Number systems and sets of numbersIB MYP Maths Extended: Revision notes

The natural numbers are the counting numbers $\mathbb{N}=\{0,1,2,3,\ldots\}$. (Some books start at $1$, so check the definition in the question.) The integers include negative whole numbers as well: $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$. Every natural number is an integer, but not every integer is natural: $-3$ is an integer but not natural. A square root can be an integer: $\sqrt{16}=4$, so $\sqrt{16}$ is natural.

A rational number can be written as a fraction $\frac{a}{b}$ with $a,b$ integers and $b\ne0$. The symbol is $\mathbb{Q}$. Rational numbers include every integer ($-3=\frac{-3}{1}$), fractions such as $\frac25$, terminating decimals such as $0.45=\frac{45}{100}=\frac{9}{20}$, and recurring decimals such as $0.\dot{3}=\frac13$. A decimal that ends or repeats in a pattern is always rational.

An irrational number cannot be written as a fraction of integers. Its decimal goes on forever without repeating. Examples: $\pi$, $\sqrt{2}$, $\sqrt{7}$, $\sqrt{48}$. The square root of an integer that is not a perfect square is always irrational. Together the rational and irrational numbers make up the real numbers, $\mathbb{R}$. Calculators and tape measures show only a finite decimal, so $1.414213562$ is a rational approximation of $\sqrt{2}$, not its exact value: $1.41^2=1.9881\ne2$.

The sets are nested: $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$. Irrational numbers sit in $\mathbb{R}$ but outside $\mathbb{Q}$. To classify a number, simplify it first, then ask: whole and non-negative (natural)? whole (integer)? a fraction or ending/repeating decimal (rational)? none of these (irrational)? Always give the smallest set. Example: $-\sqrt{25}=-5$ is an integer; $\sqrt{49}=7$ is natural; $\frac{\pi}{2}$ is irrational; $-\frac94$ is rational.

A set is a collection of elements written in curly brackets: $B=\{0,2,4,6\}$. Symbols: $x\in B$ means $x$ is an element of $B$; $x\notin B$ means it is not; $n(B)$ is the number of elements; $\varnothing$ is the empty set. Set-builder notation describes a set by a rule: $A=\{x\in\mathbb{Z}:-2\le x<3\}=\{-2,-1,0,1,2\}$ (read: all integers $x$ such that $x$ is at least $-2$ and less than $3$). The union $A\cup B$ contains elements in $A$ or $B$ (or both); the intersection $A\cap B$ contains only elements in both. List each element once.

Inequalities are shown on a number line. A closed (filled) circle means the end value is included ($\le$ or $\ge$); an open circle means it is not included ($<$ or $>$). For $\{x\in\mathbb{R}:\sqrt{2}\le x<1.5\}$, draw a closed circle at $\sqrt{2}\approx1.414$, an open circle at $1.5$ and a solid line between them. For a set of integers such as $\{-2,-1,0,1,2\}$, mark separate dots, because there are no values in between.