Rational and irrational numbers
The real numbers consist of rational and irrational numbers. Classify the exact value, rather than the number of digits on a display
- Rational numbers The numerator and denominator must both be integers, and the denominator must be non-zero. This includes every integer \(\displaystyle -15=\frac{-15}{1}\)
- Natural numbers and integers Natural numbers belong to the integers. The integers belong to the rational numbers. Being an integer does not stop a number being rational
- Decimal expansions Terminating and eventually recurring decimals are rational. An infinite decimal that never becomes recurring is irrational \(\displaystyle \begin{aligned}0.94&=\frac{47}{50}\\0.\overline{61}&=\frac{61}{99}\end{aligned}\)
- Square roots and approximations The square root of a non-square positive integer is irrational. A finite approximation is a different, rational value \(\displaystyle \begin{gathered}\sqrt{19}\text{ is irrational}\\\sqrt{19}\approx4.3589\end{gathered}\)
- Comparing fractions Use the same positive denominator, then compare the numerators. Converting forms must preserve the exact value \(\displaystyle \begin{aligned}\frac{7}{10}&=\frac{56}{80}\\\frac{11}{16}&=\frac{55}{80}\\\frac{7}{10}&>\frac{11}{16}\end{aligned}\)
Common mistakes
Confusing endless with irrational
✕\(\displaystyle 0.\overline{61}\text{ is irrational}\)✓\(\displaystyle 0.\overline{61}=\frac{61}{99}\)
Repeating forever is compatible with an exact fraction representationClassifying the symbol instead of the value
✕\(\displaystyle \sqrt{121}\text{ is irrational}\)✓\(\displaystyle \sqrt{121}=11\)
A perfect-square radicand gives an integer, so the root is rationalRounding before comparing
✕\(\displaystyle \frac{7}{10}=\frac{11}{16}\)✓\(\displaystyle \frac{7}{10}>\frac{11}{16}\)
Two exact values can round to the same decimal without being equalDifficulty levels
introDistinguish simple rational numbers from familiar non-square roots and compare fractions with equal denominators
standardDistinguish recurring and irrational values and compare fractions with different denominators
challengeCompare and order exact values written in different forms, with classification revision
examChoose a valid reason for classifying an exact number
Questions about this topic
Are negative numbers rational?+
Some are and some are not. Every negative integer is rational because it can be written over a denominator of one. A negative irrational number remains irrational
Does any pattern make an infinite decimal rational?+
No. Rational decimals terminate or eventually repeat one fixed block. A rule that produces ever longer gaps between non-zero digits is a pattern but need not produce a recurring decimal
Do I have to find a fraction for every recurring decimal?+
No. Recognising a stated recurring expansion is enough to classify it as rational. Finding its fraction is a separate conversion task
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 1: Number and financial mathematics
In this chapter
Evaluate an integer expressionFactors, multiples, HCF, LCM and primesRound to decimal places or significant figuresEstimate an expression using rounded inputsConvert between fractions, mixed numbers and decimalsClassify real numbers and compare exact valuesAdd, subtract, multiply and divide fractions and mixed numbersWrite a ratio from countsWrite ratios in simplest formComplete equivalent ratiosUse a constant rateDivide a quantity in a given ratioDecide which option is the better valueConvert between fractions, decimals and percentagesFind a percentage of a quantityExpress one quantity as a percentage of another1F Percentages and money: Find the original amount from a percentageIncrease or decrease a quantity by a percentageFind percentage changePercentage error against a reference value1G Percentage change and percentage error: Find the original amount from a percentageProfit, mark-up and discountIncome and taxSimple interestCompound interest (optional extension)