Standard form.
Keep the scale clear, whether a number is microscopic or astronomical.
Use A × 10ⁿ with 1 ≤ A < 10 · convert both ways · calculate and normalise
A compact way to write very big or tiny
Writing every zero in a very large or very small number can make it difficult to read. Standard form separates the important digits from the size of the number using a power of ten.
The first factor must be at least 1 but smaller than 10. The second factor tells us how that first number was scaled. A positive index makes it larger; a negative index makes it smaller. Check both conditions, not just the presence of a power of ten.
Standard form writes a positive number as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
Large number → positive index
6.3 is multiplied by one million.
Small number → negative index
5.2 is divided by ten thousand.
Multiply by 1000: the decimal point moves 3 places right.
Pause and explainWhy is 42 × 10³ not in standard form?
Its first factor is 42, which is not smaller than 10. Rewrite it as 4.2 × 10⁴.
Convert in both directions
Multiplying by 10 moves every digit one place towards a higher place value. Dividing by 10 moves it towards a lower place value. The decimal-point shortcut records those place-value changes.
When writing a number in standard form, first choose a coefficient between 1 and 10. Then ask which power of ten restores the original size. For a tiny number, restoring that size requires division by a power of ten, so the index is negative.
Write 0.0000781 in standard form.
- Choose the coefficient between 1 and 10: 7.81.
- 7.81 must be divided by 100 000 to give the original number.
- Therefore 0.0000781 = 7.81 × 10⁻⁵.
Write 3.06 × 10⁴ and 8.4 × 10⁻³ as ordinary numbers.
3.06 × 10⁴ = 3.06 × 10 000 = 30 600.
8.4 × 10⁻³ = 8.4 ÷ 1000 = 0.0084.
Pause and explainShould 0.0048 have a positive or negative standard-form index?
Negative. Since 4.8 must be divided by 1000, the form is 4.8 × 10⁻³.
Calculate, then normalise
A number in standard form is still an ordinary product. For multiplication, handle the coefficients together and the powers of ten together. Then check whether the resulting coefficient satisfies the standard-form interval.
Addition is different: the powers of ten name different-sized units. Rename the terms using the same power before adding their coefficients, just as you need common units before adding lengths. Never add exponents merely because the expression contains a plus sign.
Multiply (6 × 10⁵)(4 × 10⁻³).
Multiply coefficients and add exponents.
24 is too large for the coefficient, so divide it by 10 and increase the exponent by 1.
Divide (3 × 10⁴) by (6 × 10⁷).
Divide coefficients and subtract exponents.
Multiply the coefficient by 10 and decrease the exponent by 1.
Add 3.2 × 10⁵ and 4.7 × 10⁴.
Make the powers of 10 match before adding.
For subtraction: 3.2 × 10⁵ − 4.7 × 10⁴ = (3.2 − 0.47) × 10⁵ = 2.73 × 10⁵.
Pause and explainWhy must 6 × 10⁴ be rewritten before adding it to 3 × 10⁵?
It counts ten-thousands, while the other term counts hundred-thousands. Rewrite it as 0.6 × 10⁵, giving 3.6 × 10⁵.
Your turn. Check the coefficient.
For each answer, make two checks: is its size reasonable, and is its coefficient between 1 and 10? An arithmetically correct product may still need rewriting to meet the standard-form convention.
When entering values on a calculator, use brackets or its exponent-entry key correctly. Keep the meaning of a negative index clear: it describes a small positive scale factor, not necessarily a negative quantity.
Write 72 500 000 and 0.0000096 in standard form.
Reveal worked answer
Calculate (8 × 10⁶)(5 × 10⁻⁴), giving the answer in standard form.
Reveal worked answer
Calculate (1.2 × 10⁻³) ÷ (4 × 10²).
Reveal worked answer
Find 6.5 × 10⁴ − 2.8 × 10³ in standard form.