CBSE · Class 10 · Mathematics

Real Numbers

The whole chapter, 101 small steps. Each one is a single idea — read it, play with it, check it, move on.

Start from the beginning →

1.1 · Where real numbers sit

1Start here 2What are the real numbers?3Rational numbers▶ play 4Rationals stay rational▶ play 5Irrational numbers 6A few irrationals▶ play 7Together, no gaps✓ check 8What this chapter gives you 9Every integer is rational 10Finite decimals are fractions too▶ play 11No nearest neighbour 12Check: sort them✓ check 13Why any of this matters 14The two families, once more

1.2 · The Fundamental Theorem of Arithmetic

15Building numbers from primes▶ play 16Primes never run out 17The question 18Taking a number apart 19A factor tree 20Your turn to factorise✓ check 21It always works 22The Fundamental Theorem of Arithmetic 23What “unique” means✓ check 24A little history 25Putting it to work: can 4ⁿ end in 0?▶ play 26Check: can 6ⁿ end in 0?✓ check 27Reading HCF and LCM off the primes 28The HCF / LCM machine 29Worked: 6 and 20✓ check 30A handy shortcut▶ play 31Check: use the shortcut✓ check 32Three numbers at once 33But the shortcut breaks✓ check 34Spotting a composite number 35Worked: a hidden common factor✓ check 36LCM in the wild✓ check 37Section 1.2, in a breath 38What is a prime, exactly?39Two oddities✓ check 40Practice: factorise 156✓ check 41Practice: factorise 3825✓ check 42Practice: the tricky one, 7429✓ check 43Coprime numbers 44Practice: 510 and 92✓ check 45Practice: 336 and 54✓ check 46Practice: 12, 15 and 21✓ check 47Practice: 17, 23 and 29✓ check 48This has a name 49Practice: factorise 5005✓ check 50Poke: rebuild an LCM▶ play 51Poke: verify the shortcut▶ play 52Check: don’t swap the rules✓ check 53Check: coprime or not?✓ check 54Check: can 8ⁿ end in 0?✓ check 55Check: which is bigger?✓ check 56Poke: rebuild 140▶ play

1.3 · Revisiting irrational numbers

57Back to the irrationals 58The definition, precisely 59Some irrationals to hold▶ play 60First, a small helper 61Theorem 1.2 · the helper 62Why the helper is true✓ check 63The trick: proof by contradiction 64√2 · step 1: assume the opposite 65√2 · step 2: square it 66√2 · step 3: so 2 divides a 67√2 · step 4: substitute back 68√2 · step 5: the crash 69√2 is irrational✓ check 70The same machine: √3, √5, …✓ check 71Feel the endlessness▶ play 72It reaches combinations too 73Fact 1: rational ± irrational 74Fact 2: rational × irrational✓ check 75Worked: 5 − √3 is irrational 76Worked: 3√2 is irrational✓ check 77Your turn: three more✓ check 78Section 1.3, in a breath 79Why “lowest terms” is allowed 80Worked in full: √3 is irrational 81π, too▶ play 82Poke a combination▶ play 83Check: rational or not?✓ check 84That result has a number 85A quiet fact behind it 86But √4 is rational!87Check: where the proof breaks✓ check 88Worked: 1/√2 is irrational 89Poke: 3√2▶ play 90Poke: √15▶ play 91Check: one for the road✓ check

1.4 · Pulling it together

92The whole chapter, three ideas 93Idea 1 · the FTA 94Idea 2 · the helper 95Idea 3 · the irrationals 96A note to the curious 97You’ve done the chapter 98One mixed check✓ check 99Where this leads 100Self-test✓ check 101Come back anytime