Standard Form
Standard form is a compact way to write very large and very small numbers. It is useful for measurements such as tiny cell widths and very long distances.
Foundation & Higher — start with place value and powers of ten, then build towards conversions, comparisons and calculations. You do not need to know the notation before you begin.
Start with powers of ten
A power is a short way to write repeated multiplication. The small raised number is called the power or exponent.
103 means 10 × 10 × 10, which is 1000. It does not mean 10 × 3.
Look down the table. Each step divides the value by 10. The power goes down by 1.
| Power of ten | Value |
|---|---|
| 103 | 1000 |
| 102 | 100 |
| 101 | 10 |
| 100 | 1 |
| 10−1 | 0.1 — one tenth |
| 10−2 | 0.01 — one hundredth |
| 10−3 | 0.001 — one thousandth |
100 = 1 keeps this divide-by-10 pattern going. Going below zero gives tenths, hundredths and thousandths.
A negative power does not make the value negative. It tells us to divide 1 by a power of ten.
10−3 = 1 ÷ 1000 = 0.001
Multiplying by 103 makes a number 1000 times as large. Multiplying by 10−3 takes one thousandth of it: the same as dividing by 1000.
As in decimals, each neighbouring place is worth ten times as much as the place to its right.
Example Question:
Write 4500 in standard form.
Think of 4500 as 4.5 thousands. Example 1 shows how powers of ten let us write that idea briefly.
Exam Tip
Keep every significant digit, including zeros between other digits. Standard form changes how you write a number; it does not round it.
Check both parts of your final answer: the first number must be at least 1 and less than 10, and the power must give the correct size.
Method — Writing standard form
For the positive numbers in this lesson, standard form is a × 10n. The × sign means multiply.
a is the first number. It must be at least 1 and less than 10. We write this as 1 ≤ a < 10: ≤ means “less than or equal to”, and < means “less than”.
n is the power of ten. It must be an integer: a whole number, including zero or a negative whole number, such as 3, 0 or −2.
| Expression | Is it valid standard form? |
|---|---|
| 3.2 × 105 | Yes: 3.2 is between 1 and 10, and 5 is an integer. |
| 7 × 10−3 | Yes: 7 is between 1 and 10. A negative integer power is allowed. |
| 1.04 × 108 | Yes: the first number can have several decimal digits. |
| 32 × 104 | No: 32 is too large. The expression has a value, but its first number must be below 10. |
| 0.7 × 106 | No: 0.7 is too small. The first number must be at least 1. |
1. Choose the first number. Keep the digits in order and put the decimal point after the first non-zero digit.
2. Choose the power of ten. Ask what you must multiply that first number by to recover the original.
3. Check by converting back. Keep the value exactly the same.
Counting decimal-point places is a useful shortcut after you understand the place-value change. Use the check to decide the sign of the power.
Worked Examples
Example 1 — Write a familiar large number
Write 4500 in standard form.
4500 is 4.5 thousands:
4500 = 4.5 × 1000
4500 = 4.5 × 103
The first number is 4.5. Multiplying it by 1000 gives the original 4500.
Example 2 — Keep the zero placeholders
Write 405,000,000 in standard form.
The first number must be below 10, so use 4.05. Keep the zero between 4 and 5.
405,000,000 = 4.05 × 100,000,000
405,000,000 = 4.05 × 108
The power is 8 because 100,000,000 is eight factors of 10 multiplied together.
These smaller steps show the same place-value idea:
| Ordinary number | Standard form |
|---|---|
| 72,000 | 7.2 × 104 |
| 6,300,000 | 6.3 × 106 |
| 405,000,000 | 4.05 × 108 |
Example 3 — Write a small decimal
Write 0.004 in standard form.
0.004 is four thousandths:
0.004 = 4 × 0.001
0.004 = 4 × 10−3
The power is negative because we need a fraction of 1 to turn 4 into 0.004. The answer is still positive.
| Ordinary number | Same value in standard form |
|---|---|
| 0.004 | 4 × 10−3 |
| 0.00072 | 7.2 × 10−4 |
| 0.0000056 | 5.6 × 10−6 |
For 0.00072, start with 7.2 and divide by 10 four times:
7.2 ÷ 10 = 0.72
0.72 ÷ 10 = 0.072
0.072 ÷ 10 = 0.0072
0.0072 ÷ 10 = 0.00072
So 7.2 × 10−4 really does recover the original decimal.
Example 4 — Convert back in both directions
Write 3.4 × 105 as an ordinary number.
105 = 100,000, so multiply by 100,000:
3.4 × 100,000 = 340,000
Now write 6.2 × 10−4 as an ordinary number.
10−4 = 1 ÷ 10,000, so divide by 10,000:
6.2 ÷ 10,000 = 0.00062
The 6 is in the ten-thousandths place. The three zeros after the decimal point keep the earlier places empty.
Do not count zeros alone to choose a power. Think about the value of the first non-zero digit.
Comparing positive numbers
First check that both numbers are in valid standard form. Then compare their powers.
For positive numbers in valid standard form, the larger power gives the larger number. If the powers match, compare the first numbers.
Example 5 — Compare and order
Which is larger: 4 × 106 or 9 × 105?
106 is ten times 105. Even though 4 is smaller than 9, the first number uses a larger power:
4 × 106 = 4,000,000
9 × 105 = 900,000
So 4 × 106 is larger.
If the power is the same, compare directly: 5.1 × 107 is larger than 3.2 × 107 because 5.1 is larger than 3.2.
For small numbers, 2 × 10−3 = 0.002 and 8 × 10−4 = 0.0008.
So, in increasing order: 8 × 10−4, 2 × 10−3. Increasing order means smallest first.
Calculating — multiply the two parts
Multiply the first numbers together, then multiply the powers of ten.
104 has four factors of 10 and 103 has three. Together there are seven factors:
104 × 103 = 107
That is why we add the powers when multiplying powers of ten. This rule also works with zero and negative integer powers.
Example 6 — Multiply in standard form
Work out (3 × 104) × (2 × 103).
First numbers:
3 × 2 = 6
Powers of ten:
104 × 103 = 107
Combine the parts: 6 × 107.
The first number is already at least 1 and less than 10.
Example 7 — Adjust a multiplication answer
Work out (6 × 104) × (4 × 102).
6 × 4 = 24
104 × 102 = 106
Together this gives 24 × 106. The value is correct, but 24 is too large for standard form.
Write 24 as 2.4 × 10. That extra factor of 10 joins the power-of-ten part:
24 × 106 = 2.4 × 10 × 106
2.4 × 10 × 106 = 2.4 × 107
The final answer is 2.4 × 107.
Calculating — divide the two parts
Divide the first numbers, then divide the powers of ten.
107 ÷ 103 divides seven factors of 10 by three factors of 10. Cancelling those three shared factors leaves four.
107 ÷ 103 = 104
So subtract the second power from the first when dividing powers of ten. This also works with negative integer powers.
Example 8 — Divide in standard form
Work out (8 × 107) ÷ (2 × 103).
8 ÷ 2 = 4
7 − 3 = 4
107 ÷ 103 = 104
The answer is 4 × 104, or 40,000.
Example 9 — Adjust a division answer
Work out (2 × 105) ÷ (5 × 102).
2 ÷ 5 = 0.4
5 − 2 = 3
The parts give 0.4 × 103. The first number is too small.
Making 0.4 ten times larger gives 4. To keep the value unchanged, make the power-of-ten part ten times smaller:
0.4 × 103 = 4 × 102
Both equal 400.
The final answer is 4 × 102.
Example 10 — Use a small measurement
In a model, each cell is 3 × 10−5 metres wide. There are 2 × 103 cells in a straight row, touching edge to edge. Find the row’s length in standard form.
The symbol m means metres. Multiply the width of one cell by the number of cells:
Length = (3 × 10−5) × (2 × 103) m
3 × 2 = 6
−5 + 3 = −2
10−5 × 103 = 10−2
The row is 6 × 10−2 m long.
Check using ordinary numbers: each cell is 0.00003 m wide and there are 2000 cells.
0.00003 × 2000 = 0.06
6 × 10−2 = 0.06, so the two methods agree.
Adding and subtracting in standard form
For addition and subtraction, first express both numbers using the same power of ten. Then combine the first numbers. Do not add the indices: that rule belongs to multiplication.
Worked example — add 2.6 × 10⁵ and 7 × 10⁴
7 × 104 = 0.7 × 105. This intermediate expression has the same value, even though it is not yet standard form.
2.6 × 105 + 0.7 × 105
= (2.6 + 0.7) × 105
= 3.3 × 105
Subtracting uses the same idea: (2.6 − 0.7) × 105 = 1.9 × 105. Check with ordinary numbers: 260,000 − 70,000 = 190,000.
Quick practice — calculate (a) 2.4 × 10⁻³ + 6 × 10⁻⁴, (b) 8 × 10⁶ − 3.5 × 10⁶. Give standard-form answers.
(a) 6 × 10−4 = 0.6 × 10−3. Add: (2.4 + 0.6) × 10−3 = 3 × 10−3.
(b) The powers already match. Subtract 3.5 from 8: 4.5 × 106.
Using a calculator
Calculators may call standard form scientific notation. The exponent-entry function may be labelled EXP, EE or ×10x. The label depends on the model.
To enter 3.2 × 105, enter 3.2, use the exponent-entry function, then enter 5. This function supplies the “× 10 to the power of” part.
Do not enter an extra × 10 before using that function. It is not the same as a general power key.
A display such as 3.2E5 means 3.2 × 105, which is 320,000.
A display such as 6.2E−4 means 6.2 × 10−4, which is 0.00062.
For a negative exponent, use your calculator’s negative-number entry method. Check its guide if the keys are unfamiliar.
When calculating, enter each complete standard-form number before the multiplication or division sign. Check the display and estimate the answer’s size before accepting it.
Check the size of your answer
First number: at least 1 and less than 10. The power must be an integer.
Size: a positive number of 10 or more needs a positive power. A number from 1 up to, but not including, 10 uses power 0. A positive number below 1 needs a negative power.
4.7 × 106 = 4,700,000: it should be in the millions.
4.7 × 10−6 = 0.0000047: it should be very small, but still positive.
Placeholders: convert back to check every zero. Keep the original units in measurement answers.
Think Like a Mathematician
Why do 45 × 106 and 4.5 × 107 have the same value?
45 ÷ 10 = 4.5
106 × 10 = 107
The first part becomes ten times smaller. The second part becomes ten times larger. These changes balance each other.
45 × 106 = 45,000,000
4.5 × 107 = 45,000,000
Only the second expression is valid standard form, because 4.5 is at least 1 and less than 10.
This is the same idea as changing between equivalent fractions and decimals: a new way of writing a number can keep exactly the same value.
Common mistakes to avoid
First number outside the range. 0.7 is too small and 32 is too large. Adjust both parts so the value stays the same.
Negative power means negative number. It does not: 10−3 is positive 0.001.
Counting zeros instead of place value. Write the ordinary number back out. Keep zeros between digits and before a small decimal’s first non-zero digit.
Multiplying the powers. For multiplication, add the powers; multiply only the first numbers.
Adding powers in division. Subtract the second power from the first.
Stopping at 24 × 106. Its value is correct, but the final standard-form answer is 2.4 × 107.
Entering × 10 twice. EXP or EE already supplies the power-of-ten part. Check what your calculator has actually entered.
Try it yourself
Begin with the conversions. Then try the calculations. Show your working and give final standard-form answers with a first number from 1 up to, but not including, 10.
1. Write the ordinary values of 102, 100 and 10−2.
102 = 10 × 10 = 100
Following the divide-by-10 pattern from 10 gives 1, then 0.1, then 0.01.
100 = 1
10−2 = 1 ÷ 100 = 0.01
2. Which are valid standard form: 5.6 × 103, 56 × 102, 0.56 × 104 and 1 × 10−2? Explain.
5.6 × 103 and 1 × 10−2 are valid. Each first number is at least 1 and less than 10, and each power is an integer.
56 × 102 is not valid standard form because 56 is at least 10.
0.56 × 104 is not valid standard form because 0.56 is below 1.
The first three expressions all equal 5600, but only the first of those three is in standard form.
3. Write 83,000 and 602,000,000 in standard form.
83,000 = 8.3 × 10,000
83,000 = 8.3 × 104
Keep the zero between 6 and 2 in the second number:
602,000,000 = 6.02 × 100,000,000
602,000,000 = 6.02 × 108
4. Write 0.007 and 0.000093 in standard form.
0.007 = 7 × 0.001
0.007 = 7 × 10−3
0.000093 = 9.3 ÷ 100,000
0.000093 = 9.3 × 10−5
Both powers are negative because we need small fractions of 1.
5. Write 2.6 × 106 and 8.1 × 10−5 as ordinary numbers.
106 = 1,000,000
2.6 × 1,000,000 = 2,600,000
10−5 = 1 ÷ 100,000
8.1 ÷ 100,000 = 0.000081
The answers are 2,600,000 and 0.000081. The 8 is in the hundred-thousandths place.
6. Put these in increasing order: 7 × 10−3, 2 × 10−2, 4 × 10−3.
All three are valid standard form. Compare the powers first: −3 is smaller than −2.
Among the two with power −3, 4 is smaller than 7.
Increasing order: 4 × 10−3, 7 × 10−3, 2 × 10−2.
Check: their ordinary values are 0.004, 0.007 and 0.02.
7. Work out (2 × 104) × (3 × 102). Give your answer in standard form.
2 × 3 = 6
4 + 2 = 6
104 × 102 = 106
Answer: 6 × 106.
8. Work out (7 × 103) × (4 × 105). Give your answer in standard form.
7 × 4 = 28
3 + 5 = 8
The initial result is 28 × 108.
Write 28 as 2.8 × 10. The extra factor of 10 increases the power by 1.
28 × 108 = 2.8 × 109
Answer: 2.8 × 109.
9. Work out (9 × 107) ÷ (3 × 103). Give your answer in standard form.
9 ÷ 3 = 3
7 − 3 = 4
107 ÷ 103 = 104
Answer: 3 × 104.
10. Work out (3 × 104) ÷ (6 × 102). Give your answer in standard form.
3 ÷ 6 = 0.5
4 − 2 = 2
The initial result is 0.5 × 102.
Making 0.5 ten times larger gives 5, so make the power-of-ten part ten times smaller.
0.5 × 102 = 5 × 101
Answer: 5 × 101. Both forms equal 50.
11. Rewrite 0.42 × 106 in valid standard form without changing its value.
The first number 0.42 is below 1.
0.42 × 10 = 4.2
106 ÷ 10 = 105
So 0.42 × 106 = 4.2 × 105.
Check: both equal 420,000.
12. Each sheet of a material is 2 × 10−4 m thick. A stack contains 3 × 103 sheets with no gaps. Find the stack’s thickness in standard form.
Multiply the thickness of one sheet by the number of sheets.
2 × 3 = 6
−4 + 3 = −1
10−4 × 103 = 10−1
The stack is 6 × 10−1 m thick.
Check using ordinary numbers:
0.0002 × 3000 = 0.6
6 × 10−1 = 0.6, so the check agrees.
Frequently Asked Questions
Why is standard form useful?
It writes very large or very small numbers compactly, while showing their size through a power of ten. It also makes some calculations easier.
Can the first number equal 1 or 10?
1 is allowed, but 10 is not. For example, 10 × 103 must be rewritten as 1 × 104.
Why does a small positive decimal use a negative power?
Its first number is at least 1. Multiplying by a fraction of 1 makes it small enough. A negative power of ten supplies that fraction.
Does standard form round the number?
No. Keep all the digits needed to preserve the original value, unless the question separately asks you to round.
What if the original number is between 1 and 10?
Use power 0. For example, 6.4 = 6.4 × 100, because 100 = 1.
Why must I adjust some calculation answers?
The calculation may give the right value with a first number outside the allowed range. Adjust the first number and the power together so the value stays unchanged.
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