Standard Form& Calculations.

How do you convert numbers to standard form?

Standard form expresses any number in the format A × 10ⁿ where 1 ≤ A < 10 and n is an integer. To convert an ordinary number:

1. Step 1: Place the decimal point after the first non-zero digit to create A.
2. Step 2: Count how many places you moved the decimal point — this gives you n.
3. Step 3: If the original number was ≥ 10, n is positive. If it was between 0 and 1, n is negative.

Example — large number: 67,000,000 → move the decimal 7 places left → 6.7 × 10⁷.

Example — small number: 0.00042 → move the decimal 4 places right → 4.2 × 10⁻&sup4;.

How do you convert from standard form to ordinary numbers?

To convert back to an ordinary number, move the decimal point in A by n places:

• Positive n: Move the decimal n places to the right (the number gets bigger).
• Negative n: Move the decimal |n| places to the left (the number gets smaller).

Example: 3.06 × 10&sup5; → move 5 places right → 306,000.

Example: 8.1 × 10⁻³ → move 3 places left → 0.0081.

Fill in any gaps with zeros. A common exam question asks you to "write as an ordinary number" — this simply means convert out of standard form.

How do you calculate in standard form?

Multiplication

Multiply the A values together and add the powers of 10. For example: (3 × 10&sup4;) × (2 × 10³) = (3 × 2) × 10⁴⁺⁳ = 6 × 10⁷.

Division

Divide the A values and subtract the powers. For example: (8 × 10⁶) &div; (4 × 10²) = 2 × 10⁶⁻² = 2 × 10&sup4;.

Addition and subtraction

These are trickier because you must first match the powers of 10. Convert one number so both share the same exponent, then add or subtract the A values. Adjust the result back into proper standard form if necessary.

Important: After any calculation, always check that 1 ≤ A < 10. If A ≥ 10, divide A by 10 and increase n by 1. If A < 1, multiply A by 10 and decrease n by 1.

Worked Examples

Common Mistakes

Exam Tips