Real Number Class 10 Maths Chapter 1 – Real Numbers

Real Number Class 10 Maths Chapter 1 Real Numbers NCERT Solutions

Real Number Class 10 Maths Chapter 1 – NCERT Solutions for Real Numbers

Real Number Class 10 Maths Chapter 1 – All Important Formulas

1. Euclid’s Division Lemma

For any two positive integers a and b, there exist unique whole numbers q and r such that:$$a = bq + r$$a=bq+r

Where:$$0 \le r < b$$0≤r<b

• a = Dividend
• b = Divisor
• q = Quotient
• r = Remainder

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2. HCF and LCM Relationship

For any two positive integers a and b:$$HCF(a,b) \times LCM(a,b) = a \times b$$HCF(a,b)×LCM(a,b)=a×b

or$$LCM = \frac{a \times b}{HCF}$$LCM=HCFa×b​

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3. Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of prime numbers.

Example:$$24 = 2 \times 2 \times 2 \times 3$$24=2×2×2×3 $$60 = 2^2 \times 3 \times 5$$60=22×3×5

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4. Decimal Expansion of Rational Numbers

For a rational number:$$\frac{p}{q}$$qp​

where $q \ne 0$q=0,

Terminating Decimal

If the prime factorization of q is of the form:$$2^m \times 5^n$$2m×5n

then the decimal expansion terminates.

Examples:$$\frac{1}{2}=0.5$$

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Non-Terminating Recurring Decimal

If q has any prime factor other than 2 and 5, then the decimal expansion is non-terminating recurring.

Examples:$$\frac{1}{3}=0.333…$$$$\frac{2}{7}=0.285714…$$

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5. Irrational Numbers

Numbers that cannot be written in the form:$$\frac{p}{q}$$qp​

where $q \ne 0$q=0.

Examples:$$\sqrt{2},\ \sqrt{3},\ \sqrt{5},\ \pi$$2​, 3​, 5​, π

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6. Rational Numbers

Numbers that can be written in the form:$$\frac{p}{q}$$qp​

where:$$q \ne 0$$q=0

Examples:$$\frac{2}{3},\ \frac{-5}{7},\ 0.75$$32​, 7−5​, 0.75

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Quick Revision Formula Sheet

Topic	Formula
Euclid’s Division Lemma	$a=bq+r,\;0\le r<b$a=bq+r,0≤r<b
HCF × LCM	$HCF \times LCM = a \times b$HCF×LCM=a×b
LCM	$\frac{a\times b}{HCF}$HCFa×b​
Rational Number	$\frac{p}{q}, q\neq0$qp​,q=0
Terminating Decimal	Denominator = $2^m \times 5^n$2m×5n
Non-Terminating Recurring Decimal	Denominator has prime factors other than 2 and 5

Important for Exams:
The most frequently asked formulas in Chapter 1 are:

1. $a = bq + r$a=bq+r
2. $HCF \times LCM = a \times b$HCF×LCM=a×b
3. Denominator $= 2^m \times 5^n$=2m×5n ⇒ Terminating Decimal.