MYP 3 · Maths

REAL NUMBERS AND RATIO

Irrational numbers

QUESTION 1 [4 marks] — Criterion A Medium
Classify each of the following as rational or irrational, giving a brief reason:
a. $\sqrt{16}$
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Rational — $\sqrt{16}=4$, a whole number (which is rational).
b. $\sqrt{17}$
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Irrational — 17 is not a perfect square, so its square root cannot be written as a terminating or repeating decimal.
c. $\pi$
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Irrational — $\pi$'s decimal expansion never terminates or repeats (a well-known proven result).
d. $\frac{22}{7}$
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Rational — it is already expressed as a ratio of two integers, even though it is a common approximation for $\pi$.
QUESTION 2 [4 marks] — Criterion A Medium
Consider $\sqrt{30}$.
a. Between which two consecutive whole numbers does $\sqrt{30}$ lie? Justify your answer using perfect squares.
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$5^2=25$ and $6^2=36$. Since $25<30<36$, we know $5<\sqrt{30}<6$.
b. Estimate $\sqrt{30}$ correct to 1 decimal place, showing your reasoning (e.g. by testing values).
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Testing $5.5^2=30.25$ (slightly too big) and $5.4^2=29.16$ (too small). Testing $5.48^2\approx30.03$, close to 30. So $\sqrt{30}\approx5.5$ (1 d.p.).
QUESTION 3 [6 marks] — Criterion B Medium
Investigate which square roots of the whole numbers from 1 to 20 are rational, and which are irrational.
a. List the whole numbers from 1 to 20 whose square root is a whole number (i.e. a perfect square).
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$1 (=1^2), 4 (=2^2), 9 (=3^2), 16 (=4^2)$ — these are the perfect squares between 1 and 20.
b. State which of these square roots are rational, and explain your reasoning for the rest of the numbers 1–20.
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$\sqrt{1}=1$, $\sqrt{4}=2$, $\sqrt{9}=3$, $\sqrt{16}=4$ are all rational (whole numbers). Every other number from 1 to 20 (2, 3, 5, 6, 7, 8, 10–15, 17–20) is not a perfect square, so its square root is irrational.
c. State a general rule: for which whole numbers $n$ is $\sqrt{n}$ rational?
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$\sqrt{n}$ is rational exactly when $n$ is a perfect square (i.e. $n=k^2$ for some whole number $k$).
QUESTION 4 [5 marks] — Criterion C Medium
Explain why $\sqrt{2}$ cannot be written as a terminating or recurring decimal.
a. State what it would mean for $\sqrt{2}$ to be rational, in terms of being expressible as $\frac{p}{q}$.
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If $\sqrt{2}$ were rational, it could be written as a fraction $\frac{p}{q}$ in simplest form, where $p$ and $q$ are integers with no common factors.
b. It is a well-established mathematical result (proven by contradiction) that no such fraction exists for $\sqrt{2}$. Explain what this means for its decimal expansion.
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Since $\sqrt{2}$ cannot be written as any fraction of integers, it cannot be rational — and every rational number has a decimal expansion that either terminates or eventually repeats. Since $\sqrt{2}$ is not rational, its decimal expansion must go on forever without ever repeating in a pattern.
QUESTION 5 [5 marks] — Criterion D Medium
A carpenter is building a square tabletop and wants the diagonal to measure exactly $2$ m.
a. Using the relationship (diagonal)$^2 = 2\times$(side length)$^2$ for a square, find the exact side length in the form $\sqrt{k}$, then as a decimal correct to 3 decimal places.
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$$2^2 = 2s^2 \Rightarrow s^2 = 2 \Rightarrow s = \sqrt{2} \approx 1.414 \text{ m}$$
b. The carpenter's tape measure only shows millimetres (i.e. 3 decimal places in metres). Explain why the carpenter can never cut the side length with mathematically perfect accuracy, no matter how precise their tools are.
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Since $\sqrt{2}$ is irrational, its decimal expansion never terminates — so any real-world measurement (which must stop at some finite number of decimal places) can only ever be an approximation, never the mathematically exact value.