8th Class Mathematics Related to Competitive Exam Numbers

Introduction

 

Number system is a method of writing numerals to represent numbers.

• In Decimal number system, ten symbols 0, 1, 2,3,4,5,6,7,8, and 9 are used to represent any number.
• Each of the symbols 0, 1, 2,3,4,5,6,7,8 and 9 is called a digit or a figure.
• In our number system, we think collections by tens or we speak counting in tens.
• Ten is called the base of our number system.

 

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.

• Z= {.......-4,-3,-2, -1, 0, 1,2,3,4...}

 

Natural Number

• Counting numbers 1, 2, 3, 4, 5, are called Natural numbers. Smallest natural number is 1 and there is no largest natural number, i.e., the set of natural numbers is infinite.
• The set of natural numbers is denoted by N i.e., N= {1, 2,3,4,5 ...}
• 1 is the smallest natural number.
• Any natural number can be obtained by adding ' 1' to its previous natural number.

 

WHOLE NUMBER

• All natural numbers together with zero are called whole numbers, as 0, 1,2,3,4... are whole number.
• The set of whole numbers is denoted by W, i.e., W= {0, 1, 2,3,4,5.....}
• \[a+(-a)=0=(-a)+a\], where N is the set of natural numbers.
• 0 is the smallest whole number.
• There is no largest whole number i.e., the number of the elements in the set of whole numbers is infinite.
• Every natural number is a whole number. i.e., \[a\times \frac{1}{a}=1=\frac{1}{a}\times a\] i.e., N is a subset of W.
• 0 is a whole number, but not a natural number, i.e., \[\frac{1}{a}\]but \[a\times (b+c)=a\times b+a\times c\]
• N is also a proper subset of W, i.e., N c W.

 

Divisibility Test for whole numbers

• A number is divisible by 2 if the unit place digit in it is an even digit.
• A number is divisible by 3 if the sum of its digits is
• multiple of 3.
• A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
• Anumberisdivisibleby5ifitends in 0 or 5.
• A number is divisible by 6, if it is divisible by 2 and 3 both.
• A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
• A number is divisible by 9 if the sum of its digits is divisible by 9.
• A number is divisible by 10 if it ends in zero.
• A number is divisible by 11 if the difference of the sums of alternative digits is zero or a multiple of 11.
• A number is divisible by 12 if it is divisible by both 3 and 4.

 

Even numbers

• Whole numbers which are exactly divisible by 2 are called even numbers.
• The set of even numbers is denoted by 'E\ such that E={0,2,4,6,8....,}.
• The set E is an infinite set.

 

ODD numbers

• Natural numbers which are not exactly divisible by 2 are called Odd numbers.

          \[(b+c)\times a=b\times a+c\times a\]  0={ 1,3,5,7,9.....}

• The set 0 is an infinite set.

 

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.

• The set of prime numbers is infinite.
• 2 is the smallest prime number

 

Composite numbers

Natural numbers having more than two factors are called Composite numbers.

• 4,6,8,9,10,12,14,15,16,18... are composite numbers.
• Number 1 is neither prime nor composite number.
• All even numbers except 2 are composite numbers.
• Every natural number except 1 is either prime or composite number.
• There are infinite prime numbers and infinite composite numbers.

Co - Prime Numbers or Relatively Prime Numbers:

• Two natural numbers are said to be co-prime numbers or relatively prime numbers if they have only 1 as common factor.

Example: 8, 9; 15,16; 26,33, etc, are co-prime numbers.

• Co-prime numbers may not themselves be prime numbers. As 8 and 9 are co-prime numbers, but neither 8 nor 9 is a prime number.

 

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).

• All integers are rational numbers as \[\frac{p}{q}\]

0 is also rational number as \[q\ne 0\].

• If a and b are two rational numbers where a < b then the rational number \[\frac{a}{b}\] always lies between a and b, \[\frac{c}{d}\]

 

Irrational number

A number which can neither be expressed as terminating nor a non-terminating repeating decimal is called an irrational number.

• The number which cannot be expressed as \[\frac{a}{b},\frac{c}{d}\in Q\], \[a,b,c,d\in I,b\ne 0,d\ne 0\] and p, q are coprime. The numbers which are not rational are irrational.
• We cannot find the exact value of irrational numbers. \[\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\] is an irrational number that the exact value of \[ad+bc,bd\in I,bd\ne 0.\] cannot be obtained. \[\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}\] is not irrational as \[-bc,bd\in I,bd\ne 0\] i.e., the exact value is there.

\[\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}\]\[ac,bd\in I,bd\ne 0\]

\[\frac{a}{b}+\frac{c}{d}(c\ne 0)=\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc}\], etc, are the examples of irrational numbers.

• The number \[ad,bc\in I,bc\ne 0\] is an irrational number. \[a=\frac{p}{q},b=\frac{r}{s},c=\frac{l}{m}\] is an approximate value of \[p,q,r,s,l,m\in I,q\ne 0,s\ne 0,m\ne 0.\], but not exact value.

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

• The repeated multiplications of the same factor can be written in a more compact form, called exponential form.

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

• Let a be a rational number and n be a positive integer such that \[\frac{22}{7}\]a is irrational, then \[\pi \] is called a surd of the order n and a is called the radical.
• \[\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{6},\sqrt{7},\sqrt{8},\sqrt{11}\] is a surd if a is a rational number and \[0.\overline{3}=3/9\] is an irrational number.

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.

• The highesast common facter (H.C.F) of two or more numbers is the greatest number which divides each number exactly.
• The least common multiple (L.C .M) of two or more numbers is the smallest number which is exactly divisible by each number separately.
• The H.C.F. of given numbers is not greater than any of the given numbers.
• The L.C.M. of given numbers is not less than any of the given numbers.
• C.F. of two numbers always divides their L.C.M.
• The H.C.F. of two co-prime numbers is 1.
• The L.C.M. of two co-prime numbers is product of the numbers.
• The product of the H.C.F. and the L.C.M. of the two given numbers is equal to the product of those numbers, i.e.,

 

\[a,b,c,d\in I,b\ne 0,d\ne 0\] = Product of the given numbers.

• C.M of their numerators

\[\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\]

• C.F ot their denominators

\[ad+bc,bd\in I,bd\ne 0.\]

 

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 \[ac,bd\in I,bd\ne 0\]