every atom, is about 0.000 000 000 000 000 000 000 000 001 67 kg.

Try reading either number aloud. Try copying one without losing a zero. Now imagine comparing them, or using them in a calculation. Written like this, the numbers hide the one thing you need to know first: how big they are.

Scientists write them differently. The Earth's mass is $5.97 \times 1 0^{24}$ kg, and the proton's mass is $1.67 \times 1 0^{- 27}$ kg (NASA; CODATA 2022). This short way of writing is called standard form. Some books call it scientific notation. This lecture shows exactly how it works, and why it is built the way it is.

Powers of ten are place values

Start with something you already know. Each place in our number system is worth ten times the place to its right:

Power of ten	Value	Place
$1 0^{3}$	1000	thousands
$1 0^{2}$	100	hundreds
$1 0^{1}$	10	tens
$1 0^{0}$	1	ones
$1 0^{- 1}$	0.1	tenths
$1 0^{- 2}$	0.01	hundredths
$1 0^{- 3}$	0.001	thousandths

Read the table upwards, and each step multiplies by 10. Read it downwards, and each step divides by 10. Keep dividing past $1 0^{1} = 10$ , and the pattern gives $1 0^{0} = 1$ , then $1 0^{- 1} = 0.1$ , then $1 0^{- 2} = 0.01$ . A negative power of ten is not a negative number. It is a small positive number: $1 0^{- 3}$ means one thousandth. (A later lecture on indices explains negative powers fully. Here, the place-value pattern is all you need.)

So multiplying by $1 0^{5}$ means multiplying by 10 five times. Every digit moves five places to the left, into places worth 100 000 times more. Multiplying by $1 0^{- 4}$ means dividing by 10 four times. Every digit moves four places to the right.

What standard form is

Standard form

A number is in standard form when it is written as $a \times 1 0^{n}$ , where $1 \leq a < 10$ and $n$ is an integer.

An integer is a whole number that may be positive, negative or zero: …, −2, −1, 0, 1, 2, …. The condition $1 \leq a < 10$ means that $a$ has exactly one non-zero digit before the decimal point. So 4.5, 1.05 and 9.99 are allowed, but 12.4 and 0.3 are not.

The two parts do two different jobs. The power $n$ tells you the size: which place the first digit sits in. The number $a$ tells you the digits. For the Earth, $n = 24$ : the first digit, 5, sits in the $1 0^{24}$ place. For the proton, $n = - 27$ : its first digit, 1, sits in the $1 0^{- 27}$ place. One glance at the powers tells you that the Earth's mass is about 51 powers of ten bigger than the proton's.

Masses on a scale of powers of ten. On this scale, a proton, a bag of rice and the Earth all fit on one line. The power of ten gives each one its position.

Converting a large number

Take 450 000. The first non-zero digit is the 4, and it is in the hundred-thousands place, which is $1 0^{5}$ . So the power is 5. Then write the digits with the point after the first one: 4.5.

450000 = 4.5 \times 1 0^{5}

Check: $4.5 \times 100000 = 450000$ . Correct.

You may have learned a shortcut: "move the decimal point left until one digit is in front, and count the moves." It gives the same answer, because each move is one place of ten. But the place-value view tells you why the power is 5, and it lets you check your answer by sense: a number in the hundred thousands must have a power of 5.

Converting a small number

Take 0.000 37. The first non-zero digit is the 3, and it is in the ten-thousandths place, which is $1 0^{- 4}$ . So the power is −4. The digits are 3.7.

0.000 37 = 3.7 \times 1 0^{- 4}

Check: $3.7 \times 0.0001 = 0.000 37$ . Correct.

Look carefully at how many zeros there are. 0.000 37 has three zeros after the point, but the power is −4, not −3. The 3 is in the fourth place after the point. Counting zeros gives an answer that is wrong by one. Count places instead.

Explore the link between the power and the place below.

Explore — the power is the place

Choose a number a. Then change the power n. Watch which place the first digit moves into, and the ordinary number it makes.

Power: n = 5

The digits of 4.5 at different powers of ten. Each step up in the power moves both digits one place to the left.

Standard form	Ordinary number	Place of the 4
4.5 × 10⁵	450 000	hundred thousands
4.5 × 10²	450	hundreds
4.5 × 10⁰	4.5	ones
4.5 × 10⁻¹	0.45	tenths
4.5 × 10⁻³	0.0045	thousandths

Did you see it? The power is always the place of the first digit. A positive power puts it to the left of the ones place. A power of 0 leaves it in the ones place. A negative power puts it to the right of the decimal point, and the number is small but still positive.

Stop and think

Is $2.5 \times 1 0^{- 3}$ a negative number?

Show the answer

No. The negative power makes the number small, not negative. $2.5 \times 1 0^{- 3} = 0.0025$ , which is positive. A negative number has a minus sign in front of the whole thing: $- 2.5 \times 1 0^{3} = - 2500$ . The sign in front tells you whether the number is positive or negative. The sign in the power tells you whether it is bigger or smaller than 1.

Why the rule says 1 ≤ a < 10

Without the rule, every number could be written in many ways. All of these equal 4500:

45 \times 1 0^{2} = 4.5 \times 1 0^{3} = 0.45 \times 1 0^{4}

The rule picks exactly one of them: $4.5 \times 1 0^{3}$ . This matters for two reasons. First, everyone writes the same number the same way, so it is easy to read. Second, and more useful, the power alone now tells you the size. If two numbers are both in standard form, the one with the larger power is the larger number. You cannot say that about $45 \times 1 0^{2}$ and $4.5 \times 1 0^{3}$ .

This is the same idea as writing a surd in its simplest form, or a fraction in its lowest terms. Mathematics often agrees on one tidy form, so that equal things look equal.

Test yourself below. Some of these are in standard form and some are not.

Sorter — standard form or not?

Choose a card, then choose the box. Each card explains itself.

Each number, whether it is in standard form, and why:

Number	Standard form?	Reason
4.6 × 10⁸	yes	1 ≤ 4.6 < 10
46 × 10⁷	no	46 is too big: 4.6 × 10⁸
0.3 × 10⁵	no	0.3 is too small: 3 × 10⁴
7 × 10⁻²	yes	a whole number from 1 to 9 is allowed
10 × 10³	no	a must be less than 10: 1 × 10⁴
5.2 × 10⁰	yes	a power of 0 is allowed: it is 5.2
3.5 × 100	no	the 100 must be written as a power: 3.5 × 10²
9.99 × 10⁻¹²	yes	1 ≤ 9.99 < 10, and −12 is an integer

Worked examples

Worked Example 1 — The distance to the Sun

The average distance from the Earth to the Sun is about 150 000 000 km. Write this in standard form.

The first digit, 1, is in the hundred-millions place. That is $1 0^{8}$ , because 100 000 000 has eight zeros. The digits are 1.5.

150000000 = 1.5 \times 1 0^{8} km

Check: $1.5 \times 100000000 = 150000000$ . (NASA gives the Earth's average distance as $149.598 \times 1 0^{6}$ km, which rounds to $1.50 \times 1 0^{8}$ km.)

Worked Example 2 — A small number

Write 0.000 004 5 in standard form.

Count the places after the decimal point up to the first non-zero digit: 0, 0, 0, 0, 0, then 4. The 4 is in the sixth place, so the power is −6.

0.000 0045 = 4.5 \times 1 0^{- 6}

Check: $1 0^{- 6}$ is one millionth, and 4.5 millionths is 0.000 004 5.

Worked Example 3 — Nearly right

A student writes $12400 = 12.4 \times 1 0^{3}$ . The value is correct. Is it in standard form?

No, because 12.4 is not less than 10. Divide 12.4 by 10 to get 1.24, and multiply the power of ten by 10 to keep the value the same: $1 0^{3} \times 10 = 1 0^{4}$ .

12400 = 1.24 \times 1 0^{4}

A check by place value agrees: the first digit, 1, is in the ten-thousands place, $1 0^{4}$ .

Worked Example 4 — Back to ordinary numbers

Write $3.05 \times 1 0^{6}$ , $8.1 \times 1 0^{- 3}$ and $1.05 \times 1 0^{- 3}$ as ordinary numbers.

$3.05 \times 1 0^{6}$ : the 3 goes into the millions place. So the number is 3 050 000.

$8.1 \times 1 0^{- 3}$ : the 8 goes into the thousandths place, the third place after the point. So the number is 0.0081.

$1.05 \times 1 0^{- 3}$ : the 1 goes into the thousandths place. So the number is 0.001 05. Notice the zero between the 1 and the 5. It was there in 1.05, and it stays. This is the easiest zero to lose. Write every digit of $a$ after the first one, in order.

Worked Example 5 — Which is largest?

Put these in order from smallest to largest: $3.2 \times 1 0^{5}$ , $9.8 \times 1 0^{4}$ , $1.1 \times 1 0^{5}$ .

Compare the powers first. $9.8 \times 1 0^{4}$ has the smallest power, so it is the smallest, even though 9.8 is the biggest value of $a$ . The other two both have power 5, so compare their values of $a$ : 1.1 is less than 3.2.

The order is $9.8 \times 1 0^{4}$ , $1.1 \times 1 0^{5}$ , $3.2 \times 1 0^{5}$ . As ordinary numbers: 98 000, 110 000, 320 000.

Three classic mistakes

Trap 1 — Leaving a outside 1 to 10

$12.4 \times 1 0^{3}$ and $0.3 \times 1 0^{5}$ have the right values but are not in standard form. Check $a$ at the end of every answer. If it is too big or too small, move one place and change the power by one.

Trap 2 — Counting zeros instead of places

0.0003 has three zeros after the point, but it is $3 \times 1 0^{- 4}$ , because the 3 is in the fourth place. For large numbers the same mistake is rarer, but it happens: 150 000 000 has seven zeros, and its power is 8.

Trap 3 — Misreading a calculator

Many calculators and spreadsheets show $4.5 \times 1 0^{5}$ as 4.5E5 or 4.5E+05. The E means "times ten to the power". It does not mean $4. 5^{5}$ . When you copy an answer from a screen, write it in proper standard form.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

Write in standard form: (a) 7 200 000 (b) 0.000 058 (c) 640.

Hint 1
For each number, find the place of the first non-zero digit.

Hint 2
In 7 200 000, the 7 is in the millions place, $1 0^{6}$ . In 0.000 058, the 5 is in the fifth place after the point.

Full answer
(a) $7.2 \times 1 0^{6}$ . (b) $5.8 \times 1 0^{- 5}$ . (c) $6.4 \times 1 0^{2}$ . Check (b): $5.8 \times 0.000 01 = 0.000 058$ .
Write as ordinary numbers: (a) $6.02 \times 1 0^{5}$ (b) $9 \times 1 0^{- 4}$ (c) $1.005 \times 1 0^{3}$ .

Hint 1
The power tells you the place of the first digit.

Hint 2
In (c), keep both zeros of 1.005 between the 1 and the 5.

Full answer
(a) 602 000. (b) 0.0009. (c) 1005. Check (c): the 1 goes into the thousands place, and the digits 1, 0, 0, 5 follow in order.
Put these in order from smallest to largest: $4.1 \times 1 0^{- 3}$ , $3.9 \times 1 0^{- 2}$ , $5 \times 1 0^{- 3}$ , $2.2 \times 1 0^{- 4}$ .

Hint 1
Compare the powers first. With negative powers, −4 is smaller than −3.

Hint 2
Two numbers have the power −3. Compare their values of $a$ .

Full answer
$2.2 \times 1 0^{- 4}$ , $4.1 \times 1 0^{- 3}$ , $5 \times 1 0^{- 3}$ , $3.9 \times 1 0^{- 2}$ . As ordinary numbers: 0.000 22, 0.0041, 0.005, 0.039.
The Earth's mass is about $5.97 \times 1 0^{24}$ kg. Written as an ordinary whole number, how many digits does it have? A proton's mass is about $1.67 \times 1 0^{- 27}$ kg. Written as a decimal, how many zeros come between the decimal point and the 1?

Hint 1
Try smaller cases first. How many digits does $5.97 \times 1 0^{2} = 597$ have? How many zeros come after the point in $1.67 \times 1 0^{- 2} = 0.0167$ ?

Hint 2
A number with first digit in the $1 0^{n}$ place has $n + 1$ digits. A first digit in the $1 0^{- n}$ place has $n - 1$ zeros before it after the point.

Full answer
The Earth's first digit is in the $1 0^{24}$ place, and there are places $1 0^{24}, 1 0^{23}, \dots, 1 0^{0}$ below and including it: 25 places. So the number has 25 digits. The proton's 1 is in the 27th place after the point, so 26 zeros come before it. Check with a small case: $1 0^{2}$ gives 3 digits (597), and $1 0^{- 2}$ gives 1 zero (0.0167). This is exactly why standard form exists: counting 25 digits or 26 zeros by eye is slow and easy to get wrong.

The bigger idea

Standard form separates two questions that ordinary numbers mix together: how big is it? and what are its digits? The power answers the first, and $a$ answers the second. Scientists usually care about the first question before the second, so they put it where it can be seen at once.

This is why standard form is the language of science, from the size of a virus to the distance to a star. In the next lecture, it also makes calculations with such numbers surprisingly simple, because powers of ten follow the index laws.

Key concepts in this lesson

A number is in standard form when it is written as a × 10ⁿ, where a is at least 1 but less than 10, and n is an integer.
The power n is the place value of the first non-zero digit, so it tells you the size of the number at a glance.
A negative power makes a number small, not negative, because 10⁻ⁿ means one divided by 10ⁿ.
The condition on a gives every number exactly one standard form, so numbers in standard form can be compared by their powers first.
When converting, count places rather than zeros, and keep every digit of a, including zeros in the middle.