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Real Number: Formula, Theorems and Examples

Real number

Real Number

One of the fact you must know about real number is “It is named as real number because it is not imaginary.” It is a part of imaginary number.

Formally real number is defined as the number which can be expressed on number line. Any number you are thinking beyond is real number except zero and neglecting Imaginary number.

There is a unique real number corresponding to every point on the number line. Also, corresponding to each real number, there is a unique point on the number line.

There are some types of real numbers:

  1. Natural Number
  2. Whole Number
  3. Integers
  4. Rational Number
  5. Irrational Number
  6. Fraction

Formulas Related to Real Number:

  1. \(\sqrt{ab}\:=\: \sqrt{a}\sqrt{b}\)
  2. \(\sqrt{\frac{a}{b}} \:=\:\frac{\sqrt{a}}{\sqrt{b}}\)
  3. \((\sqrt{a}\:+\:\sqrt{b})(\sqrt{a}\:-\:\sqrt{b})\) = a – b
  4. (a + \(\sqrt{b}\))(a – \(\sqrt{b}\)) = a2 – b
  5. (\(\sqrt{a}\:+\:\sqrt{b})\)2 = a + 2\(\sqrt{ab}\) + b
  6. To rationalise the denominator of \(\frac{1}{\sqrt{a}\:+\:b}\) multiply it by \(\frac{\sqrt{a}\:-\:b}{\sqrt{a}\:-\:b}\), where a and b are integers.
  7. ap . bq = ap+q
  8. (ap)q = apq
  9. \(\frac{a^{p}}{a^{q}}\) = ap-q
  10. apbp = (ab)p

Euclid’s division lemma:

[A lemma is a proven statement used for proving another statement].

If there are two positive integers a and b, there exist whole numbers q and r satisfying
a = bq + r   [0 ≤ r < b].

Euclid’s division algorithm:

[An algorithm is a series of well-defined steps which gives a procedure for solving a type of problem].

This algorithm provided the way to calculate HCF.It is based on on Euclid’s division lemma.

According to this algorithm, the HCF of any two positive integers a and b, with a > b, is obtained as follows:

Step 1: Apply the division lemma to find q and r where a = bq + r, 0 ≤ r < b.

Step 2: If r = 0, the HCF is b. If r ≠ 0, apply Euclid’s lemma to b and r.

Step 3: Continue the process until the remainder is zero. The divisor at this stage will be HCF (a, b). Also, HCF(a, b) = HCF(b, r).

Fundamental Theorem of Arithmetic:

It is also known as “unique factorization theorem” or “unique prime factorization theorem”.

Every composite number can be expressed (factorised) as a product of prime numbers, and this factorisation is unique, apart from the order in which the prime factors occur.

Examples related to a real number:

Find 2 rational number between \(\frac{3}{5}\) and 1.


1 can be written as \(\frac{5}{5}\)
let’s multiply both numerator and denominator of both fractions by 2
i.e. \(\frac{6}{10}\) and \(\frac{10}{10}\)
The two rational number between them are \(\frac{7}{10}\), \(\frac{8}{10}\).

Rationalise the denominator of \(\frac{1}{\sqrt{2}}\).


As stated above to rationalize its denominator first we multiply \(\sqrt{2}\) to both numerator and denominator.
\(\frac{1}{\sqrt{2}}\) x \(\frac{\sqrt{2}}{\sqrt{2}}\)
= \(\frac{\sqrt{2}}{2}\)

More from Number System
Integers Rational Number
Tricks for Decimal Ratio and Proportion
Binomial Theorem Formulas Factorisation
Learn Set Theory Permutation and Combination
Linear Inequalities Quick Revision
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