Class 10 Mathematics — Chapter 1: Real Numbers
1. Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of prime numbers, and this prime factorisation is unique apart from the order of the factors.
For example: 32760=23×32×5×7×13
The important idea is that once the prime factors are arranged in a fixed order, there is only one prime factorisation of a number.
Why this theorem matters
Prime factorisation helps us:
- determine HCF and LCM;
- study divisibility;
- prove the irrationality of certain numbers;
- understand when rational numbers have terminating or non-terminating decimal expansions.
2. HCF and LCM by Prime Factorisation
Write each number as a product of primes.
HCF
Take the smallest power of every prime factor common to all the numbers.
LCM
Take the greatest power of every prime factor appearing in the numbers.
For two positive integers a and b: HCF(a,b)×LCM(a,b)=a×b
This relation is valid for two positive integers.
For three numbers, however, HCF×LCM=product of the three numbers
in general.
3. A Useful Divisibility Idea
Prime factorisation can prove that certain numbers cannot have particular factors.
For instance, 4n=(22)n=22n
contains only the prime factor 2. Therefore it cannot be divisible by 5, so 4n cannot end in 0.
Key principle:
If a number is divisible by a prime, that prime must occur in its prime factorisation.
4. Irrational Numbers
A number is irrational if it cannot be expressed in the form qp,p,q∈Z,q=0.
Examples include: 2,3,5,π
and certain non-terminating, non-repeating decimals.
The chapter uses the Fundamental Theorem of Arithmetic to prove that numbers such as 2,3,5 are irrational.
5. Important Theorem
If p is a prime number and p∣a2,
then p∣a
where a is a positive integer.
Meaning
If a prime divides the square of a number, that prime must already divide the original number.
This result follows from the uniqueness of prime factorisation.
6. Proving a Number is Irrational
The chapter uses proof by contradiction.
General pattern
- Assume the number is rational.
- Express it as ba, where a and b have no common factor other than 1.
- Manipulate the equation using algebra.
- Use the prime-divisibility theorem.
- Show that the same prime must divide both a and b.
- This contradicts the fact that a and b are coprime.
- Therefore, the original assumption is false, so the number is irrational.
This method is demonstrated for 2 and 3.
Example of the core contradiction for 2
Assume 2=ba
with a,b coprime.
Then 2b2=a2.
Thus 2∣a2, so by the theorem, 2∣a.
Writing a=2c eventually gives 2∣b.
Hence 2 divides both a and b, contradicting their being coprime. Therefore, 2 is irrational
The same reasoning establishes the irrationality of 3.
7. Operations Involving Rational and Irrational Numbers
The chapter recalls two important results:
- Rational + irrational = irrational
- Rational − irrational = irrational
- Non-zero rational × irrational = irrational
- Irrational ÷ non-zero rational = irrational
Why?
The basic strategy is again contradiction.
For example, if 3−5
were rational, adding 5 would make 3 rational. But 3 is irrational. Hence, 3−5 is irrational
Similarly, if 32 were rational, dividing by 3 would make 2 rational, which is impossible.
Quick Revision Sheet
| Concept | Essential result |
|---|---|
| Fundamental Theorem of Arithmetic | Every composite number has a unique prime factorisation, apart from order |
| HCF | Product of smallest powers of common prime factors |
| LCM | Product of greatest powers of all involved prime factors |
| Two numbers | HCF × LCM = product of the numbers |
| Prime divisibility | If p∣a2, then p∣a |
| Irrational number | Cannot be written as p/q, q=0 |
| Proof of irrationality | Usually by contradiction |
| Rational + irrational | Irrational |
| Rational − irrational | Irrational |
| Non-zero rational × irrational | Irrational |
Chapter at a Glance
Prime factorisation → HCF/LCM → divisibility → prime-divisibility theorem → proof by contradiction → irrationality.