Standard Form
The mass of the Sun is 2,000,000,000,000,000,000,000,000,000,000 kg. Nobody wants to count those zeros. Standard form is the fix: one tidy number, times a power of ten that says how big.
National curriculum KS3: interpret and compare numbers in standard form A × 10ⁿ, 1 ≤ A < 10, n an integer. GCSE adds calculating with it.1 · The problem
Science is full of numbers that are either enormous or tiny. Distance to the nearest star: about 40,000,000,000,000 km. Width of a human hair: about 0.00007 m. Mass of an electron: 0.000 000 000 000 000 000 000 000 000 000 911 kg.
Written out, these are hard to read, hard to compare and very easy to get wrong by one zero, which is an error of a factor of ten. Standard form (in Spanish notación científica, and in the US "scientific notation") solves all three problems at once.
2 · Powers of ten are place value
You already use powers of ten every time you read a number. Each column is ten times the one to its right. The powers and roots page gave you 10⁰ = 1 and negative powers as fractions; here is where that pays off.
| Power | Value | Column name |
|---|---|---|
| 10⁶ | 1 000 000 | millions |
| 10³ | 1 000 | thousands |
| 10² | 100 | hundreds |
| 10¹ | 10 | tens |
| 10⁰ | 1 | units |
| 10⁻¹ | 0.1 | tenths |
| 10⁻² | 0.01 | hundredths |
| 10⁻³ | 0.001 | thousandths |
The index on the 10 counts how many places the digit 1 sits away from the units column: positive to the left, negative to the right. Multiplying by 10ⁿ moves every digit n places to the left (the number gets bigger); multiplying by 10⁻ⁿ moves them n places to the right.
3 · Watch the decimal point move
Start with the number 3.7 and drag the power. The digits never change. Only the decimal point moves, one place per power of ten. Zeros fill the empty columns.
4 · The rule: A × 10ⁿ with 1 ≤ A < 10
Two parts. A carries the digits: exactly one non-zero digit before the decimal point. n carries the size: how many places the point has moved. The restriction on A is what makes the form standard: every number has exactly one way of being written, so numbers can be compared at a glance.
Type any number below. If it's not in standard form yet, the converter shows why and fixes it.
Spotting fakes
These all equal the same number, but only one is in standard form:
45 × 10⁵ and 0.45 × 10⁷ are both correct values but wrong form, and in an exam "give your answer in standard form" loses the mark. The fix is always the same trade: move the point in A, and adjust n by the same amount the other way. Make A ten times smaller, make n one bigger.
5 · The scale of things
Drag along the exponent and see what lives at each power of ten. Every step is a factor of ten, so the top of this list is 10⁴² times the bottom: a 1 with 42 zeros.
6 · Calculating in standard form
This is where the form earns its keep. Because the index laws from the powers page apply to powers of ten, multiplying and dividing huge numbers becomes small-number arithmetic plus adding or subtracting indices.
| Operation | Method | Then |
|---|---|---|
| Multiply | Multiply the A's. Add the indices. | Check A is still between 1 and 10; if not, shift the point and adjust n. |
| Divide | Divide the A's. Subtract the indices. | |
| Add / subtract | Make the indices the same first (rewrite the smaller one), then add or subtract the A's. | Re-standardise if needed. |
Calculator tip. Use the ×10ˣ key (sometimes EXP or EE), not "× 10 ^". Typing 3 ×10ˣ 8 enters 3 × 10⁸ as one number. Typing 3 × 10 ^ 8 ÷ 4 × 10 ^ 5 will divide by 4 and then multiply by 10⁵, giving an answer 10¹⁰ times too big. Put brackets round each standard-form number if you must type it the long way.
7 · Where it shows up
Physics and chemistry
Speed of light 3 × 10⁸ m/s. Avogadro's number 6.02 × 10²³. Planck's constant 6.63 × 10⁻³⁴. Every constant in the data sheet is in standard form.
Astronomy
Earth to Sun 1.5 × 10¹¹ m. Light year 9.46 × 10¹⁵ m. Observable universe 8.8 × 10²⁶ m. Comparing them is just comparing the indices.
Biology and medicine
A red blood cell is 7 × 10⁻⁶ m across; a virus about 1 × 10⁻⁷ m; a DNA strand 2 × 10⁻⁹ m wide. Drug doses in micrograms are 10⁻⁶ g.
Computing
Floating-point numbers in every programming language are standard form in binary: a mantissa and an exponent. Python prints 3e8 for 3 × 10⁸, and 1e-9 for a nanosecond.
Economics
UK GDP ≈ 2.7 × 10¹² pounds. A national debt in the 10¹² range divided by a population of 6.8 × 10⁷ is a per-person figure found by subtracting indices.
Prefixes
kilo = 10³, mega = 10⁶, giga = 10⁹, tera = 10¹²; milli = 10⁻³, micro = 10⁻⁶, nano = 10⁻⁹. Unit prefixes are standard form with names.
Worked examples
Write 0.00056 in standard form
Calculate (6 × 10⁷) × (5 × 10⁻³)
Calculate (8 × 10⁴) + (3 × 10³)
Exam-style: Light travels 3 × 10⁸ m/s. The Sun is 1.5 × 10¹¹ m away. How long does sunlight take to reach us?
Quick check
What you've learned
Standard form writes any number as A × 10ⁿ with exactly one non-zero digit before the point in A. The index n counts how far the decimal point moved: positive for big numbers, negative for small. To convert, find the first non-zero digit, place the point after it, count the move. To multiply or divide, work the A's and add or subtract the indices, then check A is still between 1 and 10. It's how every scientist, engineer and programmer writes the very large and the very small.