Irrational Numbers

Some numbers simply refuse to be written as a neat fraction, no matter how hard you try. These are the irrational numbers — the numbers that broke the ancient Greeks' picture of the universe and that still power geometry, physics and engineering today. This lesson explains exactly what makes a number irrational, how to spot one, and the misconceptions that trip students up.

Definition: An irrational number is a real number that cannot be written as p/q, where p and q are integers and q ≠ 0. Equivalently, its decimal expansion is non-terminating and non-repeating (it goes on forever with no repeating block).

Definition: A perfect square is a whole number that is the square of an integer — 1, 4, 9, 16, 25, 36, … Its square root is a whole number.

What "cannot be written as p/q" really means

A rational number's decimal either stops (like 0.75) or settles into a repeating pattern (like 0.333…). An irrational number does neither — it marches on forever with digits that never lock into a cycle. There is no fraction, however large its numerator and denominator, that equals it exactly.

The most famous example is √2. Its decimal begins 1.41421356… and never terminates or repeats. The same is true of √3 = 1.732…, √5 = 2.236…, π = 3.14159…, the golden ratio, and the constant e. You can also build an irrational on purpose, like 0.1010010001000010…, where the number of zeros keeps increasing so a repeating block can never form.

Why it matters: Irrationals are not exotic curiosities. The diagonal of a square, the circumference of a circle and the ratios in countless physics formulae are irrational. Without them, the number line would have holes — there would be points (like the diagonal length of a unit square) with no number to name them. Irrationals fill those holes and complete the real number system.

The Greeks and the scandal of √2

The Pythagoreans believed every quantity was a ratio of whole numbers. Then a member of their own school proved that the diagonal of a unit square — a length of exactly √2 — could not be such a ratio. According to legend this discovery so disturbed them that the result was kept secret. The proof is a beautiful piece of reasoning by contradiction: assume √2 = p/q in lowest terms, square to get p² = 2q², deduce that p must be even, then that q must also be even — contradicting "lowest terms." The assumption that √2 is rational collapses, so √2 must be irrational.

How to tell if a square root is irrational

This is the most exam-relevant skill in the topic, and it has one clean rule:

The square root of a positive integer is rational only if that integer is a perfect square; otherwise it is irrational.

So √4, √9, √16, √25 are rational (they equal 2, 3, 4, 5). But √2, √3, √5, √6, √7, √8, √10 are all irrational, because none of those numbers is a perfect square.

Question: Which of these are irrational — √36, √50, √81, √2?
Solution:
Step 1: Check √36 — 36 = 6², a perfect square, so √36 = 6 (rational).
Step 2: Check √50 — 50 is not a perfect square (49 and 64 are the nearest), so √50 is irrational. (Note √50 = √(25·2) = 5√2, still irrational.)
Step 3: Check √81 — 81 = 9², so √81 = 9 (rational).
Step 4: Check √2 — 2 is not a perfect square, so √2 is irrational.
Conclusion: √50 and √2 are irrational; √36 and √81 are rational.

Real-world example: π appears every time you measure anything round — the rim of a bicycle wheel, the edge of a pizza, the face of a clock. Because π is irrational, the circumference of a circle of nice whole-number radius can never be an exact whole number or simple fraction; that is why answers are written as "2π" or rounded to 3.14.

Common misconception: "Every square root is irrational." False. √9 = 3 is perfectly rational. Only the roots of non-perfect-squares are irrational.

Common misconception: "π = 22/7, so π is rational." 22/7 is only an approximation (22/7 = 3.142857…, which actually differs from π in the third decimal). The true π is irrational and no fraction equals it exactly. The same goes for 3.14 — a rounding, not the real value.

Common misconception: "A very long decimal must be irrational." Length alone proves nothing — 0.123123123… is enormously long but repeats, so it is rational. What makes a decimal irrational is being non-terminating AND non-repeating together.

√2 ≈ 1.414 sits between 1 and 2 — an irrational number with a non-terminating, non-repeating decimal, yet it has one exact point on the number line.

How can you point to a number on the number line when that number has no exact decimal — when its digits run on forever? The answer is one of the most elegant constructions in school geometry: you build the length with a right triangle and then swing it onto the line with a compass. This lesson walks through the standard NCERT construction of √2, explains why it works, and extends it to the square root spiral.

Definition: To represent a number on the number line means to mark the exact point whose distance from 0 equals that number — not an approximation, but the true point.

Definition: The Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides: hypotenuse² = side₁² + side₂².

The core idea: construct the length, then transfer it

You cannot "measure out" √2 with a ruler because 1.41421356… never ends. But Pythagoras gives you a way to create a segment whose length is exactly √2 using only segments of length 1. Once you have that exact segment, a compass lets you copy that length onto the number line. This is the trick: geometry can produce an exact irrational length even though arithmetic cannot write it as a finite decimal.

Why it matters: It proves that irrational numbers are not abstract inventions — they correspond to real, drawable points on the line. Every irrational length has a home on the number line, which is the deeper message: the line is complete, with no gaps.

Constructing √2 — step by step

Question: Construct the point representing √2 on the number line.
Solution:
Step 1: Mark O at 0 and A at 1, so OA = 1 unit along the line.
Step 2: At A, draw AB perpendicular to the line (straight up) with AB = 1 unit.
Step 3: Join OB. This is the hypotenuse of right triangle OAB, with the right angle at A.
Step 4: Apply Pythagoras: OB² = OA² + AB² = 1² + 1² = 2, so OB = √2.
Step 5: Place the compass point at O, open it to radius OB, and swing an arc down to the number line. The point where the arc meets the line is exactly √2.
Conclusion: The point where the arc lands is at distance √2 from 0 — the exact location of √2 on the number line.

The reason the arc works: every point on a circle centred at O is the same distance from O. So swinging OB (length √2) down to the line transfers that exact distance onto the line, converting a slanted segment into a horizontal position.

Extending to √3, √5, … and the square root spiral

Once you have the segment of length √2, you can keep going. At the tip of the √2 segment, draw a new perpendicular of length 1. The new hypotenuse has length √(√2² + 1²) = √(2 + 1) = √3. Add another unit perpendicular at that tip and you get √(3 + 1) = √4 = 2, then √5, √6, and so on. Each step uses the previous hypotenuse as one leg and a fresh unit segment as the other.

Drawn continuously, these triangles fan out into the beautiful square root spiral (the "spiral of Theodorus"), where each hypotenuse is √2, √3, √4, √5, … in turn. It is a visual proof that every √n can be constructed exactly with ruler and compass.

Question: How would you construct √5 by extending the spiral?
Solution:
Step 1: First build √2 as above (legs 1 and 1).
Step 2: On the √2 segment's tip, raise a unit perpendicular; the hypotenuse is √3.
Step 3: On the √3 tip, raise another unit perpendicular; the hypotenuse is √4 = 2.
Step 4: On that tip, raise one more unit perpendicular; the hypotenuse is √5.
Conclusion: The fourth hypotenuse in the spiral has length √5, which can then be swung onto the line.

Real-world example: A square floor tile measuring 1 m × 1 m has a diagonal of exactly √2 metres ≈ 1.414 m. That diagonal is a genuine, physical irrational length you can lay a measuring tape against — proof that irrational numbers describe real distances.

Common misconception: "You can't show an irrational number exactly because its decimal never ends." The decimal never ends, but the point is exact. Geometry sidesteps decimals entirely and produces the precise length via Pythagoras.

Common misconception: "The arc and the perpendicular can be any length." The construction only works if the perpendicular AB is exactly 1 unit (the same unit as OA) and the right angle is truly 90°. Get the perpendicular wrong and the hypotenuse is no longer √2.

Common misconception: "OB is the answer, so √2 is up in the air." OB is the length √2, but the number √2 lives on the number line. The compass arc is essential — it rotates that length down onto the line so you can read its position from 0.

When the ancient Pythagoreans tried to measure the diagonal of a simple square, they stumbled onto a number that broke their entire worldview — a length that no fraction could ever capture. That number was the square root of 2, and it opened the door to a vast hidden family of numbers called the irrationals.

Definition: A rational number is any number that can be written in the form p/q, where p and q are integers and q ≠ 0 (for example, 3/4, -5, 0.875 = 7/8, 1/3).

Definition: An irrational number is a real number that cannot be written in the form p/q with p, q integers and q ≠ 0. Equivalently, its decimal expansion is non-terminating and non-recurring (it goes on forever without ever settling into a repeating block).

Why irrational numbers must exist

Every measurement, ratio, and counting situation in daily life seems to be expressible as a fraction, so it is natural to assume fractions are "all the numbers there are." But geometry forces our hand. Take a square whose side is exactly 1 unit. By the Pythagoras theorem, its diagonal d satisfies d² = 1² + 1² = 2, so d = √2. This diagonal clearly has a definite length — you can draw it — yet, as the Pythagoreans discovered around 400 BCE, no fraction p/q equals it. A number that genuinely exists as a length but is not rational is, by definition, irrational. So irrationals are not a mathematical luxury; they are unavoidable the moment you allow squares, circles, and right triangles.

Why it matters: Without irrational numbers, the number line would be full of microscopic "holes" — points like √2, √3, and π would have no number sitting on them. The real numbers (rationals + irrationals together) fill every single point on the line, with no gaps. This completeness is what makes measurement, calculus, and physics possible.

Famous irrational numbers

• Square roots of non-perfect-square positive integers: √2, √3, √5, √6, √7, √8, √10, … Each of these is irrational. The pattern √2 ≈ 1.41421356…, √3 ≈ 1.73205080…, √5 ≈ 2.23606797… — none terminate, none repeat.
• π (pi) ≈ 3.14159265358979… — the ratio of a circle's circumference to its diameter. Its decimals never end and never repeat. (Note: the schoolbook value 22/7 is only an approximation of π; it is itself a rational number, not π exactly.)
• e ≈ 2.71828… and the golden ratio φ ≈ 1.61803… are other celebrated irrationals you will meet later.
• Cube roots like ∛2 and surds such as 2 + √3 are also irrational.

Locating √2 on the number line (Pythagoras construction)

The beauty of √2 is that, although it cannot be written as a fraction, it can be drawn exactly using only a ruler and compass. Method: Draw a unit square on the number line with one corner at 0. Its diagonal has length √(1² + 1²) = √2. With a compass placed at 0, swing this diagonal arc down onto the number line; the point where the arc meets the line represents √2 (a little past 1.4).

The same idea extends: from the point √2 you can erect another unit perpendicular and get √3, then √4 = 2, then √5, and so on — this beautiful spiral is called the square-root spiral (spiral of Theodorus).

Spotting irrationals from their decimals

A decimal tells you instantly which family a number belongs to:

• Terminates (stops): rational. E.g. 0.875.
• Repeats forever in a fixed block: rational. E.g. 0.333… = 1/3.
• Goes forever with no repeating block: irrational.

Real-world example: the constructed decimal 0.1010010001000010…, where the number of 0s between successive 1s keeps growing, never terminates and never falls into a repeating cycle. Hence it is irrational — and you can manufacture infinitely many irrationals this way.

Worked example.
Question: Is 0.10110111011110… (each block has one more 1 than the previous) rational or irrational?
Solution:
Step 1: Check if it terminates — it does not; the digits continue forever.
Step 2: Check for a repeating block — the pattern grows (1, then 11, then 111, …), so no fixed block ever repeats.
Conclusion: It is non-terminating and non-recurring, therefore irrational.

Operations: a few useful truths

• Rational + irrational = irrational (e.g. 3 + √2 is irrational).
• Non-zero rational × irrational = irrational (e.g. 2√3 is irrational).
• Irrational ± irrational, or irrational × irrational, may be rational or irrational — be careful. For instance √2 × √2 = 2 (rational), and (2 + √3) + (2 − √3) = 4 (rational), but √2 + √3 stays irrational.

Common misconception: "Every square root is irrational." False. √25 = 5 and √4 = 2 are perfectly rational. Only the square roots of non-perfect-square positive integers are irrational. Always check first whether the number under the root is a perfect square.

Common misconception: "π equals 22/7." False. 22/7 is just a convenient rational approximation (≈ 3.142857…); the true π ≈ 3.14159… is irrational and differs from 22/7 in the third decimal place onward.

Common misconception: "Irrational means the number is somehow not real or doesn't exist." False. Irrational numbers are genuine real numbers occupying real points on the number line; "irrational" only means "not a ratio of integers."

How do you prove that something is impossible — that no fraction in the entire universe, however large, can ever equal √2? You cannot test them one by one. Instead, mathematicians use one of the most elegant weapons in logic: proof by contradiction.

Definition: Proof by contradiction (reductio ad absurdum) is a method where you assume the opposite of what you want to prove, then show that this assumption forces a logical impossibility. Since the assumption leads to nonsense, the assumption must be false — and so the original statement must be true.

The claim and the strategy

We want to show: √2 is irrational, i.e. √2 cannot be written as p/q with integers p, q (q ≠ 0). Directly checking "no fraction works" is impossible because there are infinitely many fractions. So we flip it: assume √2 IS rational and chase the assumption until it self-destructs.

A key preliminary fact we will lean on:

Definition / Lemma: If the square of an integer is even, then the integer itself is even. (Reason: an odd number has the form 2k+1, and (2k+1)² = 4k² + 4k + 1 = 2(2k²+2k) + 1, which is always odd. So an even square can only come from an even number.) We will use this twice.

The proof, step by step

Question: Show that √2 is not a rational number.

Solution (by contradiction):

Step 1: Assume the opposite. Suppose √2 is rational. Then we can write
√2 = a/b,
where a and b are integers, b ≠ 0, and — crucially — the fraction is in lowest terms, meaning a and b share no common factor other than 1. (Any fraction can always be reduced to lowest terms, so this is allowed.)

Step 2: Square both sides to remove the root:
2 = a²/b², which rearranges to a² = 2b². …(i)

Step 3: Deduce a is even. Equation (i) says a² equals 2 times an integer, so a² is even. By our lemma, if a² is even then a is even. Therefore we can write a = 2c for some integer c.

Step 4: Substitute a = 2c into (i):
(2c)² = 2b² ⟹ 4c² = 2b² ⟹ b² = 2c².

Step 5: Deduce b is even. Equation b² = 2c² says b² is 2 times an integer, so b² is even; by the lemma again, b is even.

Step 6: The contradiction. Steps 3 and 5 show that both a and b are even, so they share the common factor 2. But in Step 1 we insisted a/b was in lowest terms, with no common factor other than 1. These two facts cannot both be true.

Conclusion: Our starting assumption — that √2 is rational — leads to a contradiction. Hence the assumption is false, and √2 is irrational. ∎

Why each step is necessary

Why insist on lowest terms? This is the engine of the whole proof. The contradiction is precisely "you said no common factor, yet I found the common factor 2." Without the lowest-terms assumption, finding a common factor would be no contradiction at all, and the proof would collapse.

Why does even square ⟹ even number? Because parity (even/odd) is preserved predictably: even × even = even, odd × odd = odd. A square can only be even if its base is even. Skipping this lemma is the most common gap in student proofs — examiners expect it stated or justified.

Why it matters: This single proof technique generalises beautifully. The identical argument (using the property of primes that if a prime p divides a², then p divides a) proves that √3, √5, √7, and indeed √p for every prime p — and more generally √n for every non-perfect-square positive integer n — is irrational. One method, infinitely many results.

Real-world example: This is exactly the discovery (attributed to the Pythagorean school, c. 5th century BCE) that a square of side 1 has a diagonal of length √2 which no whole-number ratio can measure. It shattered the Pythagorean belief that "all is number (ratio)," and historically launched the study of irrational magnitudes in geometry.

Common misconceptions corrected

Common misconception: "We assumed a and b have no common factor — isn't that assuming what we want to prove?" No. Reducing a fraction to lowest terms is always possible for any rational number; it is a harmless, standard simplification, not a hidden assumption about irrationality. We are only assuming √2 is rational; the lowest-terms form is a legitimate consequence of that.

Common misconception: "a² = 2b² already proves it's irrational." No — that equation is perfectly consistent on its own; you must drive it to the parity contradiction in Steps 3–6.

Common misconception: "If a is even, b must be odd." Not necessarily, and that is the whole point — we prove b is also even, which is exactly what produces the contradiction.

Some numbers, like √5, can never be written exactly as a fraction or as a finishing decimal — yet they have a perfectly definite, fixed place on the number line. This lesson shows you how to find that exact place using nothing but a ruler, a set square, and a compass, by turning the Pythagoras theorem into a drawing tool.

Definition: An irrational number is a real number that cannot be written as p/q where p and q are integers and q ≠ 0. √5 is irrational because 5 is not a perfect square.

The big idea: build a length, don't measure it

You cannot place √5 on the number line by measuring 2.236 cm with a ruler, because √5 = 2.2360679… never ends and never repeats, so any measurement is only an approximation. Instead we construct a line segment whose length is exactly √5, and then copy that exact length onto the number line with a compass.

The trick is the Pythagoras theorem. In a right-angled triangle, hypotenuse² = (one leg)² + (other leg)². So if we can make a right triangle whose two legs are nice whole-number lengths, the hypotenuse will automatically be the square root of (sum of squares of the legs).

We want the hypotenuse to be √5. So we need legs whose squares add to 5:

2² + 1² = 4 + 1 = 5

That is the whole secret. Legs of 2 and 1 give a hypotenuse of √5.

Step-by-step construction

Question: Represent √5 on the number line.

Solution:
Step 1: On the number line, mark O at 0 and A at 2, so OA = 2 units. (We are using the existing whole-number markings 0 and 2 — those we can place exactly.)
Step 2: At A, draw a segment AB perpendicular to the number line, with AB = 1 unit. (Use a set square or the right-angle construction.)
Step 3: Join OB. By the Pythagoras theorem,
OB² = OA² + AB² = 2² + 1² = 4 + 1 = 5, so OB = √5.
Step 4: Place the compass point at O, open it to radius OB, and draw an arc that cuts the number line at point P. Swinging the hypotenuse down onto the line keeps its length unchanged, so OP = OB = √5.
Conclusion: Point P represents √5 on the number line (P sits at about 2.236, just past the 2-mark, before 2.5).

Why the compass arc is the crucial step

A compass draws a circle, and every point on a circle is the same distance (the radius) from the centre. By centring at O with radius OB = √5, the arc carries the exact length √5 from the slanted segment OB and lays it flat along the number line. That is how a tilted, hard-to-place length becomes a precise point on the line. The arc transfers an exact length without ever measuring it.

Why it matters: This same method lets you locate √2, √3, √6, √7 — any non-perfect-square root — geometrically. It is the bridge that shows irrational numbers are not vague or "made up": each one occupies one definite point, just like 2 or 3 does. This is the visual proof that the number line is complete, with real numbers filling every gap.

Checking the construction

Question: How do we know P really is at √5 and not somewhere close?
Solution:
Step 1: √5 ≈ 2.236.
Step 2: Square it back: 2.236² = 4.999696 ≈ 5.
Conclusion: Since squaring returns (almost exactly) 5, the construction is correct; tiny error is only from rounding 2.236, not from the method.

The square-root spiral connection

The same idea, repeated, builds the beautiful square-root spiral. Start with a unit segment, raise a perpendicular of length 1: the hypotenuse is √(1²+1²) = √2. On that √2 hypotenuse, raise another perpendicular of length 1: the new hypotenuse is √(2+1) = √3. Continuing gives √4, √5, √6, … one after another. So √5 can also be reached by stacking unit perpendiculars five steps along the spiral. The "legs 2 and 1" method in this lesson is just a quicker, direct shortcut to √5.

Common misconception: "√5 ≈ 2.236, so I can just mark 2.236 on the line and that's √5." Wrong — 2.236 is a rounded value; the true point is irrational and 2.236 is slightly off. The geometric construction gives the exact point, which is its whole purpose.

Common misconception: "I need legs of √2 and √3 or some odd lengths." No — the elegance is that whole-number legs 2 and 1 already give √5, because 4 + 1 = 5. Always look for two perfect squares that add to your target.

Real-world example: Surveyors and carpenters use the very same Pythagoras logic to mark exact diagonal lengths on the ground or on a board when no ruler can give the awkward measurement directly — they create a right angle and read off the hypotenuse.

Numbers like √2, √3, √5 and π refuse to fit into any fraction, yet they are everywhere — in the diagonal of a square, the circumference of a circle, the geometry of nature. This lesson gathers the essential facts about irrational numbers: what makes them irrational, how to spot one, and how to pin them down on the number line.

Definition: A number is irrational if it CANNOT be expressed as p/q with integers p and q and q ≠ 0. Equivalently, an irrational number has a decimal expansion that is non-terminating and non-recurring (it never ends and never settles into a repeating block).

Rational vs irrational — the dividing line

A rational number can be written as a ratio of two integers: 3 = 3/1, 0.75 = 3/4, −5/7, and so on. The word "ratio" is literally inside "rational". Anything that cannot be squeezed into such a ratio is irrational.

The cleanest test is the decimal expansion:

• Terminating (e.g. 0.875) → rational.
• Non-terminating but recurring (e.g. 0.3333… or 0.142857142857…) → rational.
• Non-terminating and non-recurring (e.g. 0.1010010001…) → irrational.

Why it matters: This single test classifies any real number. If a decimal eventually repeats — even a very long block — it is rational. Only when it both never ends and never repeats is it irrational.

Which square roots are irrational?

This is the most exam-tested fact:

The square root of a positive integer is irrational UNLESS that integer is a perfect square.

• √2, √3, √5, √6, √7, √8, √10 … are irrational (these integers are not perfect squares).
• √4 = 2, √9 = 3, √16 = 4, √25 = 5 … are rational (perfect squares give whole-number roots).

So you can predict irrationality at a glance: ask "is the number under the root a perfect square?" If no, the root is irrational.

Building irrational lengths with Pythagoras

Irrational numbers are not abstract — they are genuine lengths you can draw. A right triangle with legs of length m and n has hypotenuse √(m² + n²).

• Legs 1 and 1 → hypotenuse √(1 + 1) = √2.
• Legs 2 and 1 → hypotenuse √(4 + 1) = √5.

Swinging that hypotenuse onto the number line with a compass marks the exact irrational point. This is the geometric proof that √2 and √5 are real, locatable numbers, not just symbols.

Real-world example: The diagonal of a 1 metre × 1 metre square tile is exactly √2 ≈ 1.414 metres — an irrational length you can literally hold a tape measure across, even though no fraction captures it precisely.

π — the most famous irrational

Definition: π (pi) is the ratio of a circle's circumference to its diameter; π ≈ 3.14159265…, and it is irrational.

A very common classroom value is 22/7, but this is only an approximation (22/7 = 3.142857… with the block 142857 repeating, which makes 22/7 rational). The true π is irrational: its digits go on forever with no repeating pattern. So 22/7 and 3.14 are convenient stand-ins, never the exact value.

Common misconception: "π = 22/7." False. 22/7 is rational and merely close to π. π itself is irrational and cannot equal any fraction.

Common misconception: "Every square root is irrational." False — √4, √9, √16 are rational. Only non-perfect-square roots are irrational.

Common misconception: "Irrational numbers are rare or unusual." Actually, between any two rationals lies an irrational, and between any two irrationals lies a rational. Both kinds are densely packed everywhere on the number line.

Density: they are interleaved everywhere

A subtle but powerful fact: between any two rational numbers there is an irrational number, and between any two irrational numbers there is a rational number. No matter how close together two numbers are, you can always slip a number of the other type between them. The number line is woven from both, with no smallest gap.

This is your one-page mastery sheet for irrational numbers — every idea you met, distilled and then re-explained so the facts actually make sense rather than being memorised blindly. By the end you should be able to identify, prove, and locate irrational numbers with confidence.

Definition: An irrational number cannot be written as p/q (integers p, q with q ≠ 0); its decimal expansion is non-terminating and non-recurring.

1. The defining property

Irrational numbers cannot be put in the form p/q, and their decimals never end and never repeat. Contrast this with rationals, whose decimals either stop (0.875) or fall into a repeating block (0.333…). The "no fraction" and "no repeating decimal" descriptions are two sides of the same coin — each implies the other.

2. Which numbers are irrational

The square root of a non-perfect-square positive integer is irrational: √2, √3, √5, √7, √10, … Also π is irrational. The reason √2 is irrational (not just "it looks messy") is a genuine logical proof, covered next.

3. The proof that √2 is irrational (proof by contradiction)

This is the classic must-know argument.

Question: Prove that √2 is irrational.
Solution:
Step 1: Assume the opposite — suppose √2 is rational, so √2 = a/b where a and b are integers with no common factor (the fraction is in lowest terms) and b ≠ 0.
Step 2: Square both sides: 2 = a²/b², so a² = 2b². This means a² is even, which forces a to be even (the square of an odd number is odd). Write a = 2c.
Step 3: Substitute: (2c)² = 2b² → 4c² = 2b² → b² = 2c². So b² is even, which forces b to be even too.
Step 4: But now both a and b are even — they share a common factor 2. This contradicts our assumption that a/b was in lowest terms.
Conclusion: The assumption "√2 is rational" leads to a contradiction, so it must be false. Therefore √2 is irrational.

Why it matters: This shows irrationality is a provable property, not a guess. The same "both must be even" engine proves √3, √5, etc. are irrational.

4. Locating irrationals geometrically

Using a right triangle with legs 1 and 1, the hypotenuse is √(1+1) = √2, which we swing onto the number line. With legs 2 and 1, the hypotenuse is √(4+1) = √5. A compass arc, centred where the segment starts and opened to the hypotenuse length, drops the exact irrational length onto the number line — turning a tilted segment into a precise point.

5. The square-root spiral

Adding unit perpendiculars one after another generates √2, √3, √4, √5, … each new hypotenuse being √(previous² + 1). This visual chain shows the whole family of square roots living on a single elegant spiral.

6. The big common error

Question: Is √16 irrational?
Solution:
Step 1: Check if 16 is a perfect square: 4² = 16. Yes.
Step 2: So √16 = 4, an integer.
Conclusion: √16 is rational. Not every square root is irrational — √4 = 2, √9 = 3, √16 = 4 are all rational. Only non-perfect-square roots are irrational.

Common misconception: "All roots are irrational." Wrong — perfect-square roots are whole numbers.
Common misconception: "If a decimal looks long and messy, it must be irrational." Not necessarily — 1/7 = 0.142857142857… looks messy but repeats, so it is rational.

Cover the answer, recall, then check. 7 cards on irrational numbers.

Q1. What is an irrational number?
A1. A number that cannot be written in the form p/q with integers p, q (q ≠ 0); its decimal is non-terminating and non-repeating.

Q2. Give three examples of irrational numbers.
A2. √2, √3 and π (also 0.101001000... type numbers).

Q3. Is √4 rational or irrational?
A3. Rational, because √4 = 2, which can be written as 2/1.

Q4. Is the sum of a rational and an irrational number rational or irrational?
A4. Irrational (e.g. 2 + √3 is irrational).

Q5. How can you represent √2 on the number line?
A5. By constructing a right triangle with both legs 1 unit; the hypotenuse is √2, which is then marked with a compass.

Q6. What kind of decimal expansion do irrational numbers have?
A6. Non-terminating and non-repeating.

Q7. Is π equal to 22/7?
A7. No — 22/7 is only an approximation; π is irrational and 22/7 is rational.