What are the integers?

Positive and negative whole numbers and zero, written \mathbb{Z}.

What is a rational number?

A number that can be written as a fraction \frac{a}{b} of two integers, with b\ne0.

What is an irrational number?

A number that cannot be written as a fraction; its decimal never ends or repeats, for example \pi and \sqrt{2}.

Order the sets \mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R} using \subset.

\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}

Is \sqrt{16} rational or irrational?

Rational: \sqrt{16}=4, a natural number.

Is \sqrt{7} rational or irrational?

Irrational: 7 is not a perfect square.

Is 0.\dot{3} rational?

Yes, 0.\dot{3}=\frac13. A recurring decimal is rational.

What do A\cup B and A\cap B mean?

Union: elements in A or B (or both). Intersection: elements in both.

The number of elements in set A.

Read \{x\in\mathbb{Z}:-2\le x<3\}.

All integers x with x\ge-2 and x<3, which is \{-2,-1,0,1,2\}.

Closed circle or open circle for x<3 on a number line?

Open circle at 3, because 3 is not included.

Number systems and sets of numbersIB MYP Maths Extended: Flashcards

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What are the natural numbers?

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What are the natural numbers?: The counting numbers $\{0,1,2,3,\ldots\}$ , written $\mathbb{N}$ .
What are the integers?: Positive and negative whole numbers and zero, written $\mathbb{Z}$ .
What is a rational number?: A number that can be written as a fraction $\frac{a}{b}$ of two integers, with $b\ne0$ .
What is an irrational number?: A number that cannot be written as a fraction; its decimal never ends or repeats, for example $\pi$ and $\sqrt{2}$ .
Order the sets $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$ using $\subset$ .: $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$
Is $\sqrt{16}$ rational or irrational?: Rational: $\sqrt{16}=4$ , a natural number.
Is $\sqrt{7}$ rational or irrational?: Irrational: $7$ is not a perfect square.
Is $0.\dot{3}$ rational?: Yes, $0.\dot{3}=\frac13$ . A recurring decimal is rational.
What does $x\in A$ mean?: $x$ is an element of set $A$ .
What do $A\cup B$ and $A\cap B$ mean?: Union: elements in $A$ or $B$ (or both). Intersection: elements in both.
What does $n(A)$ mean?: The number of elements in set $A$ .
Read $\{x\in\mathbb{Z}:-2\le x<3\}$ .: All integers $x$ with $x\ge-2$ and $x<3$ , which is $\{-2,-1,0,1,2\}$ .
Closed circle or open circle for $x<3$ on a number line?: Open circle at $3$ , because $3$ is not included.

Exam questions on Number systems and sets of numbers

Consider the numbers in the set $S=\{-3,\ 0,\ \frac{2}{5},\ \sqrt{16},\ \sqrt{7},\ 0.\dot{3},\ \pi\}$ .
Write down the elements of $S$ that are rational but not integers.2 marks
Let $A=\{x\in\mathbb{Z}:-2\le x<3\}$ and $B=\{0,\ 2,\ 4,\ 6\}$ .
Write down $A\cup B$ and state the value of $n(A\cup B)$ .2 marks
Consider the numbers $\sqrt{49}$ , $\sqrt{48}$ , $-\frac{9}{4}$ , $0.45$ , $\frac{\pi}{2}$ and $-\sqrt{25}$ .
For each of $\sqrt{49}$ , $-\sqrt{25}$ and $\frac{\pi}{2}$ , state the smallest of the sets $\mathbb{N}$ , $\mathbb{Z}$ and $\mathbb{Q}$ that contains it, or state that it is irrational.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).