Chapter 12 — Miscellaneous Topics
Number Types and Number Sets
Every number you work with in MYP mathematics belongs to one or more of the standard number sets. Understanding where a number fits helps you answer classification questions on the exam.
The Number Sets
| Symbol | Name | Description | Examples |
|---|---|---|---|
| N | Natural numbers | The counting numbers | 1, 2, 3, 4, 5, ... |
| Z | Integers | Natural numbers, their negatives, and zero | $..., -3, -2, -1, 0, 1, 2, 3, ...$ |
| Q | Rational numbers | Expressible as a fraction of two integers | $7, -3, 0.5, 0.75, 0.333...$ |
| Irrational numbers | Numbers that cannot be written as a fraction of two integers | $\sqrt{2}, \sqrt{7}, \pi , e$ | |
| R | Real numbers | All rational and irrational numbers together | Every number on the number line |
Set Containment
Every natural number is an integer, every integer is rational, and every rational number is real:
N ⊂ Z ⊂ Q ⊂ R
Rational Numbers — Key Facts
A number is rational if it can be expressed as a fraction of two integers (with denominator not equal to zero).
This includes:
- All integers, because every integer can be written as a fraction with denominator 1
- All fractions of integers
- All terminating decimals, because they can be converted to fractions
- All recurring (repeating) decimals, because they can also be converted to fractions
Examples of rational numbers:
$$7 = \frac{7}{1} \qquad \text{(integer written as fraction)}$$
$$-3 = \frac{-3}{1} \qquad \text{(negative integer)}$$
$$\frac{3}{4} = 0.75 \qquad (fraction = \text{terminating decimal })$$
$$\frac{1}{3} = 0.333... \qquad (fraction = \text{recurring decimal })$$
$$\frac{22}{7} \approx 3.142857142857... \qquad (fraction = \text{recurring decimal })$$
Irrational Numbers — Key Facts
Numbers that cannot be written in rational form are called irrational numbers.
- Surds such as √2 and √7 are irrational
- $\pi (\text{which is } 3.14159265...) \text{is irrational }$
- e (which is 2.71828...) is irrational
- Non-recurring, non-terminating decimals such as 0.12233344445... are irrational
The decimal expansion of an irrational number goes on forever without repeating.
How to Classify a Number
| Ask yourself | If YES → | If NO → |
|---|---|---|
| Is it a counting number (1, 2, 3, ...)? | N (and also Z, Q, R) | Keep checking |
| Is it a whole number or its negative (including zero)? | Z (and also Q, R) | Keep checking |
| Can it be written as a fraction of two integers? | Q (and also R) | It is irrational (still R) |
Example 1: Classify each number. State ALL sets it belongs to.
| Number | Sets | Reason |
|---|---|---|
| 5 | N, Z, Q, R | Natural number |
| $-3$ | Z, Q, R | Integer, not natural |
| $\displaystyle \frac{2}{3}$ | Q, R | Rational, not an integer |
| $\sqrt{9} = 3$ | N, Z, Q, R | Simplifies to a natural number |
| $\sqrt{5}$ | R only | Irrational (5 is not a perfect square) |
| 0 | Z, Q, R | Integer; by convention 0 is not in N |
| $\pi $ | R only | Irrational |
| $-0.75$ | Q, R | Rational (equals −3 over 4), not an integer |
Part 1: Radicals (Surds)
1.1 What Is a Radical?
A radical is a number written using the radical sign √. The expression √a is read as "the square root of a."
Some radicals simplify to rational numbers:
- $\sqrt{4} = 2, \sqrt{9} = 3, \sqrt{16} = 4, \sqrt{25} = 5$
These are rational because the number under the radical sign is a perfect square.
Other radicals do NOT simplify to a nice fraction or terminating decimal:
- $\sqrt{2} = 1.41421356..., \sqrt{3} = 1.73205080..., \sqrt{5} = 2.23606797...$
These are irrational — their decimal expansions go on forever without repeating. Irrational radicals are also called surds.
| Type | Examples | Why? |
|---|---|---|
| Rational radical | $\sqrt{4}, \sqrt{9}, \sqrt{25}, \sqrt{1/4}$ | Perfect squares — simplify to a fraction |
| Irrational radical (surd) | $\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}$ | Not perfect squares — infinite non-repeating decimal |
Пояснение: Что такое радикал
Радикал (корень) — это способ записи числа через знак √. Главное: если под корнем полный квадрат (4, 9, 25...), результат рациональный; если нет (2, 3, 7...) — иррациональный (сурд). На экзамене часто просят определить, рациональное число или нет — именно этот признак и нужен.
1.2 Square Roots — Definition
The square root of a (written √a) is the positive number that, when squared, gives a.
$$\sqrt{a} \times \sqrt{a} = a \qquad \text{(where a is non-negative)}$$
Key points:
- For √a to exist (in real numbers), $\text{we need } a \geq 0.$
- The square root is always non-negative: √9 = 3, not ±3.
❌ Common mistake: Writing √9 = ±3. The symbol √ always means the positive root. When solving x² = 9, we get x = ±3 — but that is because there are two solutions to the equation, not because √9 has two values.
✅ Correct: √9 = 3 (always positive). If x² = 9, then x = ±√9 = ±3.
Пояснение: Квадратный корень — определение
Квадратный корень √a — это ВСЕГДА положительное число. Частая ошибка на экзамене: написать √9 = ±3. Нет! √9 = 3. Знак ± появляется только когда решаешь уравнение x² = 9, потому что у уравнения два решения. Также помни: √a существует только при a ≥ 0 (в действительных числах).
Example 1: Evaluate each expression.
$$(a) \sqrt{49} = 7$$
$$(b) \sqrt{1/4} = \frac{1}{2}$$
$$(c) -\sqrt{25} = -5 \qquad \text{(the minus is outside the radical)}$$
Example 2: State whether each statement is true or false, and explain why.
(a) Statement: √0 = 0. This is True because 0 × 0 = 0.
(b) Statement: √(−4) = −2. This is False because the square root of a negative number is not real.
(c) Statement: (√7)² = 7. This is True by the definition of square root.
1.3 Radicals on a Number Line
We can locate surds on a number line using Pythagoras' theorem. The idea: if we construct a right-angled triangle whose hypotenuse has length √n, we can transfer that length onto the number line.
The method:
If n = a² + b², then a right-angled triangle with legs a and b has hypotenuse √n (by Pythagoras).
| Surd | Decomposition | Triangle legs |
|---|---|---|
| $\sqrt{2}$ | $1^{2} + 1^{2} = 2$ | 1 and 1 |
| $\sqrt{5}$ | $1^{2} + 2^{2} = 5$ | 1 and 2 |
| $\sqrt{8}$ | $2^{2} + 2^{2} = 8$ | 2 and 2 |
| $\sqrt{10}$ | $1^{2} + 3^{2} = 10$ | 1 and 3 |
| $\sqrt{13}$ | $2^{2} + 3^{2} = 13$ | 2 and 3 |
$\text{Construction steps } (for \sqrt{2}):$
- Draw a number line and mark 0 and 1.
- At the point 1, draw a perpendicular of length 1 upwards.
- Connect the origin (0) to the top of the perpendicular — this line has length √(1² + 1²) = √2.
- Using a compass centred at 0 with radius √2, draw an arc that meets the number line. This point is √2.
Example 3: How would you construct √5 on a number line?
We need two numbers whose squares add to 5: $1^{2} + 2^{2} = 5.$
- From the origin, mark the point 1 on the number line.
- At the point 1, draw a perpendicular of length 2 upwards.
- The line from the origin to the top of the perpendicular has length √(1² + 2²) = √5.
- Use a compass to transfer this length onto the number line.
Example 4: Between which two consecutive integers does √7 lie?
Since 2² = 4 and 3² = 9, and 4 < 7 < 9, we have 2 < √7 < 3.
More precisely: √7 ≈ 2.646, so √7 is between 2 and 3, closer to 3.
Example 5: Between which two consecutive integers does √40 lie?
Since 6² = 36 and 7² = 49, and 36 < 40 < 49, we have 6 < √40 < 7.
Пояснение: Радикалы на числовой прямой
Этот раздел показывает, что иррациональные числа — настоящие точки на числовой прямой, а не абстракция. Метод с теоремой Пифагора (построить прямоугольный треугольник с нужной гипотенузой) иногда встречается на экзамене как задача на построение. Практический навык: уметь быстро определить, между какими целыми числами лежит √n — сравни n с ближайшими полными квадратами.
1.4 Operations with Radicals — Adding and Subtracting
We add and subtract like radicals the same way we combine like terms in algebra.
In algebra: 3x + 2x = 5x (same variable → combine coefficients).
With radicals: 3√2 + 2√2 = 5√2 (same radicand → combine coefficients).
The rule:
$$a\sqrt{n} + b\sqrt{n} = (a + b)\sqrt{n}$$
$$a\sqrt{n} - b\sqrt{n} = (a - b)\sqrt{n}$$
You can only combine radicals with the same number under the radical sign (same radicand).
❌ Common mistake: √(a + b) = √a + √b — this is WRONG! You cannot split a radical over addition or subtraction.
✅ Check: √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. Since 5 ≠ 7, the split does not work.
Example 6: Simplify each expression.
$$(a) 3\sqrt{5} + 4\sqrt{5} = 7\sqrt{5}$$
$$(b) 5\sqrt{3} - \sqrt{3} = 4\sqrt{3}$$
$$(c) 2\sqrt{7} + 3\sqrt{7} - \sqrt{7} = 4\sqrt{7}$$
Example 7: Simplify $6\sqrt{2} - 3\sqrt{3} + 2\sqrt{2} + 5\sqrt{3}$
Group like radicals:
$$= (6\sqrt{2} + 2\sqrt{2}) + (-3\sqrt{3} + 5\sqrt{3})$$
$$= 8\sqrt{2} + 2\sqrt{3}$$
This cannot be simplified further because √2 and √3 are unlike radicals.
Example 8: Simplify $3\sqrt{5} - 2\sqrt{5} + 4\sqrt{2} - \sqrt{2}$
$$= (3\sqrt{5} - 2\sqrt{5}) + (4\sqrt{2} - \sqrt{2})$$
$$= \sqrt{5} + 3\sqrt{2}$$
Example 9: Show that $\sqrt{7 - 2} \neq \sqrt{7} - \sqrt{2}$
$$\sqrt{7 - 2} = \sqrt{5} \approx 2.236$$
$$\sqrt{7} - \sqrt{2} \approx 2.646 - 1.414 = 1.232$$
Since 2.236 ≠ 1.232, we confirm that $\sqrt{7 - 2} \neq \sqrt{7} - \sqrt{2}$
This illustrates the critical rule: you cannot distribute the radical sign over addition or subtraction.
Пояснение: Сложение и вычитание радикалов
Работает точно как с переменными: 3√2 + 2√2 = 5√2, но √2 + √3 нельзя упростить (разные «подкоренные»). Главная ловушка на экзамене: √(a + b) НЕ равно √a + √b — это проверяют почти каждый год. Проверь себя: √(9+16) = √25 = 5, а √9 + √16 = 3 + 4 = 7. Не равны!