# NOTES

## 1. REAL NUMBERS

• Rational number: The number, which is written in the form of p/q where p,q are integers q not equal to zero is called a rational number. It is denoted by Q.

• Irrational numbers:- the number, which is not rational is called an irrational number. It is denoted by Q’ or S.

• Euclid division lemma:- For any positive integers a and b, then q, r are integers exists uniquely satisfying the rules a = bq + r, 0 ≤ r < b.

• Prime number:- The number which has only two factors 1 and itself is called a prime number. (2, 3, 5, 7 …. Etc.)

• Composite number:- the number which has more than two factors is called a composite number. (4, 6, 8, 9, 10,… etc.)

• Co-prime numbers:- Two numbers  said to be co-prime numbers if they have no common factor except 1. [Ex: (1, 2), (3, 4), (4, 7)…etc.]

• To find HCF, LCM by using prime factorization method:  H. C.F= product of the smallest power of each common prime factors of given numbers. L.C.M = product of the greatest power of each prime factor of given numbers.

• In p/q, if prime factorisation of q is in form 2m 5n, then p/q is terminating decimal. Otherwise non-terminating repeating decimal.
• Decimal numbers with the finite no. of digits is called terminating Decimal numbers with the infinite no. of digits is called non-terminating decimal. In a decimal, a digit or a sequence of digits in the decimal part keeps repeating itself infinitely. Such decimals are called non-terminating repeating decimals.

• ‘p’ is a prime number and ‘a’ is a positive integer, if p divides a2, then p divides a.

• If ax = N then x = ${log_{a}}^{N}$

(i) log (xy) = log(x) + log(y)  (ii) log (x/y) = log( x) – log( y) (iii) log (xm ) = m log (x

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# Math problems asked in the CBSE board and entrance examinations.

## 1. RATIONAL AND IRRATIONAL NUMBERS

Natural numbers: counting numbers 1, 2, 3… called Natural numbers. The set of natural numbers is denoted by N.

N = {1, 2, 3…}

Whole numbers: Natural numbers including 0 are called whole numbers. The set of whole numbers denoted by W.

W = {0, 1, 2, 3…}

Integers: All positive numbers and negative numbers including 0 are called integers. The set of integers is denoted by I or Z.

Z = {…-3, -2, -1, 0, 1, 2, 3…}

Rational number: The number, which is written in the form of, where p, q are integers and q ≠ o is called a rational number. It is denoted by Q.

∗ In a rational number, the numerator and the denominator both can be positive or negative, but our convenience can take a positive denominator.

Ex: – $\inline&space;\fn_jvn&space;-\frac{2}{3}$ can be written as $\inline&space;\fn_jvn&space;\frac{-2}{3}=\frac{2}{-3}$  but our convenience we can take $\inline&space;\fn_jvn&space;\frac{-2}{3}$

Equal rational numbers:

For any 4 integers a, b, c, and d (b, d ≠ 0), we have $\inline&space;\fn_jvn&space;\frac{a}{b}=\frac{c}{d}$ ⇒ ad = bc

The order of Rational numbers:

If  are two rational numbers such that b> 0 and d > 0 then $\inline&space;\fn_jvn&space;\frac{a}{b}>&space;\frac{c}{d}$ ⇒ ad > bc

The absolute value of rational numbers:

The absolute value of a rational number is always positive. The absolute value of  $\inline&space;\fn_jvn&space;\frac{a}{b}$ is denoted by $\inline&space;\fn_jvn&space;\left&space;|&space;\frac{a}{b}&space;\right&space;|$ .

Ex: – absolute value of $\inline&space;\fn_jvn&space;-\frac{2}{3}=\frac{2}{3}$

To find rational number between given numbers:

• Mean method: – A rational number between two numbers a and b is $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{a&space;+&space;b}{2}$

Ex: – insert two rational number between 1 and 2

1 <  $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{1&space;+&space;2}{2}$ < 2   ⟹     1 <  $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{3}{2}$  < 2

1 <  $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{3}{2}$ $\inline&space;\dpi{120}&space;\fn_jvn&space;<&space;\frac{\frac{3}2{+2}}{2}$< 2   ⟹   1 < $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{3}{2}<&space;\frac{7}{4}$ $\dpi{120}&space;\fn_jvn&space;<$  2

To rational numbers in a single step: –

Ex:- insert two rational number between 1 and 2

To find two rational numbers, we 1 and 2 as rational numbers with same denominator 3

(∵ 1 + 2 = 3)

1 =   $\fn_jvn&space;\frac{1\times&space;3}{3}$  and 2 = $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{2\times&space;3}{3}$

$\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{3}{3}\left&space;(&space;1&space;\right&space;)<&space;\frac{4}{3}<&space;\frac{5}{3}<&space;\frac{6}{3}\left&space;(&space;2&space;\right&space;)$

Note: – there are infinitely many rational numbers between two numbers.

The decimal form of rational numbers

∗ Every rational number can be expressed as a terminating decimal or non-terminating repeating decimal.

Converting decimal form into $\dpi{120}&space;\fn_jvn&space;\frac{p}{q}$  form:

1.Terminating decimals: –

1.2 =$\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{12}{10}=\frac{6}{5}$

1.35 =$\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{135}{100}=\frac{27}{20}$

2.Non-Terminating repeating decimals: –

Irrational numbers:

• The numbers which are not written in the form of  $\dpi{120}&space;\fn_cm&space;\frac{p}{q}$, where p, q are integers, and q ≠ 0 are called rational numbers. Rational numbers are denoted by QI or S.
• Every irrational number can be expressed as a non-terminating decimal or non-repeating decimal.

Ex:- $\dpi{120}&space;\fn_cm&space;\sqrt{2},\,&space;\sqrt{5},\pi$ and so on.

• Calculation of square roots:
• There is a reference of irrationals in the calculation of square roots in Sulba Sutra.
• Procedure to finding $\dpi{120}&space;\fn_cm&space;\sqrt{2}$ value:

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