Chapter 1.3 — Exercise 1.2 — Irrational Numbers

The Test for √n

• Non-perfect-square roots are irrational. √2, √3, √5, √6, √7, √8 all qualify. But √1 = 1, √4 = 2, √9 = 3 are rational, because 1, 4, and 9 are perfect squares.
• Non-terminating, non-recurring decimals are irrational by definition — no repeating pattern means no p/q form exists.
• If n is a natural number and not a perfect square, √n is irrational. One rule, infinitely many irrational numbers.

Listing the square roots of 1 through 9 in order makes the pattern obvious rather than abstract: √1 = 1, √2 = 1.4142135623731…, √3 = 1.7320508075689…, √4 = 2, √5 = 2.2360679…, √6 = 2.4494897…, √7 = 2.6457513…, √8 = 2.8284271…, √9 = 3. Only the perfect squares — 1, 4, 9 — settle into a whole number. Every value in between refuses to terminate or repeat, no matter how many digits get computed.

π's Special Case

π is defined as C/d, the ratio of a circle's circumference to its diameter, which looks like it should be a fraction of two measured lengths. The problem is that no common unit exists that measures both C and d exactly — measure precisely enough and at least one of the two turns out to be irrational. π = 3.14159265358979323846… never terminates and never repeats. The familiar 22/7 used in calculations is only a convenient rational approximation, not the exact value.

This is a genuinely strange fact worth sitting with: π comes from measuring two lengths of an actual, physical circle, yet the ratio between them can never be captured exactly by any pair of whole numbers. Every calculator value of π, every 22/7, every 3.14 is a rounded stand-in for a number that has no exact fractional address at all.

Classifying Six Numbers — Question 1

• √27 — 27 isn't a perfect square → irrational.
• √441 — 441 = 21² → rational (equals 21 exactly).
• 30.232342345… — non-terminating, non-repeating → irrational.
• 7.484848… — non-terminating but the block "48" repeats → rational.
• 11.2132435465 — terminates → rational.
• 0.3030030003… — the gap between 3s keeps growing, so it never settles into a repeating block → irrational.

Question 2 asks for four examples each: rationals such as −3/5, 12.25, 7.232323…, √16, or 0, and irrationals such as π, 12.010120123…, 7.11121314…, or √19. Notice that √16 belongs on the rational list and √19 on the irrational one — the radical sign alone never decides the answer; whether the number underneath it is a perfect square always does.

Manufacturing an Irrational Number Between Two Rationals

Just as rationals are dense between any two values, so are irrationals — and you can construct one on demand rather than search for it. Convert both endpoints to decimals, then invent digits between them that visibly never repeat.

Question 3: between 5/7 = 0.714285̄ and 7/9 = 0.7̄, the number 0.72723724… works — its digit pattern never locks into a cycle. Question 4: between 0.70 and 0.77, both 0.71712713… and 0.7486549823… qualify, and neither is the only valid answer.

A second, more systematic method also exists: if a and b are positive rationals and a × b is not a perfect square, then √(ab) is irrational and lies between them. Between 3 and 4, that gives √(3 × 4) = √12 = 2√3.

Between 1/4 and 2/3, the same method applies to non-integers just as well: converting to decimals first gives 1/4 = 0.25 and 2/3 = 0.6̄, and infinitely many non-repeating decimals — 0.313233343536…, 0.51512513514…, 0.6010010000100001… among them — sit in that gap. There's no single correct irrational number to find here, only infinitely many valid ones, which is exactly the point the exercise is making about how densely irrationals are packed alongside the rationals.

Long Division Toward √5 and √7

Questions 5 and 6 compute square roots to several decimal places by long division rather than a calculator — grouping digits in pairs from the decimal point, finding the largest fitting integer at each stage, then doubling the running quotient to build the next divisor.

• √5 ≈ 2.236, to 3 decimal places.
• √7 ≈ 2.645751, to 6 decimal places.

Neither expansion shows any sign of repeating, which is exactly what the theory predicts for a non-perfect-square root.

Long division for a square root works differently from ordinary division: digits are paired up starting from the decimal point (so 5.000000 becomes 5 . 00 00 00), and at each stage the running quotient is doubled to form the start of the next trial divisor, which is then completed and multiplied to fit under the current remainder. It's slower than ordinary long division, but it's exact at every step — no digit produced is ever a rounded guess.

Constructing √10 With a Right Triangle

Question 7 asks for √10 to be marked on the number line geometrically rather than estimated. Since 10 = 9 + 1 = 3² + 1², a right triangle with legs 3 and 1 has hypotenuse exactly √10 by the Pythagorean theorem. Drawing that triangle with the base leg starting at 0 on the number line, then swinging the hypotenuse down with a compass, marks √10 precisely — landing between 3 and 4, closer to 3.

This trick works for any n that can be split into a sum of two perfect squares — √13 = √(3² + 2²), √5 = √(2² + 1²), and so on. Numbers that resist this exact split, like √7, need a different construction, which is exactly what the square root spiral in the next exercise is built to handle.

True, False, and the Real Number Line

The union of every rational and every irrational number is the set of real numbers (R) — everything encountered from here through Class 10 lives inside it. Question 9's true/false check reinforces the boundaries: every irrational number is real (true); every rational number is real (true); a real number need not be rational (true — √7 is real but irrational); √n is rational whenever n is a perfect square (true); √n is irrational whenever n isn't (true); but the claim that all real numbers are irrational is false, since 7/5 is real and perfectly rational.

These classification instincts carry forward directly. Exercise 1.3 gives irrational numbers like the square roots built here a visual home on the number line through successive magnification and the square root spiral, and Class 10 revisits √2 and √3 to prove their irrationality formally by contradiction, rather than relying on a non-repeating decimal as evidence.