1. Real Numbers

6 topics1 h

Chapter 1 of NCERT Mathematics for Class 10: Real Numbers. Two sections — Factorising into primes and Proving a number irrational. Six videos, 1 h in all.

Worked answers to the 10 exercise questions in this chapter: Exercise 1.1 · Exercise 1.2

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Part 1

Factorising into primes

14 minWhy a composite number has one prime factorisation and no other32760 opens into primes 47 different ways at the top; chase every one to the bottom, and 15015 distinct factor trees come out, every single one ending at the same eight primes.14 min · 13 partsहिंदी12 minReading HCF and LCM straight off two factorisationsHCF and LCM are usually taught as two rules that happen to sit next to each other: take the lower index for one, the higher index for the other. They are not two rules. They are one sentence about fitting inside, read once downward and once upward - and the explanation derives both from that sentence rather than asking you to remember which way round they go.12 min · 13 partsहिंदी14 minWhy the HCF-times-LCM shortcut works for two numbers but breaks for threeFor two numbers, HCF times LCM is their product - and that is not a coincidence, it is what happens when a column holding two indices hands both of them back. For three numbers the same argument stops working, and it stops working in a way you can predict to the exact factor before you divide anything: the middle index is the one that gets dropped.14 min · 16 partsहिंदी

Part 2

Proving a number irrational

11 minA prime dividing a square must divide the number itself4 divides 36 and 9 divides 36, yet neither divides 6. A divisor of a square is not generally a divisor of the number — the claim survives only when that divisor is prime.11 min · 13 partsहिंदी13 minAssume the square root of 2 is a fraction, and watch the assumption collapse99 over 70 squares to 9801 over 4900, one short of the 9800 it would need to equal 2 exactly. The gap can be shrunk forever and it never closes — getting closer is not arriving.13 min · 14 partsहिंदी14 minRe-running the same contradiction on other prime square rootsThe proofs that the square roots of 3 and 5 are irrational are not new proofs. They are one proof with the prime left as a slot. And the step everyone names as the reason it only works for primes is not that step at all - it holds for 6 too, on all 600 numbers tested. What actually forces primes is somewhere else, and the explanation goes and finds it.14 min · 14 partsहिंदी

What you will be able to do

  • Generate composite numbers by multiplying a chosen set of primes with repetition, and explain why that process alone never proves the reverse direction
  • State the divisibility test in index form: one number divides another when every prime index in the first is at most the matching index in the second
  • Verify the two-number identity on a worked pair and state what it lets you compute
  • Write the square of a number as a product of primes, given the primes of the number, and state which primes appear and how often
  • State what has to be shown to call a number irrational, and explain why it is a claim about every fraction at once
  • Extract the template from the square-root-of-2 proof: name every place the number 2 entered, and replace it with a slot

What this chapter assumes you already know

  • Prime and composite numbers, and why 1 counts as neither
  • Divisibility tests for 2, 3, 5, and trial division by primes up to the square root to decide whether a number is prime
  • Index notation: writing repeated factors as powers, and multiplying powers of the same base
  • Building a factor tree from earlier classes, and reading its leaves
  • That a number ending in 0 is divisible by both 2 and 5
  • HCF and LCM from earlier classes by listing factors, listing multiples, or common division, and what each word means
  • Index notation, including that a prime absent from a number can be written with index 0
  • Multiplying prime powers back into an ordinary number

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "The theorem just says numbers can be factorised."
  • "Different factor trees give different answers."
  • "3803 × 3607 is a prime factorisation."
  • "A factorisation is unique, full stop."
  • "4ⁿ never ends in 0 because I checked the first six powers."
  • "1 is prime, so it can go in the factorisation."
  • "HCF means multiply the small numbers, LCM means multiply the big ones."
  • "A prime in only one of the numbers goes into the HCF as well."