Standard form

Standard form writes a number as A × 10ⁿ, where 1 ≤ A < 10 and n is a whole number. It's a short way to write very big and very small numbers.

1. Put the decimal point after the first non-zero digit to get A.
2. Count how many places the point moved: that is n.
3. n is positive for a big number (10 or more) and negative for a small one (less than 1).

Example: write 4 500 000 in standard form.

1. A = 4.5
2. The point moves 6 places.
3. 4.5 × 10⁶

Example: write 0.000 37 in standard form.

1. A = 3.7
2. The point moves 4 places, and the number is small.
3. 3.7 × 10⁻⁴

To change a number in standard form back to an ordinary number, move the decimal point n places: right if n is positive, left if n is negative. Fill gaps with zeros.

Example: write 2.06 × 10⁵ and 8 × 10⁻³ as ordinary numbers.

1. 2.06 × 10⁵: move the point 5 places right: 206 000
2. 8 × 10⁻³: move the point 3 places left: 0.008

To put numbers in standard form in order:

1. Compare the powers of 10 first: the bigger the power, the bigger the number.
2. If the powers are the same, compare the values of A.

Example: order 3.1 × 10⁴, 9.6 × 10³ and 2.5 × 10⁴, smallest first.

1. 9.6 × 10³ has the smallest power.
2. 2.5 × 10⁴ is less than 3.1 × 10⁴.
3. 9.6 × 10³, 2.5 × 10⁴, 3.1 × 10⁴

You can multiply and divide without a calculator by dealing with the numbers and the powers separately.

1. Multiply (or divide) the A numbers.
2. Add the powers to multiply; subtract them to divide.
3. If the new A isn't between 1 and 10, adjust it.

Example: work out (3 × 10⁴) × (6 × 10⁵).

1. 3 × 6 = 18
2. 10⁴ × 10⁵ = 10⁹
3. 18 × 10⁹ = 1.8 × 10¹ × 10⁹ = 1.8 × 10¹⁰

Example: work out (8.4 × 10⁶) ÷ (2.1 × 10⁻²).

1. 8.4 ÷ 2.1 = 4
2. Subtract the powers: 6 − (−2) = 8, so 10⁶ ÷ 10⁻² = 10⁸
3. 4 × 10⁸

To add or subtract, the powers must match, so the easiest way is to write the numbers out in full.

1. Change both numbers to ordinary numbers.
2. Add or subtract.
3. Change the answer back to standard form.

Example: work out 3.2 × 10⁴ + 5.6 × 10³.

1. 32 000 + 5600
2. = 37 600
3. 3.76 × 10⁴

On a calculator, use the × 10ˣ key (or EXP) to enter a power of 10, and brackets around each number.

Example: use a calculator to work out (6.4 × 10⁸) ÷ (1.6 × 10³).

1. Type ( 6.4 × 10ˣ 8 ) ÷ ( 1.6 × 10ˣ 3 ) =
2. The display shows 400 000
3. 4 × 10⁵