Number

📌 Key points

1. Integers are whole numbers (including 0 and negatives); rational numbers can be written as ab with a and b integers; irrational numbers such as √2 and π cannot.
2. A prime number has exactly two factors, 1 and itself, so 1 is not prime. Any whole number can be written as a product of primes, e.g. 360 = 2³ × 3² × 5.
3. HCF: multiply the prime factors the numbers share, using the lowest powers. LCM: multiply every prime factor, using the highest powers.
4. Index laws: a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m−n, (a^m)ⁿ = a^mn, a⁰ = 1 and a⁻ⁿ = 1aⁿ.
5. A power of 12 means square root and 13 means cube root; for 27 to the power 23, take the cube root first (3), then square it (9).
6. Standard form is A × 10ⁿ where A is at least 1 but less than 10 and n is an integer, e.g. 0.00052 = 5.2 × 10⁻⁴.
7. The first significant figure is the first non-zero digit: 0.04072 = 0.0407 (3 s.f.); zeros after it still count, so 0.0060 has 2 s.f.
8. A length of 7.4 cm to 1 d.p. has lower bound 7.35 cm and upper bound 7.45 cm. For the largest a − b use upper a − lower b; for the largest a ÷ b use upper a ÷ lower b.
9. Surds: √a × √b = √(ab), so √12 = √4 × √3 = 2√3. Rationalise 1√2 by multiplying top and bottom by √2 to get √22.
10. A recurring decimal is rational: 0.272727… = 2799 = 311.

📖 Key terms

Prime number

A number with exactly two factors: 1 and itself.

Factor

A whole number that divides exactly into another number.

Multiple

The result of multiplying a number by a whole number.

Reciprocal

1 divided by the number; a number times its reciprocal is 1.

Standard form

A × 10ⁿ, where A is at least 1 and less than 10, and n is an integer.

Significant figures

The digits of a number counted from the first non-zero digit.

Upper and lower bounds

The largest and smallest values a rounded number could have been.

Irrational number

A number that cannot be written as a fraction of two integers, e.g. √2, π.