Counting, then filling in the gaps
Mathematics did not arrive with the real numbers already in place. Each new set was invented because the old one could not answer a question that people insisted on asking.
You start with the natural numbers ℕ = {1, 2, 3, …}, which are enough for counting sheep. Then someone asks what is left when you take 3 sheep from 3 sheep, and you need zero, giving the whole numbers W = {0, 1, 2, 3, …}. Then someone owes money, and you need the integers ℤ = {…, −2, −1, 0, 1, 2, …}. Then three people share two loaves, and you need the rational numbers ℚ, every number that can be written as p/q with p and q integers and q ≠ 0.
At that point the system looks complete. It is not. The Pythagoreans discovered, to their considerable distress, that the diagonal of a square with side 1 cannot be written as any fraction at all. That number is √2, and numbers like it are called irrational. Put the rationals and the irrationals together and you get ℝ, the real numbers — every point on the number line, with nothing missing.
Switch to Locate √2 and watch the compass swing the diagonal of a unit square onto the line. That landing point is exactly √2 — irrational numbers sit at precise positions even though their decimals never end.
Telling a rational from an irrational
You do not need to guess. Write the number as a decimal and look at what the tail does. There are exactly three possibilities, and the first two are the same case in disguise.
| Decimal behaves like… | Example | Verdict |
|---|---|---|
| It stops (terminates) | 3/8 = 0.375 | Rational |
| It repeats for ever in a block | 2/3 = 0.666…, 1/7 = 0.142857… | Rational |
| It runs on and never repeats | √2 = 1.41421…, π = 3.14159… | Irrational |
Why a repeating decimal must be a fraction
Let x = 0.7777…. Then 10x = 7.7777…. Subtract the first from the second and every decimal place cancels: 9x = 7, so x = 7/9. The same trick converts any recurring decimal to a fraction, which is a complete proof that recurring decimals are rational.
The properties you are expected to name
The paper asks you to state which property has been used, so learn them by name rather than by feel. All of these hold for addition and for multiplication on ℝ.
| Property | Addition | Multiplication |
|---|---|---|
| Closure | a + b is real | a × b is real |
| Commutative | a + b = b + a | ab = ba |
| Associative | (a+b)+c = a+(b+c) | (ab)c = a(bc) |
| Identity | a + 0 = a | a × 1 = a |
| Inverse | a + (−a) = 0 | a × 1/a = 1 (a ≠ 0) |
| Distributive | a(b + c) = ab + ac | — |
The two that fail
Subtraction and division are not commutative and not associative: 7 − 3 ≠ 3 − 7 and (12 ÷ 6) ÷ 2 ≠ 12 ÷ (6 ÷ 2). And zero has no multiplicative inverse, which is the real reason division by zero is forbidden.
Radicals and surds
A radical is the n-th root sign; a surd is a radical whose value is irrational. So √9 is a radical but not a surd, because it equals 3. √5 is a surd.
Three rules do all the work. They follow directly from the laws of exponents, because a root is a fractional power.
Simplify √50 + √18 − √8.
- Split each radical so that one factor is a perfect square:
50 = 25 × 2,18 = 9 × 2,8 = 4 × 2.Pull out the largest square factor you can see; if you pull out a small one you will just have to do it again. √50 = 5√2,√18 = 3√2,√8 = 2√2.Using √(ab) = √a·√b with the square factor first.- Now every term carries the same surd, so they are like terms:
5√2 + 3√2 − 2√2.You can only add radicals that are identical after simplifying — √2 and √3 never combine. - Collect:
(5 + 3 − 2)√2 = 6√2.
6√2
Rationalising a denominator
A surd in the denominator is regarded as unfinished work, and the paper takes a mark for it. Getting rid of it is a one-line trick: multiply top and bottom by something that squares the surd away. Since you are multiplying by a fraction equal to 1, the value never changes.
For a single surd, multiply by that surd. For a two-term denominator such as 3 + √2, multiply by its conjugate 3 − √2, because (a+b)(a−b) = a² − b² removes the root.
Rationalise the denominator of 6 / (√5 − √3).
- The conjugate of
√5 − √3is√5 + √3. Multiply top and bottom by it.Changing the sign between the two terms is all "conjugate" means here. - Denominator:
(√5 − √3)(√5 + √3) = 5 − 3 = 2.Difference of two squares. The cross terms cancel, which is the whole point. - Numerator:
6(√5 + √3).Nothing to simplify yet — leave the bracket intact until the last step. - So the expression is
6(√5 + √3) / 2 = 3(√5 + √3).Cancel the common factor 2. Leaving the answer as 6(√5+√3)/2 loses a mark for not simplifying.
3(√5 + √3), or 3√5 + 3√3
Before you leave this chapter
- ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ. Every natural number is real; almost no real number is natural.
- Terminating or recurring decimal → rational. Non-terminating and non-recurring → irrational.
- A surd is an irrational radical. √16 is not a surd; √17 is.
- Simplify a radical by pulling out the largest perfect-square factor before you do anything else.
- Never leave a surd in a denominator. Multiply by the surd, or by the conjugate if the denominator has two terms.