Why This Matters for Grade 10#
Negative numbers and number classification appear in almost every Grade 10 topic:
- Exponents: $(-2)^2 = 4$ but $-2^2 = -4$ — the bracket trap costs marks every exam
- Equations: Solving $x^2 = 9$ gives $x = \pm 3$ — both positive AND negative
- Functions: The sign of $a$ determines whether a parabola opens up or down
- Analytical Geometry: Gradients can be positive, negative, zero, or undefined
- Surds (Grade 11): Knowing rational vs irrational is essential for surd simplification
1. The Real Number System#
Every number you encounter in Grade 10 is a real number ($\mathbb{R}$). Real numbers are classified into a hierarchy:
| Type | Symbol | Examples | Definition |
|---|---|---|---|
| Natural | $\mathbb{N}$ | $1, 2, 3, \ldots$ | Counting numbers (start at 1) |
| Whole | $\mathbb{N}_0$ | $0, 1, 2, 3, \ldots$ | Natural numbers + zero |
| Integer | $\mathbb{Z}$ | $\ldots, -2, -1, 0, 1, 2, \ldots$ | Whole numbers + negatives |
| Rational | $\mathbb{Q}$ | $\frac{1}{2},\; -3,\; 0.75,\; 0.\overline{3}$ | Can be written as $\frac{a}{b}$ where $a, b \in \mathbb{Z}$ and $b \neq 0$ |
| Irrational | $\mathbb{Q}'$ | $\sqrt{2},\; \pi,\; \sqrt{5}$ | Cannot be written as a fraction — infinite non-repeating decimal |
| Real | $\mathbb{R}$ | All of the above | Every number on the number line |
The Hierarchy#
$$\mathbb{N} \subset \mathbb{N}_0 \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$$Every natural number is also a whole number, which is also an integer, which is also rational, which is also real. Irrational numbers sit alongside rational numbers inside $\mathbb{R}$.
Worked Example 1 — Classifying Numbers#
Classify each number: $-7$, $\frac{3}{4}$, $\sqrt{16}$, $\sqrt{7}$, $0$, $\pi$, $0.\overline{142857}$
| Number | Simplified | Classification |
|---|---|---|
| $-7$ | $-7$ | Integer, Rational, Real |
| $\frac{3}{4}$ | $0.75$ | Rational, Real (not integer) |
| $\sqrt{16}$ | $4$ | Natural, Whole, Integer, Rational, Real |
| $\sqrt{7}$ | $2.6457\ldots$ (non-repeating) | Irrational, Real |
| $0$ | $0$ | Whole, Integer, Rational, Real |
| $\pi$ | $3.14159\ldots$ (non-repeating) | Irrational, Real |
| $0.\overline{142857}$ | $\frac{1}{7}$ | Rational, Real |
Key for exams: When a question says “solve for $x$, $x \in \mathbb{Z}$”, it means integer answers only. $x \in \mathbb{N}$ means positive integers only.
How to Tell Rational from Irrational#
A number is rational if its decimal either:
- Terminates: $0.75$, $3.125$
- Repeats: $0.\overline{3}$, $0.\overline{142857}$
A number is irrational if its decimal goes on forever without repeating: $\sqrt{2} = 1.41421356\ldots$
The square root test: $\sqrt{n}$ is rational only if $n$ is a perfect square ($1, 4, 9, 16, 25, \ldots$). Otherwise it’s irrational.
2. Operations with Negative Numbers#
Addition & Subtraction#
- $5 + (-3) = 5 - 3 = 2$ (adding a negative = subtracting)
- $-4 + (-2) = -6$ (adding two negatives → bigger negative)
- $-7 - (-3) = -7 + 3 = -4$ (subtracting a negative = adding)
- $-3 + 8 = 5$ (start at $-3$, move 8 to the right)
Memory aid: Two negatives next to each other (like $- (-3)$) become positive. Think: “the opposite of the opposite.”
Multiplication & Division Sign Rules#
| Signs | Result | Example |
|---|---|---|
| $(+) \times (+)$ | $+$ | $3 \times 4 = 12$ |
| $(-) \times (-)$ | $+$ | $(-3) \times (-4) = 12$ |
| $(+) \times (-)$ | $-$ | $3 \times (-4) = -12$ |
| $(-) \times (+)$ | $-$ | $(-3) \times 4 = -12$ |
Same signs → positive. Different signs → negative. The same rule applies to division.
Worked Example 2 — Mixed Operations#
$(-2)(3) + (-4)(-5) - (-6)$
$= -6 + 20 + 6$
$= 20$
Worked Example 3#
$\frac{(-3)(-8)}{(-2)(3)} = \frac{24}{-6} = -4$
Count the negatives: 2 negatives on top → positive. 1 negative on bottom → still one negative total → answer is negative.
3. Squares, Cubes & Negatives — THE Big Trap#
This is where most marks are lost in Grade 10 exponents. The bracket determines everything:
| Expression | Meaning | Value |
|---|---|---|
| $(-3)^2$ | $(-3) \times (-3)$ | $+9$ |
| $-3^2$ | $-(3 \times 3)$ | $-9$ |
| $(-2)^3$ | $(-2) \times (-2) \times (-2)$ | $-8$ |
| $-2^3$ | $-(2 \times 2 \times 2)$ | $-8$ |
| $(-1)^{100}$ | $(-1) \times (-1) \times \ldots$ (100 times) | $+1$ |
| $(-1)^{99}$ | $(-1) \times (-1) \times \ldots$ (99 times) | $-1$ |
The Rule#
Brackets around the negative mean the negative is PART of the base. It gets raised to the power.
No brackets means the negative is separate — the power applies to the number only, then the negative is applied after.
Even vs Odd Powers#
| Power | Result (with brackets) | Why |
|---|---|---|
| Even ($2, 4, 6, \ldots$) | Positive | Negative pairs cancel: $(-)(-)=+$ |
| Odd ($1, 3, 5, \ldots$) | Negative | One negative left over after pairing |
Worked Example 4#
Simplify: $(-2)^4 - 2^4 + (-3)^3$
$= 16 - 16 + (-27)$
$= 0 - 27 = -27$
Worked Example 5#
Simplify: $-(-1)^{50} + (-1)^{51}$
$(-1)^{50} = 1$ (even power) → $-(-1)^{50} = -1$
$(-1)^{51} = -1$ (odd power)
$= -1 + (-1) = -2$
4. Square Roots — A Subtle Distinction#
$$\sqrt{9} = 3 \qquad \text{(positive only — the principal root)}$$$$\text{But solving } x^2 = 9 \text{ gives } x = \pm 3$$These are different questions:
- $\sqrt{9}$ asks: “what positive number squares to 9?” → Answer: $3$
- $x^2 = 9$ asks: “what numbers square to 9?” → Answer: $x = 3$ or $x = -3$
Exam rule: The $\sqrt{\phantom{x}}$ symbol always means the positive root. The $\pm$ only appears when you solve a quadratic.
Worked Example 6#
$\sqrt{25} + \sqrt{16} = 5 + 4 = 9$ (NOT $\pm 5 + \pm 4$)
Worked Example 7#
Solve: $x^2 = 49$
$x = \pm\sqrt{49} = \pm 7$
So $x = 7$ or $x = -7$.
5. Absolute Value#
The absolute value $|x|$ is the distance from zero on the number line — always positive (or zero).
$$|5| = 5, \qquad |-5| = 5, \qquad |0| = 0$$Solving Absolute Value Equations#
Since $|x|$ represents distance, $|x| = 3$ means “$x$ is 3 units from zero”:
$$|x| = 3 \quad \Rightarrow \quad x = 3 \text{ or } x = -3$$Worked Example 8#
Solve: $|x - 2| = 5$
$x - 2 = 5$ or $x - 2 = -5$
$x = 7$ or $x = -3$
Worked Example 9#
Solve: $|2x + 1| = 7$
$2x + 1 = 7$ or $2x + 1 = -7$
$2x = 6$ or $2x = -8$
$x = 3$ or $x = -4$
When There’s No Solution#
$|x| = -2$ has no solution — distance is never negative.
$|3x - 5| = -1$ has no solution — same reason.
Absolute Value Properties#
| Property | True or false? |
|---|---|
| $\|x\| \geq 0$ | Always true |
| $\|{-x}\| = \|x\|$ | Always true |
| $\|x \cdot y\| = \|x\| \cdot \|y\|$ | Always true |
| $\|x + y\| = \|x\| + \|y\|$ | NOT always true |
The last one is a common trap: $|3 + (-5)| = |-2| = 2$, but $|3| + |-5| = 8$.
6. Rounding#
The Rule#
- Next digit is 5 or more → round up
- Next digit is 4 or less → round down
Worked Example 10#
$3.456$ to 2 decimal places = $3.46$ (the 6 rounds the 5 up)
$3.454$ to 2 decimal places = $3.45$ (the 4 keeps the 5 as is)
$12.995$ to 2 decimal places = $13.00$ (the 5 rounds up)
Significant Figures#
$0.004523$ to 2 significant figures = $0.0045$ (leading zeros don’t count as significant)
$34\,567$ to 3 significant figures = $34\,600$
The Finance Rounding Rule#
Always round money to 2 decimal places (cents), but keep full precision during calculations and only round the final answer. Rounding intermediate steps introduces rounding errors that accumulate.
🚨 Common Mistakes#
| Mistake | Why it’s wrong | Fix |
|---|---|---|
| $-3^2 = 9$ | $-3^2 = -(3^2) = -9$. Only $(-3)^2 = 9$ | Check for brackets around the negative |
| $\sqrt{9} = \pm 3$ | $\sqrt{9} = 3$ (positive only). Solving $x^2 = 9$ gives $\pm 3$ | $\sqrt{\phantom{x}}$ = positive root; $\pm$ comes from solving |
| $5 - (-3) = 2$ | $5 - (-3) = 5 + 3 = 8$ | Subtracting a negative = adding |
| $\|a + b\| = \|a\| + \|b\|$ | Try $a = 3$, $b = -5$: $\|{-2}\| = 2 \neq 8$ | Absolute value of a sum ≠ sum of absolute values |
| $\sqrt{4} + \sqrt{9} = \sqrt{13}$ | $2 + 3 = 5 \neq \sqrt{13} \approx 3.6$ | You cannot add under the root sign |
| $\sqrt{5}$ is rational | $\sqrt{5} = 2.236\ldots$ (non-repeating, non-terminating) | Only $\sqrt{\text{perfect square}}$ is rational |
| Rounding $2.35$ to $2.3$ | The 5 rounds UP: answer is $2.4$ | “5 or more → round up” |
💡 Pro Tips for Exams#
1. The Bracket Check#
Every time you see a negative with an exponent, ask: “Is the negative inside brackets?”
- YES → the negative is part of the base
- NO → apply the power first, then the negative sign
2. Number Classification Questions#
Work from the most specific to the most general: Is it Natural? Whole? Integer? Rational? Irrational? Real? The first “yes” gives you all the categories above it too.
3. The Square Root Rule#
$\sqrt{n}$ is rational $\Leftrightarrow$ $n$ is a perfect square. Memorise the perfect squares up to $15^2 = 225$.
| $n$ | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $\sqrt{n}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
🔗 Related Grade 10 topics:
🏠 Back to Fundamentals | ⏮️ Fractions & Decimals | ⏭️ Basic Algebra