Subject 1 — Mathematics1.1

Numbers & Arithmetic

Where this fitsStage 1

UnlocksFractions, Decimals & Percentages Factors, Multiples & Divisibility Exponents & Roots

The Number Sets

Every number you will ever use belongs to one of these sets. Each set contains the previous one.

Natural Numbers ℕ

The counting numbers. Positive integers starting from 1 (or 0, depending on convention).

\mathbb{N} = \{1, 2, 3, 4, \ldots\}

Integers ℤ

Natural numbers extended with zero and negative counterparts.

\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}

Rational Numbers ℚ

All numbers expressible as a ratio of two integers, where the denominator is non-zero.

\mathbb{Q} = \left\{ \frac{p}{q} \;\middle|\; p, q \in \mathbb{Z},\; q \neq 0 \right\}

Irrational Numbers

Numbers that cannot be expressed as a ratio of integers. Their decimal expansions are non-terminating and non-repeating.

\sqrt{2},\quad \pi,\quad e

Real Numbers ℝ

The union of all rational and irrational numbers — every point on the number line.

\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}^c

Absolute Value

The absolute value of a number is its distance from zero on the number line — always non-negative.

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Example

Computing absolute values:

|-7| = 7 \qquad |3| = 3 \qquad |0| = 0

Order of Operations

When an expression has multiple operations, evaluate them in this order: Parentheses → Exponents → Multiplication & Division → Addition & Subtraction.

Example

Evaluate step by step:

3 + 2 \times (8 - 5)^2 = 3 + 2 \times 9 = 3 + 18 = 21

Multiplication and division are evaluated left to right with equal precedence, as are addition and subtraction.

← Mathematics

Subject 1 — Mathematics1.1

Numbers & Arithmetic

Where this fitsStage 1

UnlocksFractions, Decimals & Percentages Factors, Multiples & Divisibility Exponents & Roots

The Number Sets

Every number you will ever use belongs to one of these sets. Each set contains the previous one.

Natural Numbers ℕ

The counting numbers. Positive integers starting from 1 (or 0, depending on convention).

\mathbb{N} = \{1, 2, 3, 4, \ldots\}

Integers ℤ

Natural numbers extended with zero and negative counterparts.

\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}

Rational Numbers ℚ

All numbers expressible as a ratio of two integers, where the denominator is non-zero.

\mathbb{Q} = \left\{ \frac{p}{q} \;\middle|\; p, q \in \mathbb{Z},\; q \neq 0 \right\}

Irrational Numbers

Numbers that cannot be expressed as a ratio of integers. Their decimal expansions are non-terminating and non-repeating.

\sqrt{2},\quad \pi,\quad e

Real Numbers ℝ

The union of all rational and irrational numbers — every point on the number line.

\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}^c

Absolute Value

The absolute value of a number is its distance from zero on the number line — always non-negative.

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Example

Computing absolute values:

|-7| = 7 \qquad |3| = 3 \qquad |0| = 0

Order of Operations

When an expression has multiple operations, evaluate them in this order: Parentheses → Exponents → Multiplication & Division → Addition & Subtraction.

Example

Evaluate step by step:

3 + 2 \times (8 - 5)^2 = 3 + 2 \times 9 = 3 + 18 = 21

Multiplication and division are evaluated left to right with equal precedence, as are addition and subtraction.