Every number you have ever met — and even the strange ones like √2 and π — lives somewhere on a single straight line called the number line. This chapter teaches you to name them, place them, and play with them!
Natural & Whole
Counting numbers 1, 2, 3… ; add 0 and you get whole numbers.
Integers & Rationals
Negatives join in; any number written as p/q is rational.
Irrationals
Numbers like √2 and π that never become a clean fraction.
Real numbers
Rationals + irrationals together fill the whole number line.
1. The families of numbers
Numbers are grouped into families that fit inside one another. Natural numbers (N) are the counting numbers 1, 2, 3, 4… . When we include 0 we get the whole numbers (W) = 0, 1, 2, 3… . Add the negatives and we get the integers (Z) = …−3, −2, −1, 0, 1, 2, 3… . So every natural number is a whole number, and every whole number is an integer. The families are nested like boxes inside boxes.
2. Rational numbers (Q)
A number is rational if it can be written in the form p/q where p and q are integers and q is not 0. Examples: 3/4, −5/7, 6 (because 6 = 6/1), and 0 (because 0 = 0/1). Between any two rational numbers there are infinitely many more rationals — for example, the average (a + b) ÷ 2 always lies between a and b. So rationals are “dense” on the number line, yet they still leave tiny gaps!
3. Decimal expansions
Every rational number has a decimal form that either terminates (ends), like 1/2 = 0.5 or 7/8 = 0.875, or is non-terminating but recurring (a block of digits repeats forever), like 1/3 = 0.333… or 1/7 = 0.142857142857… . The repeating block is shown with a bar over it. The key rule: a fraction in lowest terms terminates only when the denominator’s prime factors are made of just 2s and 5s.
4. Irrational numbers
An irrational number cannot be written as p/q. Its decimal expansion is non-terminating and non-recurring — it goes on forever with no repeating pattern. Famous examples are √2 = 1.41421356…, √3, √5, and π = 3.14159265… . The square root of any positive integer that is not a perfect square is irrational. There are infinitely many irrationals filling the gaps the rationals leave behind.
5. Real numbers and the number line
All rationals together with all irrationals form the real numbers (R). The big truth of this chapter: every real number corresponds to exactly one point on the number line, and every point on the line is exactly one real number. So the line has no gaps at all. We can even locate √2 geometrically: draw a right triangle with both legs of length 1, its hypotenuse is √2; swing that length onto the line with a compass and you have marked √2 precisely. This is called the successive magnification / square-root spiral idea.
6. Representing decimals on the line
To show a number like 3.765 on the line we use successive magnification: zoom into the gap between 3 and 4, then between 3.7 and 3.8, then between 3.76 and 3.77, narrowing in until the point appears. This shows that even a long decimal has a definite home on the line.
7. Operations on real numbers
You can add, subtract, multiply and divide real numbers. Sum or difference of a rational and an irrational is always irrational (e.g. 2 + √3). Product or quotient of a non-zero rational and an irrational is also irrational. But two irrationals can combine to give a rational, e.g. √2 × √2 = 2, or (3 + √5) + (3 − √5) = 6.
8. Laws of exponents for real numbers
For a positive real base a and rational powers p, q the familiar rules hold: ap × aq = ap+q ; (ap)q = apq ; ap ÷ aq = ap−q ; ap × bp = (ab)p. We also extend roots: a1/n means the n-th root of a, so 81/3 = 2.
9. Rationalising the denominator
We dislike a root in the denominator. To remove it we rationalise: multiply top and bottom by a suitable factor. For 1/√a multiply by √a/√a. For 1/(a + √b) multiply by the conjugate (a − √b), using the identity (a + √b)(a − √b) = a² − b, which has no root. Rationalising never changes the value of the fraction — it only rewrites it in a tidier form that is easier to add, compare, or use in further calculation.
10. Why all this matters
This chapter is the foundation of all later algebra and geometry. Knowing that the number line is “complete” (no gaps) lets us measure any length, plot any point, and solve equations whose answers are roots. Whenever you simplify a surd, convert a recurring decimal, or apply an exponent rule, you are using the ideas built here. Master these basics and the rest of Class 9 mathematics becomes far easier.
- Rational number: p/q, q ≠ 0, p and q integers.
- Terminating decimal ↔ denominator has only 2s and 5s as prime factors.
- √a × √b = √(ab) ; √a ÷ √b = √(a/b).
- (a + √b)(a − √b) = a² − b (conjugate identity).
- (√a + √b)(√a − √b) = a − b.
- ap × aq = ap+q ; (ap)q = apq ; a1/n = n-th root of a.
- Rational + Irrational = Irrational (always).
Express the recurring decimal 0.3636… (i.e. 0.36 repeating) as a fraction p/q.
- Let x = 0.363636…
- The repeating block “36” has 2 digits, so multiply by 100: 100x = 36.363636…
- Subtract the first equation from the second: 100x − x = 36.3636… − 0.3636…
- This gives 99x = 36.
- So x = 36/99 = 4/11 (dividing top and bottom by 9).
Rationalise the denominator of 1 ÷ (3 + √2) and simplify.
- The denominator is 3 + √2, so its conjugate is 3 − √2.
- Multiply numerator and denominator by the conjugate: [1 × (3 − √2)] ÷ [(3 + √2)(3 − √2)].
- Use the identity (a + √b)(a − √b) = a² − b: denominator = 3² − 2 = 9 − 2 = 7.
- Numerator stays 3 − √2.
- So the expression becomes (3 − √2) ÷ 7.
Remember the family order with “Næve Wizards Zap Quirky Real monsters” → N (natural) ⊂ W (whole) ⊂ Z (integers) ⊂ Q (rational) ⊂ R (real). Each box sits inside the next bigger one!
The biggest mistake: calling π rational because 22/7 “equals” it. 22/7 is only an approximation; π itself is irrational. Also, never forget that q ≠ 0 in p/q — division by zero is undefined!
Q1. Find two irrational numbers between 2 and 3.
Answer: Step 1 — note 2 = √4 and 3 = √9. Step 2 — any √(non-perfect-square) between 4 and 9 works. Step 3 — choose √5 = 2.236… and √7 = 2.645… . Both are non-terminating, non-recurring, so both are irrational and lie between 2 and 3. Two answers: √5 and √7.
Q2. Show that 0.999… (9 repeating) is equal to 1.
Answer: Step 1 — let x = 0.999… . Step 2 — one repeating digit, so multiply by 10: 10x = 9.999… . Step 3 — subtract: 10x − x = 9.999… − 0.999… , giving 9x = 9. Step 4 — so x = 1. Therefore 0.999… = 1 exactly.
Q3. Simplify (3 + √5)(3 − √5) and state whether the result is rational or irrational.
Answer: Step 1 — this is of the form (a + √b)(a − √b) = a² − b. Step 2 — here a = 3, b = 5, so the result = 3² − 5 = 9 − 5 = 4. Step 3 — 4 = 4/1, which is p/q form, so it is rational. Answer: 4 (rational).
Q4. Evaluate 641/2 + 271/3 − 161/4.
Answer: Step 1 — 641/2 = √64 = 8. Step 2 — 271/3 = cube root of 27 = 3. Step 3 — 161/4 = fourth root of 16 = 2. Step 4 — add and subtract: 8 + 3 − 2 = 9. Answer: 9.
- ✅ N ⊂ W ⊂ Z ⊂ Q ⊂ R — the families nest inside each other.
- ✅ Rational = p/q (q ≠ 0); decimals terminate or recur.
- ✅ Irrational decimals never terminate and never repeat (√2, π).
- ✅ Every real number is exactly one point on the number line, and vice-versa.
- ✅ Rationalise denominators using the conjugate (a + √b)(a − √b) = a² − b.
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