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.

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**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.

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**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\[\text{a}=\text{bq}+\text{r},0\le \text{r}<\text{b}\].

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**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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