Scientific Notation (Standard Form)

Why we need scientific notation

Some numbers are gigantic. The Sun is about 149,600,000,000 metres from Earth. Others are minuscule: a hydrogen atom is roughly 0.0000000001 metres wide. Writing all those zeros is slow, error-prone, and hard to read. Scientific notation — also called standard form — is a compact way to write such numbers using powers of ten.

This lesson builds on exponents and powers, so if powers like 10³ are unfamiliar, read that first and come back.

The form: a × 10ⁿ

Every number in scientific notation looks like:

a × 10ⁿ

with two rules:

• a is a number from 1 up to (but not including) 10 — so it has exactly one non-zero digit before the decimal point.
• n is a whole number (positive, negative, or zero) telling you the power of ten.

For example, 149,600,000,000 becomes 1.496 × 10¹¹, and 0.0000000001 becomes 1 × 10⁻¹⁰. Much tidier.

Powers of ten: a quick reminder

Power	Value	In words
10³	1,000	one thousand
10²	100	one hundred
10¹	10	ten
10⁰	1	one
10⁻¹	0.1	one tenth
10⁻²	0.01	one hundredth
10⁻³	0.001	one thousandth

A positive power makes a number bigger; a negative power makes it smaller. The size of the power equals how many places the decimal point moves.

Converting a large number to standard form

The method: place the decimal point after the first non-zero digit, then count how many places the decimal moved.

Worked example: Write 5,000 in standard form.

1. Put the decimal after the first digit: 5.000, which is 5.
2. Count how many places the point moved from the end of 5,000: from 5000. to 5. is 3 places to the left.
3. Moving left means a positive power: 5 × 10³.

Worked example: Write 62,400 in standard form.

1. First non-zero digit is 6, so write 6.24 (drop trailing zeros).
2. The decimal moved from 62400. to 6.2400 — that is 4 places left.
3. Result: 6.24 × 10⁴.

Check: 6.24 × 10⁴ = 6.24 × 10,000 = 62,400. Correct.

Converting a small number to standard form

For small decimals, the decimal point moves to the right, giving a negative power.

Worked example: Write 0.0007 in standard form.

1. The first non-zero digit is 7, so the front number is 7.
2. Move the decimal point from 0.0007 to 7.0 — that is 4 places to the right.
3. Moving right gives a negative power: 7 × 10⁻⁴.

Worked example: Write 0.000056 in standard form.

1. First non-zero digit is 5; front number is 5.6.
2. Count places from 0.000056 to 5.6: that is 5 places right.
3. Result: 5.6 × 10⁻⁵.

Converting back to an ordinary number

Just reverse the process: the power tells you how far and which way to move the decimal point.

Worked example: Write 3.2 × 10⁵ as an ordinary number.

1. Positive power 5 → move the decimal 5 places right.
2. 3.2 → 320000 (adding zeros as needed).
3. Result: 320,000.

Worked example: Write 8 × 10⁻³ as an ordinary number.

1. Negative power 3 → move the decimal 3 places left.
2. 8 → 0.008.
3. Result: 0.008.

Spotting incorrect standard form

Remember the rule 1 ≤ a < 10. Watch for these mistakes:

• 12 × 10³ is wrong because 12 is not less than 10. Fix it: 12 × 10³ = 1.2 × 10⁴.
• 0.5 × 10⁶ is wrong because 0.5 is less than 1. Fix it: 5 × 10⁵.

Multiplying and dividing in standard form

This is where standard form shines. To multiply, multiply the front numbers and add the powers (this uses the law 10ᵐ × 10ⁿ = 10ᵐ⁺ⁿ). To divide, divide the front numbers and subtract the powers.

Worked example (multiply): (2 × 10³) × (4 × 10²).

1. Multiply the fronts: 2 × 4 = 8.
2. Add the powers: 3 + 2 = 5.
3. Result: 8 × 10⁵ (= 800,000).

Worked example (divide): (6 × 10⁸) ÷ (3 × 10²).

1. Divide the fronts: 6 ÷ 3 = 2.
2. Subtract the powers: 8 − 2 = 6.
3. Result: 2 × 10⁶.

Worked example (needs tidying): (5 × 10⁴) × (4 × 10³).

1. Multiply fronts: 5 × 4 = 20.
2. Add powers: 4 + 3 = 7, giving 20 × 10⁷.
3. But 20 is not less than 10, so rewrite 20 = 2 × 10¹: 20 × 10⁷ = 2 × 10¹ × 10⁷ = 2 × 10⁸.
4. Final answer: 2 × 10⁸.

That last step — fixing the front number back into the range 1 to 10 — is a very common requirement, so always check it.

Where it is used

Scientific notation is the everyday language of science and engineering. Astronomers use it for distances between stars; biologists for the size of cells and viruses; chemists for the number of atoms in a sample (around 6.02 × 10²³); and computer engineers for storage and processing speeds. Calculators display it too — a screen showing 4.5E9 means 4.5 × 10⁹. Knowing standard form lets you read and compute with the extreme numbers that describe the real universe.

Practice activity

1. Write 9,200 in standard form.
2. Write 0.00031 in standard form.
3. Convert 7.5 × 10⁴ to an ordinary number.
4. Convert 2 × 10⁻⁵ to an ordinary number.
5. Calculate (3 × 10⁵) × (2 × 10⁴).
6. Calculate (8 × 10⁹) ÷ (4 × 10³).

Answers:

1. 9.2 × 10³ (decimal moves 3 places left).
2. 3.1 × 10⁻⁴ (decimal moves 4 places right).
3. 75,000.
4. 0.00002.
5. Fronts: 3 × 2 = 6; powers: 5 + 4 = 9 → 6 × 10⁹.
6. Fronts: 8 ÷ 4 = 2; powers: 9 − 3 = 6 → 2 × 10⁶.

Summary

Scientific notation (standard form) writes a number as a × 10ⁿ where 1 ≤ a < 10. Large numbers get positive powers; small decimals get negative powers; the power equals how many places the decimal point moves. To multiply, multiply the fronts and add the powers; to divide, divide the fronts and subtract the powers — then tidy the front number back into the range 1 to 10. This compact form makes the enormous and the tiny numbers of science easy to write and calculate with.