“9781009761277c02_p000-000_ProofPDF” — 2026/3/11 — 6:38 — page 40 — #2 2A Irrational numbers and surds Irrational numbers U N SA C O M R PL R E EC PA T E G D ES √ √ When we apply Pythagoras’ theorem, we often obtain numbers such as 2. The numbers 2 and √ √3 are examples of irrational numbers and so is π, the number that arises from circles. Note that 2 means the positive square root of 2. p Recall that a rational number is a number that can be written as , where p is an integer and q is a q non-zero integer. A real number is a point on the number line. Every rational number is real but, as we have seen, not every real number is rational. A real number that is not rational is called irrational. −4 −3 −2 π √2 −√3 −1 0 1 2 3 4 As we have mentioned, every real number is a point on the number line and, conversely, every point on the number line is real. Surds can always be approximated by decimals, but working with exact values enables us to see important relationships and gives insights that would be lost if we approximated everything. Surds √ We can take the nth root of any positive number a. The nth root of a, written as n a, is the positive √ number whose nth power is a. Thus, n a = b is equivalent to the statement bn = a. √ A surd is an irrational number in the form of n a, where a is a rational number and n is an integer. √ √ √ √ √ 5 3 3 4 Hence, √ 3, 5 and 7 are surds. On the other hand, 8 = 2 and 81 = 3, so they are not surds. Note that π is not a surd, as π is irrational. Approximations to surds can be found using a calculator. Example 1 √ √ √ √ 3 Arrange the surds 8, 23, 2 and 60 in order of size on the number line. Use your calculator to check the answer. Solution We estimate the surds by comparing the number under the square root with the closest perfect squares. 4<8<9 √ √ 4< 8< 9 √ 2< 8<3 16 < 23 < 25 √ √ √ 16 < 23 < 25 √ 4 < 23 < 5 √ 1<2<4 √ √ 1< 2< 4 √ 1< 2<2 √ Uncorrected 3rd sample pages • Cambridge University Press & Assessment © • Evans, et al 2026 • 978-1-009-76127-7 • (03) 8671 1400 40 ICE-EM MATHEMATICS YEAR 10