# 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:**

- Natural Number
- Whole Number
- Integers
- Rational Number
- Irrational Number
- Fraction

**Formulas Related to Real Number:**

- \(\sqrt{ab}\:=\: \sqrt{a}\sqrt{b}\)
- \(\sqrt{\frac{a}{b}} \:=\:\frac{\sqrt{a}}{\sqrt{b}}\)
- \((\sqrt{a}\:+\:\sqrt{b})(\sqrt{a}\:-\:\sqrt{b})\) = a – b
- (a + \(\sqrt{b}\))(a – \(\sqrt{b}\)) = a
^{2}– b - (\(\sqrt{a}\:+\:\sqrt{b})\)
^{2}= a + 2\(\sqrt{ab}\) + b **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.- a
^{p }. b^{q}= a^{p+q} - (a
^{p})^{q}= a^{pq} - \(\frac{a^{p}}{a^{q}}\) = a
^{p-q} - a
^{p}b^{p }= (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.*

**Solution:**

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 \(\frac{1}{\sqrt{2}}\).*

**Solution:**

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

\(\frac{1}{\sqrt{2}}\) x \(\frac{\sqrt{2}}{\sqrt{2}}\)

= \(\frac{\sqrt{2}}{2}\)

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