Quadratic Functions: Background You’ll Need 2

Giselle Miller

Quadratic Functions: Background You’ll Need 2

Identify and understand the differences between rational numbers (like fractions and whole numbers) and irrational numbers (like pi and square roots)

Exploring Number Types: Rational, Irrational, and Real Numbers

Numbers in mathematics are sorted into different types such as, rational, irrational, and real numbers. Rational numbers are fractions with integers on top and bottom, like [latex]½[/latex] or [latex]-3/4[/latex]. Irrational numbers can’t be neatly written as fractions because their decimals go on endlessly without repeating—think of [latex]π[/latex] or the square root of [latex]2[/latex]. Both of these types are part of the real numbers, which make up the number line we use for all basic math. This page will guide you through these concepts, starting with rational numbers.

You should already know about the other number types – counting numbers, whole numbers and integers.

Counting numbers, also known as natural numbers, are the numbers we use to count items: [latex]1, 2, 3,[/latex] and so on. They are a subset of whole numbers, which extend counting numbers to include [latex]0[/latex], forming the set ([latex]0, 1, 2, 3,[/latex] …).
Integers further broaden this scope by incorporating their negative counterparts, resulting in an uninterrupted sequence (… [latex]-3, -2, -1, 0, 1, 2, 3,[/latex] …). These foundational elements serve as the groundwork for rational numbers, since any counting number, whole number, or integer can be expressed as a fraction with one as the denominator.

Rational Numbers

rational numbers

A rational number is a number that can be written in the form [latex]{\Large\frac{p}{q}}[/latex], where [latex]p[/latex] and [latex]q[/latex] are integers and [latex]q\ne o[/latex].

Rational numbers are the counts and measures we encounter in everyday life. Whether it’s in dividing a pizza into equal slices (fractions) or measuring the distance between two points (decimals), these numbers are all around us. Each can be expressed as a fraction, with both the numerator and denominator being whole numbers and the denominator never being zero. Let’s put this into practice and express the following values as ratios of two integers.

Irrational Numbers

irrational number

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.

Let’s summarize a method we can use to determine whether a number is rational or irrational. If the decimal form of a number:

stops or repeats, the number is rational.
does not stop and does not repeat, the number is irrational.

Real Numbers

real number

Real numbers are numbers that are either rational or irrational.

Determine whether each of the numbers in the following list is a

whole number
integer
rational number
irrational number
real number

[latex]-7,\Large\frac{14}{5}\normalsize ,8,\sqrt{5},5.9,-\sqrt{64}[/latex]

Number	Whole	Integer	Rational	Irrational	Real
[latex]-7[/latex]		[latex]\quad\checkmark[/latex]	[latex]\quad\checkmark[/latex]		[latex]\quad\checkmark[/latex]
[latex]\Large\frac{14}{5}[/latex]			[latex]\quad\checkmark[/latex]		[latex]\quad\checkmark[/latex]
[latex]8[/latex]	[latex]\quad\checkmark[/latex]	[latex]\quad\checkmark[/latex]	[latex]\quad\checkmark[/latex]		[latex]\quad\checkmark[/latex]
[latex]\sqrt{5}[/latex]				[latex]\quad\checkmark[/latex]	[latex]\quad\checkmark[/latex]
[latex]5.9[/latex]			[latex]\quad\checkmark[/latex]		[latex]\quad\checkmark[/latex]
[latex]-\sqrt{64}[/latex]		[latex]\quad\checkmark[/latex]	[latex]\quad\checkmark[/latex]		[latex]\quad\checkmark[/latex]