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Classify real numbers and recognise undefined forms

Start a set of 8
Step 1 of 5

What should happen before the real-number set or status is chosen?

\[\sqrt{\left(-7\right)^{2}}\]
Year 11, VCE Units 1 & 2, Chapter 1, 4 difficulty levels, No calculatorBuild a worksheet

Classifying real numbers

Evaluate an expression exactly before deciding its real-number set or status. Use the smallest set that contains the value, and check domain restrictions before assigning a real value

\[\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R},\quad\mathbb{R}\setminus\mathbb{Q}=\text{irrational numbers}\]
  1. Use the smallest containing set Use the positive counting numbers first, then the integers, then the rational numbers; irrational values belong to the real numbers but not the rational numbers \(\displaystyle 5\in\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R},\quad\sqrt{2}\in\mathbb{R}\setminus\mathbb{Q}\)
  2. Reduce rational values exactly An integer-valued fraction and a recurring decimal are rational even when their written forms differ \(\displaystyle \frac{6}{3}=2\in\mathbb{Q},\quad0.\overline{3}\in\mathbb{Q}\)
  3. Use the principal square root For a non-negative radicand, the square-root symbol means the non-negative root \(\displaystyle \sqrt{(-5)^{2}}=\sqrt{25}=5\)
  4. Check the real domain A zero denominator is undefined, and an even root of a negative real number is not real \(\displaystyle \frac{4}{0}\text{ is undefined},\quad\sqrt{-9}\text{ is not real}\)

Worked example

Classify the exact expression using the smallest standard real-number set

\[\sqrt{72}\]
  1. \(\displaystyle 72=36\times2\)Extract the largest square factor from 72.
  2. \(\displaystyle \sqrt{72}=\sqrt{36\times2}=6\sqrt2\)The square root of 36 is 6, and 2 remains under the root
  3. \(\displaystyle 6\sqrt2\in\mathbb{R}\setminus\mathbb{Q}\)Because 2 is not a perfect square, 6 times the square root of 2 is irrational

Common mistakes

Principal root versus both signs
✕\(\displaystyle \sqrt{49}=\pm7\)✓\(\displaystyle \sqrt{49}=7\)
The square-root symbol denotes the non-negative principal root, so it does not mean both signs
Recurring decimal omitted from the rationals
✕\(\displaystyle 0.\overline{3}\notin\mathbb{Q}\)✓\(\displaystyle 0.\overline{3}\in\mathbb{Q}\)
Every recurring decimal can be written as a ratio of integers, so it is rational
Zero denominator
✕\(\displaystyle \frac{5}{0}=0\)✓\(\displaystyle \frac{5}{0}\text{ is undefined}\)
Division by zero does not produce a real numeric value

Difficulty levels

introClassify a simplified integer, rational or irrational value
standardEvaluate an exact expression before choosing its smallest containing set
challengeDistinguish nested exact forms, undefined expressions and non-real even roots
examSelect the only true membership or containment statement

Questions about this topic

Why is a recurring decimal rational?+

A recurring decimal can be written as a fraction whose numerator and denominator are integers, so it belongs to the rational numbers

Why is the square root of 49 not both 7 and negative 7?+

The square-root symbol denotes the non-negative principal root. Both signs describe the solutions of an equation such as x squared equals 49.

What is the difference between undefined and non-real?+

An expression is undefined when its operation is not allowed, such as division by zero. An even root of a negative real number is defined over the complex numbers but is not a real number

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 1: Number, sets and proof

In this chapter

Classify real numbers and recognise undefined formsExplain number-system structureConvert between fractions and decimals, and between mixed and improper formsConstruct a fraction from a recurring decimalWork with sets and interval notationCount subsets and power-set elementsExpress a number in prime-factor index formInterpret propositions and quantifiersReason with direct and indirect proof methodsBuild and check mathematical induction arguments