# 10th Class Mathematics Real Numbers

Real Numbers

Category : 10th Class

Real Numbers

• 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$q\ne 0$are called irrational numbers. In other words, numbers which are not rational are called irrational numbers.
• Real numbers: The rational numbers and the irrational numbers together are called real numbers.

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$\text{a}=\text{bq}+\text{r},0\le \text{r}<\text{b}$.

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

• Algorithm: An' algorithm is a process of solving particular problems.
• Euclid's division algorithm: is used to find the greatest common divisor (G.C.D.) or Highest Common Factor (H.C.F.) of two numbers.
• 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:
• Apply the division lemma to find 'q' and 'r' where$a=bq+r,0\le r<b$.
• lf$\text{r}=0$,then H.C.F. is b. If$\text{r}\ne \text{0}$, 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 unique prime factorization theorem.

Note: (i) The order in which the prime factors occur may differ. In general, any composite number x, can be expressed as a product of prime numbers as shown below.

$x={{p}_{1}}{{p}_{2}}{{p}_{3}}...........{{p}_{n}}$where${{p}_{1}},{{p}_{2}},{{p}_{3}},...........{{p}_{n}}$are primes in ascending order.

• If 'p' is a prime, 'a' is a positive integer, and if 'p' divides${{a}^{2}}$, then 'p' divides 'a'. Also, if 'p' divides${{a}^{3}}$, then 'p' divides 'a'.
• If 'a' is a terminating decimal, then 'a' can be expressed as $\frac{p}{q}(q\ne 0)$, where 'p' and 'q' are co primes and the prime factorization of q is of the form ${{2}^{m}}{{5}^{n}}$, (where m and n are whole numbers.).
• If $\frac{p}{q}$is a rational number and q is not of form ${{2}^{m}}{{5}^{n}}(m,n\in W)$, then $\frac{p}{q}$has a non-terminating repeating decimal expansion.
• C.F. of two numbers is the product of the smallest power of each common prime factor in the numbers.
• M. of two numbers is the product of the greatest power of each prime factor involved in the numbers.
• For any two numbers 'a' and 'b', L.C.M. (a, b) x H.C.F. $\text{(a},\text{ b)}=\text{a}\times \text{b}$.

That is, the product of two numbers is equal to the product of their LC.M. and H.C.F.

• but,$H.C.F.(p,q,r)\times L.C.M.(p,q,r)\ne pqr$, where p, q and r are positive integers.
• If H.C.F. (a, b) = 1, then 'a' and 'b' are said to be co-prime or relatively prime.
• If $\text{a}=\text{bq}+\text{r}$ and r < b, then H.C.F. (a, b) - H.C.F. (b, r).
• For all a, b, m e N, H.C.F. (am, bm) = m [H.C.F. (a, b)].

In other words, if 'a', 'b' and 'm' are natural numbers, then H.C.F. (am, bm) is m times H.C.F. of 'a' and 'b'.

• C.F. of three numbers is the H.C.F. of the H.C.F. of any two numbers and the third number. i.e., H.C.F. (p, q, r) = H.C.F. [H.C.F. (p, q), r].
• C.M. of two co-prime numbers or two prime numbers is the product of the numbers.
• C.M. of fractions or rational numbers $=\frac{A}{B}$, where A = L.C.M. of numerators and B = H.C.F. of denominators, i.e., L.C.M. of$\frac{a}{b}$and$\frac{c}{b}=\frac{L.C.M.\,of\,a\,and\,c}{H.C.F.\,of\,b\,and\,d}=\frac{L.C.M.(a,c)}{H.C.F.(b,d)}$
• C.F. of fractions or rational numbers$=\frac{p}{q}$ where P = H.C.F. of numerators and Q = L.C.M. of denominators, i.e., H.C.F. of$\frac{a}{b}$and$\frac{c}{d}=\frac{H.C.F.\,of\,a\,and\,c}{L.C.M.\,of\,b\,and\,d}=\frac{H.C.F.(a,c)}{L.C.M.(b,d)}$

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##### Notes - Real Numbers

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