Project Work

Mathematics

TWO KIND OF NUMBERS

Rational Numbers Irrational Numbers

RATIONAL NUMBERSA rational number is a real number that can be written as a ratio of two integers.

A rational number written in decimal form is terminating or repeating.

EXAMPLES OF RATIONAL NUMBERS-81.3333…- 3/4161/23.56

IRRATIONAL NUMBERSAn irrational number is a number that cannot be written as a ratio of two integers.

Irrational numbers written as decimals are non-terminating and non-repeating.

EXAMPLES OF IRRATIONAL NUMBERS

Square roots of non-perfect “squares”

Pi

π√17

Integers One of the subsets of

rational numbers

WHAT ARE INTEGERS?Integers are the whole numbers and their opposites.

Examples of integers are i. 6ii. -12iii. 0iv. 186v. -934

INTEGERSIntegers are rational numbers because they can be written as fraction with 1 as the denominator

WHOLE NUMBERSWhole numbers one of the subsets of rational numbers

WHAT ARE WHOLE NUMBERS?

Whole numbers are rational numbers because they can be written as fraction with 1 as the denominator

Examples Of whole numbers:i. 1ii. 2iii. 3

WHOLE NUMBERSWhole numbers are rational numbers because they can be written as fraction with 1 as the denominator

COMMUTATIVITYWhole NumbersOperation Numbers Remarks

Addition 0+7=7+0=7,2+3=3+2=5For any two whole numbers a and b,a+b = b+a

Addition is commutative.

Subtraction 4-6 ≠ 6-4 Subtraction is not commutative

Multiplication 2×6 = 6×2 = 12,4×5 = 5×4 = 20

Multiplication is commutative

Division 7÷3 ≠ 3÷7 Division is not commutative

COMMUTATIVITYIntegersOperation Numbers Remarks

Addition -6+5 = (-1)-7+(-5) = (-12)

Addition is commutative

Subtraction 7-5 = 25-7= (-2)

Subtraction is not commutative

Division 5÷8 = 8÷5=

Division is not commutative

Multiplication 5×8 = 40(-5) ×(-8)= 40

Multiplication is commutative

COMMUTATIVITY Rational Numbers Addition-i. +()= =

Subtraction-

We find that subtraction is not commutative.

COMMUTATIVITY Multiplication-

So, we find that multiplication is commutative.

Division-

So, we find that division is not commutative.

ASSOCIATIVITY Whole numbers-Operation Numbers Remarks

Addition 7+(2+5)=(7+2)+5LHS= 7+(7)=14RHS= (9)+5=14

So, we find addition is associative with whole numbers.

Subtraction 8-(9-11)=11-(9-8)LHS= 8-2=6RHS= 11-1=10

So, we find subtraction is associative with whole numbers.

Division 3÷(2÷9)≠ 9÷(3÷2)LHS= 3÷[]=RHS= 9÷[=

So, we find division is associative with whole numbers.

Multiplication

7×(2×5)=(7×2) ×5LHS= 7×10=70RHS= 14×5=70

So, we find multiplication is associative with whole numbers.

ASSOCIATIVITY IntegersOperation

Numbers Remarks

Addition (-2)+[3+(-4)]=[(-2)+3]+(-4)LHS=(-2)+(-1)=(-3)RHS=1+(-4)=(-3)

Addition is associative.

Subtraction

5-(7-3)= (5-7)-3LHS= 5-10= (-5)RHS= (-2)-3= (-5)

Subtraction is not associative.

Division [(-10)2](-5)=(-10)[2(-5)]LHS=(-5)(-5)= (-1)RHS= (-10)=(-25)

Division is not associative.

Multiplication

5×[(-7) ×(-8)]=[5((-7)]×(-8)LHS=5×56=280RHS=(-35)×(-8)=280

Multiplication is associative.

ASSOCIATIVITY Rational numbers- Addition-

Addition is associative with rational numbers.

Subtraction-

[ Subtraction is associative with rational

numbers.

ASSOCIATIVITY Division-

LHS= = =

Division is associative with rational number.

ASSOCIATIVITY Multiplication-

( Multiplication is associative with

rational numbers.

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