Standard form is a compact language for numbers that are awkward to write in full. It is especially useful when mathematics reaches astronomy, microscopic measurements, scientific data and repeated multiplication by powers of ten.
The structure is:
a × 10ⁿ, where 1 ≤ a < 10 and n is an integerThe first factor carries the significant digits. The power of ten records scale.
Large numbers
Write 4,730,000 in standard form. Move the decimal point until the first factor lies between 1 and 10:
4,730,000 = 4.73 × 10⁶The decimal moved six places to the left, so the exponent is +6.
Small numbers
Write 0.000082 in standard form:
0.000082 = 8.2 × 10⁻⁵The negative exponent records a scale smaller than 1.
Why the first factor must be between 1 and 10
For example, 47.3 × 10⁵ represents the same value as 4.73 × 10⁶, but it is not in standard form because 47.3 is not between 1 and 10.
The normalised first factor gives a consistent representation, making comparisons and calculations easier.
Multiplication
Multiply the coefficients and combine powers of ten using index laws.
(3 × 10⁴)(2 × 10⁵)
= 6 × 10⁹If the coefficient leaves the 1-to-10 interval, normalise it.
(6 × 10⁷)(4 × 10³)
= 24 × 10¹⁰
= 2.4 × 10¹¹Division
(8 × 10⁹) ÷ (2 × 10³)
= 4 × 10⁶The coefficients divide; the exponents subtract.
Addition and subtraction need matching powers
You cannot safely add coefficients until the powers of ten match.
3.2 × 10⁵ + 4.7 × 10⁴
= 3.2 × 10⁵ + 0.47 × 10⁵
= 3.67 × 10⁵This is place value in compressed form.
Comparing numbers
When coefficients are in standard form, compare exponents first. For positive numbers, the larger exponent usually indicates the larger number. If exponents match, compare coefficients.
Thus 8.1 × 10⁷ is larger than 9.9 × 10⁶ despite 9.9 being the larger coefficient.
Calculator notation
Calculators may display 3.6E8 or 3.6 EXP 8. This means 3.6 × 10⁸. The E or EXP notation is a compact calculator representation, not a different mathematical concept.
Order of magnitude
Standard form makes scale visible immediately. A bacterium measuring about 2 × 10⁻⁶ m and a road measuring 2 × 10³ m differ by nine powers of ten.
This ability to compare scale is one reason standard form is so useful in science and engineering.
Common errors
Wrong exponent sign. Numbers greater than 10 usually use positive exponents; numbers between 0 and 1 use negative exponents.
Coefficient outside the required range. 32 × 10⁴ must become 3.2 × 10⁵.
Adding exponents during addition. Exponent laws for multiplication do not transfer to addition.
Moving the decimal without preserving value. Every decimal movement must be balanced by the power of ten.
Diagnostic table
| Observed mistake | Likely issue | Repair |
|---|---|---|
| Writes 42 × 10⁵ as final answer | Normalisation missing | Force coefficient into 1 ≤ a < 10 |
| Uses + exponent for 0.0007 | Scale direction | Check whether number is above or below 1 |
| Adds exponents in a sum | Index law overgeneralised | Rewrite with common power first |
Practice
- Write 68,200,000 in standard form.
- Write 0.00000345 in standard form.
- Calculate (4 × 10⁶)(3 × 10⁴).
- Calculate (9 × 10⁸) ÷ (3 × 10²).
- Calculate 5.4 × 10⁵ + 7 × 10⁴.
Answers
1. 6.82 × 10⁷. 2. 3.45 × 10⁻⁶. 3. 1.2 × 10¹¹. 4. 3 × 10⁶. 5. 6.1 × 10⁵.
Connected routes
Continue to Indices: Why the Laws Work for the exponent structure beneath these calculations. Return to the Mathematics Learning Hub for the wider Secondary Mathematics route.