Are these Real Numbers notes free?

Yes. The full 4-page Real Numbers sheet is free inside CapCompass, which runs in your browser with no account needed.

What does the Real Numbers revision sheet cover?

Prime factorisation, The Fundamental Theorem of Arithmetic, every composite number, primes collected as powers, and 12, 18 the same way, Turning a number into primes, Prime factors hanging from the number, HCF, LCM, HCF and LCM from the factorisation, smallest power of each shared prime, largest power of every prime seen.

Is this for Class 10 CBSE Mathematics?

It is written from the NCERT syllabus, so it serves CBSE and any board that follows NCERT for Mathematics.

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Real Numbers — Class 10 CBSE Mathematics revision notes

A 4-page visual revision sheet for Real Numbers, written for Class 10 CBSE Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.

Real Numbers revision notes, page 1 of 4 — A preview of page 1 — all 4 pages open full size in the app.

Real Numbers in one look

primes are the building blocks, and some numbers refuse to be fractions

The Fundamental Theorem of Arithmetic

every composite number n=p_1× p_2× p_3×…× p_k
primes collected as powers 3825=3^2×5^2×17
and 12, 18 the same way 12=2^2×3, 18=2×3^2

Turning a number into primes

a prime has exactly two factors, 1 and itself. 1 is neither prime nor composite
2 is the only even prime, so every even composite carries at least one 2
keep dividing by the smallest prime that fits: 2, then 3, then 5, then 7
stop once the quotient is itself prime: 3825 → 1275 → 425 → 85 → 17

HCF and LCM from the factorisation

smallest power of each shared prime HCF(12,18)=2×3=6
largest power of every prime seen LCM(12,18)=2^2×3^2=36
the product law, two numbers only HCF(a,b)× LCM(a,b)=a× b
check it 6×36=216=12×18

HCF and LCM: the special forms

a and b share no factor, HCF=1 — LCM is the plain product
a divides b exactly — HCF is the smaller, LCM the larger
a=b — both collapse to the number itself
for 2, 3 and 4 — product law fails

Page 1 of 4. The full sheet, all 4 pages, opens free in the app.

What this Real Numbers sheet covers

Prime factorisation
The Fundamental Theorem of Arithmetic
every composite number
primes collected as powers
and 12, 18 the same way
Turning a number into primes
Prime factors hanging from the number
HCF
LCM
HCF and LCM from the factorisation
smallest power of each shared prime
largest power of every prime seen
the product law, two numbers only
check it
HCF and LCM: the special forms
HCF and LCM of 24 and 90, by factorising
Where HCF and LCM actually turn up
Three numbers at once
factorise all three
and the product law is no help
Two bells, one meeting
Why a power of 6 can never end in 0
ending in 0 means a multiple of
but 6^n splits into 2s and 3s only
so no 5 ever joins the list
4^n is shut out for the same reason
Numbers that share nothing
Checking the two answers you wrote
Symbols
Revisiting Irrational Numbers
Rational vs irrational
Rational
Irrational
Why √2 is irrational
Mixing rational with irrational
The same proof, dressed differently
The one line both proofs lean on
Primes worth knowing by sight
There is no last prime
suppose the primes stopped at p
now look one place further up
each prime on the list leaves remainder
so a prime outside the list must exist
Traps that cost marks
Tricks that save time

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