The Family of Numbers
Every number you'll use in this course belongs to a family, and each group is actually contained inside the next one:
Loading diagram…
Read this diagram as "is contained inside." Every natural number is also a whole number, every whole number is also an integer, and so on — but not the other way around.
A common exam trap: "ℕ ⊂ W" (every natural number is a whole number) is true — but "ℝ ⊂ W" (every real number is a whole number) is false, since ℝ is the biggest group here, not a small one contained in W.
Prime, Even, and Odd Numbers
- A prime number has exactly two factors: 1 and itself (e.g. 2, 3, 5, 7, 11, 13 — note that 1 itself is not prime).
- An odd number is one that cannot be divided by 2 without a remainder.
- An even number can be divided by 2 with no remainder.
The Division Algorithm
For any two whole numbers a and b (b ≠ 0), you can always write:
where q is the quotient and r is the remainder (with ).
Worked example: Divide 237 by 25. 25 × 9 = 225, and 237 − 225 = 12. So the quotient is 9 and the remainder is 12.
Greatest Common Divisor (GCD)
The GCD of two numbers is the largest number that divides both of them exactly, with no remainder. For 237 and 25 above, the GCD is 1 — meaning they share no common factor other than 1.
Properties of Real Numbers
The additive identity is the unique number that doesn't change a number when added to it — that's 0 (e.g. 5 + 0 = 5). By contrast, the multiplicative identity is 1 (e.g. 5 × 1 = 5).
Why This Matters for Your Exams
This topic is mostly about recognising which family a number belongs to, and correctly applying two simple procedures (division algorithm, GCD). The number-family diagram above answers most of the "true or false" style questions instantly.