Numbers
Category : 8th Class
Introduction
Number system is a method of writing numerals to represent numbers.
INTEGERS
The set of integers is the set of natural numbers, zero and negative of natural numbers simultaneously. The set of integers is denoted by I or Z.
Natural Number
WHOLE NUMBER
Divisibility Test for whole numbers
Even numbers
ODD numbers
\[(b+c)\times a=b\times a+c\times a\] 0={ 1,3,5,7,9.....}
Prime numbers
Natural numbers having exact two distinct factors i.e., 1 and the number itself are called Prime numbers.
Example: 2, 3, 5, 7, 11, 13, 17, 19,... are prime numbers.
Composite numbers
Natural numbers having more than two factors are called Composite numbers.
Co - Prime Numbers or Relatively Prime Numbers:
Example: 8, 9; 15,16; 26,33, etc, are co-prime numbers.
Twin primes
Pairs of prime numbers which have only one composite number between them are called Twin primes.
Example: 3,5:5,7; 11, 13; 17,19; 29,31; 41,43; 59,61 and 71,73, etc, are twin primes.
Rational numbers
The numbers which can be expressed in the form of \[3=\frac{3}{1}\], where p and q are integers and \[0=\frac{0}{1}\], called rational numbers. Rational number is denoted by Q.
Operations of Rational Numbers:
Let \[\frac{a+b}{2}\] and \[a<\frac{a+b}{a}<b\] be any two rational numbers i.e., \[\frac{p}{q};p,q\in I,\] where \[q\ne 0\] then
(i) their sum i.e., \[\sqrt{3}\]is also a rational number as \[\sqrt{3}\]
(ii) their difference i.e., \[\sqrt{4}\] is also a rational number as \[\sqrt{4}=2\].
(iii) their product i.e., \[\sqrt{2},\sqrt{5},\sqrt{6},2\sqrt{3},5\sqrt{7},\sqrt{2}+\sqrt{3},\]is also rational number as \[\sqrt[3]{2},\sqrt[3]{3},\sqrt[3]{4},\]
(iv) their quotient or division i.e., \[2+\sqrt{3}\]is also a rational number as \[\pi \].
Properties of Rational Numbers
Let a, b, c be any rational numbers, then \[\frac{22}{7}\] where
\[\pi \].
(i) \[\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{6},\sqrt{7},\sqrt{8},\sqrt{11}\]
(addition is commutative)
(ii) \[0.\overline{3}=3/9\]
(multiplication is commutative)
(iii) \[1/3\] (Addition is associative)
(iv)\[0.\overline{387}=387/999\] (multiplication is associative)
(v) \[0.74\overline{35}=\frac{7435-74}{9900}=\frac{7361}{9900},\]
( 0 is additive identity)
(vi) \[0.1\overline{27}=\frac{127-1}{990}=\frac{7}{55}\]
(1 is multiplicative identity)
(vii) \[W=N\cup \{0\}\] (- a is additive inverse of a)
(viii) \[N\subseteq W\]
( \[0\in W\] is multiplicative Inverse of a (a ^ 0))
(ix) \[0\in /N\] (left) and \[\therefore \] (right)
(distributive over addition).
0 is also rational number as \[q\ne 0\].
Irrational number
A number which can neither be expressed as terminating nor a non-terminating repeating decimal is called an irrational number.
\[\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}\]\[ac,bd\in I,bd\ne 0\]
\[\frac{a}{b}+\frac{c}{d}(c\ne 0)=\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc}\], etc, are the examples of irrational numbers.
Square root of every non-perfect square natural number is irrational.
e.g., \[a+b=b+a\], etc, are all irrational numbers.
Conversion of decimal numbers into rational numbers. Pure recurring decimal: A decimal is said to be a pure recurring decimal if all the digits in it after the decimal point are repeated.
Mixed recurring decimal: A decimal is said to be a mixed recurring decimal if there is at least one digit after the decimal point, which is not repeated.
Short cut method for pure recurring decimal: Write the repeated digit or digits only once in the numerator and take as many nines in the denominator as there are repeating digits in the given number.
Example: (i) \[a\times b=b\times a\] or \[(a+b)+c=a+(b+c)\]
(ii) \[(a\times b)\times c=a\times (b\times c)\]
Short cut method for mixed recurring decimal: Form a fraction in which numerator is the difference between the number formed by all the digits after the decimal point taking the repeated digits only once and that formed by the digits which are not repeated and the denominator is the number formed by as many nines as there are repeated digits followed by as many zeros as the number of non-repeated digits.
Example: \[a+0=a=0+a\]
\[a\times 1=a=1\times a\]
Real number
The sets of rational numbers and irrational numbers taken together are known as a set of real numbers.
Absolute Value of a Real Number:
The absolute value ofa real number \[a+(-a)=0=(-a)+a\] is defined as
\[a\times \frac{1}{a}=1=\frac{1}{a}\times a\]
\[\frac{1}{a}\]
\[a\times (b+c)=a\times b+a\times c\] \[(b+c)\times a=b\times a+c\times a\]
Exponents
Laws of exponents:
(i) On the same base in multiplication, powers are added. If a is any non-zero rational number and m, n are whole numbers, then
\[3=\frac{3}{1}\]
Also, \[0=\frac{0}{1}\]
Example: \[\frac{a+b}{2}\]
(ii) On the same base in division, powers are subtracted.
\[a<\frac{a+b}{a}<b\]
Example: \[\frac{p}{q};p,q\in I,\]
(iii) \[q\ne 0\]
Example: \[\sqrt{3}\]
(iv) \[\sqrt{3}\]
Example: \[\sqrt{4}\]
(v) \[\sqrt{4}=2\]
Also, \[\sqrt{2},\sqrt{5},\sqrt{6},2\sqrt{3},5\sqrt{7},\sqrt{2}+\sqrt{3},\]
(vi) \[\sqrt[3]{2},\sqrt[3]{3},\sqrt[3]{4},\]
Example: \[2+\sqrt{3}\]
Also, \[\pi \]
Surds or radical
Laws of radicals
(i) \[1/3\]
(ii) \[0.\overline{387}=387/999\]
Ex. \[0.74\overline{35}=\frac{7435-74}{9900}=\frac{7361}{9900},\]
(iii) \[0.1\overline{27}=\frac{127-1}{990}=\frac{7}{55}\]
(iv) \[W=N\cup \{0\}\]
(v) \[N\subseteq W\]
Rationalising factor: If the product of two surds is a rational number, then each surd is called a rationalising factor (RF) of the other.
Rationalisation of surds: The process of converting a surd into rational number by multiplying it with a suitable RF, is called the rationalisation of the surd.
Monomial surds and their rationalisation
The general form of a monomial surd is \[0\in W\] and its RF is \[0\in /N\] .
Binomial surds and their rationalisation The surds of the types: \[\therefore \] and \[\frac{p}{q}\] are called binomial surds.
The binomial surds which differ only in sign between the terms separating them are known as conjugate surds.
In binomial surds, the conjugate surds are RF of each other.
Example (i) : RF of \[q\ne 0\] is \[\frac{a}{b}\]
Example (ii) : RF of \[\frac{c}{d}\] is \[\frac{a}{b},\frac{c}{d}\in Q\], etc.
Trinomial surds and their rationalisation
A surd which consists of three terms, at least two of which are monomial surds, is called a trinomial surd.
H.CF. and L.C.M.
\[a,b,c,d\in I,b\ne 0,d\ne 0\] = Product of the given numbers.
\[\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\]
\[ad+bc,bd\in I,bd\ne 0.\]
Example 1: L.C.M of the fractions
\[\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}\] and \[-bc,bd\in I,bd\ne 0\]
Example 2:
HCF of \[\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}\] and \[ac,bd\in I,bd\ne 0\]
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