# 10th Class Mathematics Real Numbers Real Number

Real Number

Category : 10th Class

REAL NUMBER

FUNDAMENTALS

• Rational numbers: Numbers which can be written in the form of$\frac{p}{q}(q\ne 0)$, where p and q are integers, are called rational numbers.

Note: Every terminating decimal and non-terminating repeating decimal can be expressed as a rational number,

• Irrational numbers: Numbers which cannot be written in the form of $\frac{p}{q}$where p and q are integers and q ^ 0 are called irrational, numbers. Numbers which are not rational are called irrational numbers.
• Real numbers: The rational numbers and the irrational numbers together are called real numbers. Both rational & irrational numbers real line on number line.

Note: Any number that can be represented on a number line is called a real number.

• Lemma: A proven statement which is used to prove another statement is called a lemma.
• Euclid’s division lemma: For any two positive integers ‘a’ and ‘b’, there exist whole numbers ‘q’ and ‘r’ such that $a=bq+r,0\le r<b$

This is an extension of the idea:

Dividend =        Divisor x            quotient +         Remainder

(a)                      (b)                      (q)                      (r)

Remainder ‘r’ is always less than divisor (b) (This is basic principle of mathematics).

Note: Euclid’s division algorithm is stated only for positive integers, but can be extended/or all negative integers.

• Algorithm: An algorithm is a process of solving particular problems.
• Euclid’s division algorithm is used to find the Highest Common Factor (H.C.F.) of two numbers.

• Following is the procedure for finding H.C.F. using Euclid’s division algorithm: Suppose the two positive numbers are ‘a’ and ‘b’, such that a > b. Then the H.C.F. of ‘a’ and ‘b’ can be found by following the steps given.

(a) Apply the division lemma to find ‘q’ and ‘r’ where $a=bq+r,0\le r<b$.

(b) If r = 0, then$H.C.F.\,\,is\,\,b.\,\,If\,\,r\ne 0$, then apply Euclid's lemma to find ‘b’ and ‘r’.

• Continue steps (a) and (b) till r = 0. The divisor at this state will be H.C.F. (a, b). Also, H.C.F. (a, b) = H.C.F. (b, r).

• Fundamental theorem of Arithmetic: Every composite number can be expressed as a unique product of prime numbers. This is also called the prime factorization theorem.

Note: (i) The order in which the prime factors occur is immaterial.

In general, any composite number x, can be expressed as a product of prime numbers

Elementary Question: 1

Find HCF of 6 and 16.

Also verify that HCF of 18 and 48 is 3 times HCF of 6 and 16.

Sol.: 6 and 16: $6=2\times 3$

$16=2\times 2\times 2\times 2\text{ }\therefore {{\left( HCF \right)}_{1}}=2$

• and 48:    $18=2\times 3\times 3;$       $48=2\times 2\times 2\times 2\times 3$

$\therefore {{(HCF)}_{2}}=2\times 3=6;$  $\therefore {{\left( HCF \right)}_{2}}=\text{ }3\times {{\left( HCF \right)}_{1}}$

Elementary Question: 2

Do the above problem by Euclid is division algorithm.

• C.M. of $\frac{a}{b}$ and $\frac{c}{d}=\frac{L.C.M.\,\,of\,\,a\,\,and\,\,c}{H.C.F.\,\,of\,\,b\,\,and\,\,d}=\frac{L.C.M.\,\,(a,c)}{H.C.F.\,\,(b,d)}.$
• Some Important Result on Natural Numbers
• $1+2+3+----------+n=\frac{n(n+1)}{2}$
• ${{1}^{2}}+{{2}^{2}}+{{3}^{2}}+----------+{{n}^{2}}=\frac{n(n+1)(2n+1)}{6}$
• ${{1}^{3}}+{{2}^{3}}+{{3}^{3}}+----------+{{n}^{3}}={{\left[ \frac{n(n+1)}{2} \right]}^{2}}$
• ${{x}^{n}}+{{y}^{n}}$ is divisible by $\left( x\text{ }+y \right)$if n is odd.
• ${{x}^{n}}-{{y}^{n}}$is divisible by $\left( x\text{ }-\text{ }y \right)$ for all values of n.
• If a number is divisible by m and n, then it is always divisible by the LCM of m and n.
• ${{x}^{n}}-{{y}^{n}}$is divisible by $\left( x\text{ }+y \right)$ if n is even.

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