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CBSE Class 10 · Mathematics

Real Numbers

Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.

This lesson follows the official chapter “Real Numbers” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 10
  • Medium level
  • 8 concepts

1The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic says a whole number greater than 1 writes as a product of primes in only one way, apart from order. 84 = 2² × 3 × 7. A different-looking product that uses the same primes is the same write shuffled, not a second theorem.

12 = 2×6 is not finished — 6 is not prime.

Figure. A composite has one prime writing except for order. 84 is 2 squared times 3 times 7 — those primes, those powers, no other set.

How it works

  1. Factor until every factor is primeThe write.
  2. Collect powers2² × 3 × 7.
  3. Say this list of primes is unique up to orderThe theorem.

Prime write of 84

Write 84 as a product of primes.

  • Split84 = 2 × 42 = 2 × 2 × 21
  • 213 × 7
  • Collect2² × 3 × 7

Pro tip. Stop only when every factor is prime.

2Revisiting Irrational Numbers

An irrational at this class is a real that is not a/b: √2, √3 as taught. A nonzero rational times an irrational stays irrational (as taught). √4 is 2 — not this drawer. Bounding 1.4 < √2 < 1.5 is honest; writing √2 = 22/7 is the π-steal.

A long decimal you have not tested is not yet proved irrational.

Figure. √2 sits between 1 and 2. It is a place on the line, not a whole-number tick and not a ratio of two integers.

How it works

  1. Name a taught irrational√2 or √3.
  2. Refuse a fake-close fractionA box, not equality.
  3. If you multiply by a nonzero rational, keep it irrational as taughtThe revisit.

3A definition is a test you can run

Prime factorisation is a test: every factor prime, and the multiset of primes is the write. If a composite remains, the theorem has not run.

A pretty factor-tree that stops at 9 is not finished.

Figure. Euclid’s lemma is a test: divide 48 by 18, get two full 18s and a remainder 12 that is smaller than 18. The remainder must stay strictly less than the divisor.

How it works

  1. Write the productThe object.
  2. Check each factor is primeThe test.
  3. Then uniqueness has meaningThe definition ran.

4Name the given before the unknown

The given is the whole number (or the √-write). The unknown is the prime product or the drawer. Copy 84 before you drop a 2.

Using 48 because it “looks like 84” is a silent given.

Figure. Write the two given integers first. The HCF is the unknown that those two numbers determine; do not start from a guessed answer.

How it works

  1. Copy the numberThe given.
  2. Name prime-write or rational/irrationalThe unknown.
  3. Then factor or sortGiven first.

5One worked case is enough at this class

One prime-write is enough. A page of the same 84 clone does not add a new uniqueness.

Ten twins of 2²×3×7 are still one idea.

Figure. One Euclid run: replace (a, b) by (b, a mod b) until the remainder is 0. The last non-zero remainder is 6, the HCF of 48 and 18.

How it works

  1. Factor one n > 1The case.
  2. A second only as a checkOptional.
  3. StopThis class.

6A miss: swapping the school name for the picture

A barcode photo is a setting. The maths is a unique prime product. Stripes are not primes.

A lock-icon is not FTA.

Figure. 7/5 is a ratio and writes as a terminating decimal. √2 is a place on the line that is not a ratio of two integers — do not swap those names.

How it works

  1. Keep the photo as a storyA setting.
  2. Write the prime productThe maths.
  3. Then uniquenessDo not swap.

7Check by the opposite action or the opposite test

Check a prime-write by multiplying back, and check “irrational” by asking whether a/b was shown. Expanding 2²×3×7 must return 84.

A factor-tree that never multiplies back is unchecked.

Figure. The check for two positives: HCF times LCM rebuilds the product. 6 times 36 is 216, and 12 times 18 is 216. Unequal bars mean a factor was dropped.

How it works

  1. State 2²×3×7 or “√2 is irrational”The claim.
  2. Multiply back, or refuse a fake-closeThe check.
  3. Mend if they disagreeHonest.

8Keep the claim at this chapter, not the next

This chapter’s claim stops at unique primes and a school irrational-sort. It does not prove every real is constructible, and it does not steal the polynomial chapter’s zeroes.

A page that already “solves” quadratics here misses the next official chapter.

Figure. This chapter locates √2 and proves it is not a ratio. Completeness and decimal expansions of every real wait for a later book. Keep the claim here.

How it works

  1. Keep FTA and the irrational drawerThis chapter.
  2. Leave zeroes and graphs for polynomialsThe next.
  3. Do not treat the map as already answeredClaim-size.
84 as primes is
  1. 2² × 3 × 7
  2. 2 × 42 as finished
  3. √2

Every factor prime; 42 is not.

Notes

  • The official chapter title is “Real Numbers”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • n > 1 has a unique prime factorisation (up to order)

Recap

Hold these pegs from the official chapter “Real Numbers”. The wording is ExamMaster’s teaching, not a textbook recap.

The Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic says a whole number greater than 1 writes as a product of primes in only one way, apart from order.
Revisiting Irrational Numbers
An irrational at this class is a real that is not a/b: √2, √3 as taught.
A definition is a test you can run
Prime factorisation is a test: every factor prime, and the multiset of primes is the write.
Name the given before the unknown
The given is the whole number (or the √-write).
One worked case is enough at this class
One prime-write is enough. A page of the same 84 clone does not add a new uniqueness.
A miss: swapping the school name for the picture
A barcode photo is a setting. The maths is a unique prime product. Stripes are not primes.

Practise Real Numbers

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