01Cirrationals

=** Irrational Numbers **= toc

Rational Numbers
A ** Rational number ** is a number which __can__ be __**written as a fraction**__:
 * all whole numbers are rational (they can be written as the number over 1)
 * whole numbers are called ** integers **. (pronounced with a soft "g")
 * We use the letter ** Z ** to represent the set of all integers.
 * all fractions are rational
 * all finite decimals are rational - they can be written as a fraction (see Decimals)
 * all recurring decimals are rational - they can be written as a fraction (see Decimals)

Notice that the word __**rational**__ has "ratio" in it. Any fraction can be written as a ratio and some people call fractions, ratios.

We use the letter ** Q ** to represent the set of all rational numbers.

** Example **
math \\ . \qquad 9 = \dfrac{9}{1} \qquad 9 \textit{ is rational} \\. \\ . \qquad 0.3 = \dfrac{3}{10} \qquad 0.3 \textit{ is rational} \\. \\ . \qquad 0.\overline{12} = 0.12121212 ... = \dfrac{4}{33} \qquad 0.\overline{12} \textit{ is rational} math

Integers
The** integers ** (with a soft "g" like in George) are the whole numbers, both positive and negative.

We use ** Z ** to represent the set of all integers.
 * ** Z – ** is the set of negative integers
 * ** Z + **is the set of positive integers
 * ** Z + ** are also sometimes called the Natural Numbers (or ** N **)
 * Zero (0) is an integer but not included in ** Z – ** or ** Z + **

Irrational Numbers
An ** Irrational number ** is a number which can __not__ be written as a fraction.

Any irrational number will be equivalent to an infinite decimal with no repeating pattern.

The following are irrational:
 * special numbers like pi and e
 * surds (see below)
 * decimals with no repeating pattern

Irrational numbers are the __complement__ of **Q** (the set of rational numbers) {Recall: **complement** means "everything outside of the set". In this case: everything that is not rational}

so we represent the set of irrational numbers with ** Q' ** (or sometimes simply with ** I **)

** Example **
math \\ . \qquad \pi = 3.141592654... \qquad \pi \textit{ is irrational} \\. \\ . \qquad \sqrt{7} = 2.645751311 ... \qquad \sqrt{7} \textit{ is irrational} math

Real Numbers

 * Real numbers ** consists of __every__ number that can be shown on the number line. Real numbers include __all__:
 * rational numbers (Q)
 * irrational numbers (Q')

We use the letter ** R ** to represent the set of all real numbers.

math . \qquad 9, \; 0.3, \; \pi, \; \sqrt{7} \quad \textit{ are all real} math

{There are numbers which are __not__ real. They are called __**imaginary numbers**__. Those of you who do Specialist Maths in Year 12 will learn about imaginary numbers.}

Surds
A ** surd ** is the root of a number (eg square root or cube root, etc) that does __not__ have a finite decimal answer.

Surds are __**irrational**__.

If you convert a surd into a decimal, you get an infinite decimal with no repeating pattern, therefore it can not be written as a fraction.

For example: math \\ . \qquad \sqrt{7} = 2.645751311 ... \qquad \sqrt{7} \textit{ is a surd because the decimal is infinite and not recurring} \\ .\\ . \qquad \sqrt{16} = 4 \qquad \qquad \qquad \quad \sqrt{16} \textit{ is not a surd because the decimal is finite} \\. \\ . \qquad \sqrt[3]{10} = 2.15443469... \qquad \sqrt[3]{10} \textit{ is a surd} \\. \\ . \qquad \sqrt{0.25} = 0.5 \qquad \qquad \qquad \sqrt{0.25} \textit{ is not a surd} math

Note: If you are asked to find the **__exact__** answer, write the surd and do not change into decimal form.

Tree Diagram of Numbers
The connection between all of the different types of numbers within the Real number system can be shown in the following tree diagram.

Source: MathsQuest 10, 3rd edition, (JacarandaPlus)

Scientific Calculator
All scientific calculators have a square root button. Just press the square root button, then enter the number.

Most have a cube root button. For the calculator in the screen shot on the right, you have to press 2nd function then the x 3 button.

All scientific calculators have a button that allows finding the nth root (eg cube root, fourth root, etc). For the calculator in the screen shot on the right, you have to press 2nd function then the y x button.

math \text{To find } \sqrt[4]{10} \; : \;\; \text{ type } 4 \text{ then press 2nd } y^x \text{ then } 10 math



Casio Classpad
On the Classpad,
 * check your calculator is in Decimal mode (circled in red)
 * open the Virtual Keyboard (by pressing the blue Keyboard button)
 * then go to the 2D tab (circled in red)
 * select the square root or nth root icons from the first screen. (circled in purple)
 * enter the values in their appropriate slots and press EXE

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