Lesson 4.2 · 30 min

Standard Form

Writing very large and very small numbers as $A \times 1 0^{n}$ , converting back, and calculating with them.

In this lesson you will learn to…

Write a large number in standard form
Write a small number in standard form using a negative power of 10
Convert a number in standard form back to an ordinary number
Spot numbers that are not in standard form, and calculate with standard form

Why this matters

The mass of the Moon is about $73\,500\,000\,000\,000\,000\,000\,000$ kg. A proton’s mass is about $0.000\,000\,000\,000\,000\,000\,000\,001\,673$ g. Numbers like these are easy to miscopy and impossible to compare at a glance. Scientists write them as

$7.35 \times 10^{22} \text{ kg} \qquad \text{and} \qquad 1.673 \times 10^{-24} \text{ g}$

The power tells you the size

Number	Standard form
8000	$8 \times 10^3$
7 000 000	$7 \times 10^6$
420	$4.2 \times 10^2$
3.6	$3.6 \times 10^0$
0.003	$3 \times 10^{-3}$
0.000 05	$5 \times 10^{-5}$

For any number of 10 or more the power is positive. For numbers less than 1 it is negative. The bigger the power, the bigger the number — so to compare two numbers in standard form, compare the powers first.

Converting back

$3.81 \times 10^2$ means move the digits 2 places left: $381$ . $6 \times 10^{-5}$ means move the digits 5 places right: $0.000\,06$ .

On a calculator

Most calculators have a standard form key marked $\times 10^x$ , EXP or EE. To enter $2.3 \times 10^{-4}$ you type 2.3, press the key, then type -4. Calculators often show answers in standard form when they get very large or small — make sure you can read them.

Adding and subtracting. You cannot just add the front numbers when the powers are different. Convert to ordinary numbers first: $4.2 \times 10^{-3} - 0.001 = 0.0042 - 0.001 = 0.0032 = 3.2 \times 10^{-3}$ .

Key rules and formulas

The shape of standard form
$A \times 1 0^{n}, 1 \leq A < 10, n an integer$
The first number must be at least 1 and less than 10. The power of 10 says how many places the digits have moved.
Large numbers — positive power
$81432 = 8.1432 \times 1 0^{4}$
Put the decimal point after the first non-zero digit, then count how many places it has to move to get back to the original number.
Small numbers — negative power
$0.000 07845 = 7.845 \times 1 0^{- 5}$
For numbers less than 1 the power is negative. Count how many places the digits move to the left to make a number between 1 and 10.
Not standard form
$54.3 \times 1 0^{7}$ (first number too big), $0.03 \times 1 0^{8}$ (first number too small) and $7.1 \times 1 0^{3} \times 1 0^{2}$ (two powers) are not in standard form.

Worked examples

Worked example 1

Writing large numbers in standard form

Write a 81 432 b 91 285 000 in standard form.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Writing small numbers in standard form

Write a 0.0526 b 0.000 078 45 in standard form.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Fixing a number that is not quite in standard form

Emil writes

62 \times 100000

62 \times 1 0^{5}

. Write it correctly in standard form.

Try it yourself first, then reveal the solution one step at a time.

Worked example 4

Multiplying in standard form

Work out

(3 \times 1 0^{4}) \times (8 \times 1 0^{7})

, giving the answer in standard form.

Try it yourself first, then reveal the solution one step at a time.

Practice: standard form

11 questions

Eleven questions. Where there are two boxes, type the front number in the first and the power of 10 in the second.

Question 1 of 11Fill the gapsFoundation
Write each number in standard form.
$6300 =$ Gap a1 $\times 1 0^{□}$ Gap a2 $170000 =$ Gap b1 $\times 1 0^{□}$ Gap b2 $416 =$ Gap c1 $\times 1 0^{□}$ Gap c2
Question 2 of 11Fill the gaps
Write each small number in standard form.
$0.0074 =$ Gap a1 $\times 1 0^{□}$ Gap a2 $0.000 13 =$ Gap b1 $\times 1 0^{□}$ Gap b2 $0.023 56 =$ Gap c1 $\times 1 0^{□}$ Gap c2
Question 3 of 11Choose all that apply
Select every number that is correctly written in standard form.
Select every correct answer — there may be more than one.
$9.99 \times 1 0^{- 12}$ $0.03 \times 1 0^{8}$ $54.3 \times 1 0^{7}$ $7.1 \times 1 0^{3} \times 1 0^{2}$ $5.4 \times 1 0^{7}$ $1 \times 1 0^{- 6}$
Question 4 of 11Complete the table
Write each number in full.
Standard form Ordinary number
3 × 10⁷ Row 1, Ordinary number
3.81 × 10² Row 2, Ordinary number
1.628 × 10⁴ Row 3, Ordinary number
6 × 10⁻⁵ Row 4, Ordinary number
4 × 10⁻³ Row 5, Ordinary number
Question 5 of 11Multiple choice
Emil says: To write $62 \times 100000$ in standard form, you write $62 \times 1 0^{5}$ . What is the correct answer?
Question 6 of 11Put in order
Put these numbers in order, smallest first.
Put these in the right order. Drag them, or use the arrow buttons.
1. $7.2 \times 1 0^{4}$
2. $3.2 \times 1 0^{4}$
3. $8.6 \times 1 0^{3}$
4. $4.1 \times 1 0^{- 3}$
5. $3.2 \times 1 0^{- 2}$
Question 7 of 11Fill the gaps
A factory makes $270000000$ sweets every week. How many is that in a year (52 weeks)? Give your answer in standard form.
Gap a $\times 1 0^{□}$ Gap b
Question 8 of 11Fill the gaps
The diameter of the Milky Way is about $9.46 \times 1 0^{20}$ m. How many kilometres is this, in standard form?
Gap a $\times 1 0^{□}$ Gap b km
Question 9 of 11Step by step
Calculate in standard form one step at a time.
Work out $(8.2 \times 1 0^{3}) \div (2 \times 1 0^{- 6})$ .
1. Divide the front numbers: $8.2 \div 2$ .
  Step 1 answer
Question 10 of 11Fill the gapsChallenge
The mass of a red blood cell is about $9 \times 1 0^{- 14}$ kg. What is this in grams, in standard form?
Gap a $\times 1 0^{□}$ Gap b g
Question 11 of 11True or false
$32000 = 3.2 \times 1 0^{4}$

Standard form	Ordinary number
3 × 10⁷	Row 1, Ordinary number
3.81 × 10²	Row 2, Ordinary number
1.628 × 10⁴	Row 3, Ordinary number
6 × 10⁻⁵	Row 4, Ordinary number
4 × 10⁻³	Row 5, Ordinary number

In this lesson you will learn to…

Why this matters

The power tells you the size

Converting back

On a calculator

Key rules and formulas

The shape of standard form

Large numbers — positive power

Small numbers — negative power

Not standard form

Worked examples

Practice: standard form