In competitive exams, we will see the game of numbers. so we will describe here numbers, its classifications and all types of aptitude numerical which are contentiously asked in competitive exams.
List of Mathematical Operations of numbers are as follows:
1. Addition,
2. Subtraction,
3. Multiplication or Product
4. Division
For representing these operation in an equation we require some more operations like
5. Bracket,
6. Of (of is a form of multiplication)
Addition ( + ):
Addition of two numbers have some rules
1. If both numbers have same sign then the numbers will be added and result will also have the same sign
(a) 4 + 6 = +10
(b) ( -8) + ( -6) = ( -14)
2. If both numbers have different sign then the smaller numbers will be subtracted from larger numbers and result will have sign of larger number.
(a) 4 + ( -11) = { - (11 - 4)} = ( -7)
(b) ( -6) + 20 = +(20 - 6) = +14
Properties of Addition:
(i) Sum of two odd numbers is an even number always.
For Example: 5+9 = 14
(ii) Sum of two even number is also an even number always.
For Example: 8+22 = 30
(iii) Sum of an even and an odd number is an odd number always.
For Example: 12+9 = 21
(iv) Sum of two rational numbers is a rational number always.
For Example: 5+9 = 14
(v) Sum of a rational number and an irrational number is an irrational number always.
For Example:4+(2+ √3)= 6+ √3(vi) Sum of two irrational numbers can be rational sometimes and irrational sometimes.
For Example: (2 - √3) + (2+ √3) = 4 [irrational + irrational = rational]
For Example: (2 - √3) + (2+ 2√3) = (4 + √3) [irrational + irrational = irrational]
Subtraction ( - ):
Subtraction of two numbers have some rules
1. If both numbers have different sign then the numbers will be added and result will have the sign of first number
(a) ( -4) - (+6) = (-10)
(b) ( +8) - ( -6) = +14
2. If both numbers have same sign then the smaller numbers will be subtracted from larger numbers and result will have sign of larger number after bracket, and addition operations are performed.
(a) ( -4) - ( -11) = ( -4) + 11 = + (11 - 4) = +7
(b) ( -6) - ( -4) = ( -6) + 4 = - (6-4) = ( -2)
Properties of Subtraction:
(i) Difference of two odd numbers is an even number always.
For Example: 9 - 5 = 4
(ii) Difference of two even number is also an even number always.
For Example: 22 - 8 = 14
(iii) Difference of an even and an odd number is an odd number always.
For Example: 12 - 9 =3
(iv) Difference of two rational numbers is a rational number always.
For Example: 9 - 5 = 4
(v) Difference of a rational number and an irrational number is an irrational number always.
For Example:4 - (2 + √3)= 2 - √3(vi) Difference of two irrational numbers can be rational sometimes and irrational sometimes.
For Example: (2 - √3) - (4 - √3) = (- 2) [irrational - irrational = rational]
For Example: (2 - √3) - (4 + √3) = (- 2 - 2√3) [irrational - irrational = irrational]
Multiplication or Product ( x or * ):
Multiplication of two numbers have some rules
1. If both numbers have same sign then result will have positive (+) sign.
(a) 4 x 6 = +24
(b) ( -8) x ( -6) = +48
2. If both numbers have different sign then result will have negative (-) sign.
(a) 4 x ( -11) = ( -44)
(b) ( -6) x 20 = ( -120)
3. If more than two numbers are multiplied in a mathematics, then
(i) if number of negative (-) numbers are even then result will have positive (+) sign.
(ii) if number of negative (-) numbers are odd then result will have negative (-) sign.
(a) 4 x ( -2) x ( -3) x 5 x ( -5) x ( -4) x 3 = +7200
(b) 4 x ( -2) x 3 x 5 x ( -5) x ( -4) x 3 = ( -7200)
Properties of Multiplication:
(i) Product of two odd numbers is an odd number always.
For Example: 9 x 5 = 45
(ii) Product of two even number is an even number always.
For Example: 10 x 4 = 40
(iii) Product of an even and an odd number is an even number always.
For Example: 4 x 5 = 20
(iv) Product of two rational numbers is a rational number always.
For Example: 9 x 5 = 45
(v) Product of a rational number and an irrational number is an irrational number always.
For Example:4 x (2 + √3)= 8 x 4√3(vi) Product of two irrational numbers can be rational sometimes and irrational sometimes.
For Example: √3 x √3 = 3 [irrational x irrational = rational]
For Example: (2 - √3) x (4 + √3) = (5 - 2√3) [irrational x irrational = irrational]
Division ( ÷ or / ):
Multiplication of two numbers have some rules
1. If both numbers have same sign then result will have positive (+) sign.
(a) 10 ÷ 2 = +5
(b) ( -10) ÷ ( -2) = +5
2. If both numbers have different sign then result will have negative (-) sign.
(a) 10 ÷ ( -2 )= ( -5)
(b) ( -10) ÷ 2 = ( -5)
3. If more than two numbers are multiplied in a mathematics, then
(i) if number of negative (-) numbers are even then result will have positive (+) sign.
(ii) if number of negative (-) numbers are odd then result will have negative (-) sign.
Properties of Division:
(i) Division of two odd numbers is either a fraction or an odd integer.
For Example: 9 ÷ 5 = 1.8, 15 ÷ 3 = 5
(ii) Division of two even number is either a fraction or an even integer.
For Example: 10 ÷ 4 = 2.5, 24 ÷ 8 = 3
(iii) Division of an even and an odd number is always a fraction.
For Example: 25 ÷ 8 = 3.125 46 ÷ 5 = 9.2
(iv) Division of two rational numbers cab be a rational or an irrational number.
For Example: 9 ÷ 4 = 2.25, 50 ÷ 47 = 06382978723404255319148
(v) Division of a rational number and an irrational number is an irrational number always.
For Example:(8 + 4√3) ÷ 4 = 2 + √3(vi) Division of two irrational numbers can be rational sometimes and irrational sometimes.
For Example: 4√3 ÷ √3 = 4 [irrational ÷ irrational = rational]
For Example: (4 + √3) x (2 - √3) = (18 + 6√3) [irrational ÷ irrational = irrational]
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