Q.1 What are different kinds of numbers? Explain all types of numbers.
Q.2 Write all the properties of integers with example.
Answers
Answer:
And 1:- Natural Numbers
The Natural (or Counting numbers ) are 1,2,3,4,5 etc. There are infinity Natural Numbers. Hence the set of Natural Numbers will be {1,2,3,4,5,…..N}.
The addition of two Natural numbers is a Natual Number and the product of two numbers is Natural Numbers. Though this may not true for Subtraction and division.
Whole Numbers
The whole Numbers are the Natural numbers with 0.
Integers
The integers are the set of real numbers including natural numbers, their additive inverses and zero.
{…, -5, -4,-3, -2, -1, 0, 1, 2, 3, 4, 5…..}
The set of Integers is represented by J or Z.
Rational Numbers
Rational Numbers are those numbers which can be expressed as the ratio between two integers.
eg. 8.27 can be written as 827/100.
The Irrational Numbers
An Irrational number is a number that can be written as ratio or fraction in the form of decimals.
Golden ratio numbers are
1+5/√2=1.61803398874989...
π=3.14159265358979...
Real Numbers
Real numbers are a combination of both rational numbers and Irrational numbers.
The Complex Numbers
The complex numbers are the set { a+bi | a and b are real numbers}, where
i i is the imaginary unit,
√ −1
Ans 2:.
Property 1: Closure Property
Among the various properties of integers, closure property under addition and subtraction states that the sum or difference of any two integers will always be an integer i.e. if x and y are any two integers, x + y and x − y will also be an integer.
Example 1: 3 – 4 = 3 + (−4) = −1;
(–5) + 8 = 3,
The results are integers.
Closure property under multiplication states that the product of any two integers will be an integer i.e. if x and y are any two integers, xy will also be an integer.
Example 2: 6 × 9 = 54 ; (–5) × (3) = −15, which are integers.
Division of integers doesn’t follow the closure property, i.e. the quotient of any two integers x and y, may or may not be an integer.
Example 3: (−3) ÷ (−6) = ½, is not an integer.
Property 2: Commutative Property
The commutative property of addition and multiplication states that the order of terms doesn’t matter, the result will be the same. Whether it is addition or multiplication, swapping of terms will not change the sum or product. Suppose, x and y are any two integers, then
⇒ x + y = y + x
⇒ x × y = y × x
Example 4: 4 + (−6) = −2 = (−6) + 4;
10 × (−3) = −30 = (−3) × 10
But, subtraction (x − y ≠ y − x) and division (x ÷ y ≠ y ÷ x) are not commutative for integers and whole numbers.
Example 5: 4 − (−6) = 10 ; (−6) – 4 = −10
⇒ 4 − (−6) ≠ (−6) – 4
Ex: 10 ÷ 2 = 5 ; 2 ÷ 10 = 1/5
⇒ 10 ÷ 2 ≠ 2 ÷ 10
Property 3: Associative Property
The associative property of addition and multiplication states that the way of grouping of numbers doesn’t matter; the result will be same. One can group numbers in any way but the answer will remain same. Parenthesis can be done irrespective of the order of terms. Let x, y and z be any three integers, then
⇒ x + (y + z) = (x + y) +z
⇒ x × (y × z) = (x × y) × z
Example 6: 1 + (2 + (-3)) = 0 = (1 + 2) + (−3);
1 × (2 × (−3)) =−6 = (1 × 2) × (−3)
Subtraction of integers is not associative in nature i.e. x − (y − z) ≠ (x − y) − z.
Example 7: 1 − (2 − (−3)) = −4; (1 – 2) – (−3) = −2
1 – (2 – (−3)) ≠ (1 − 2) − (−3)
Property 4: Distributive Property
The distributive property explains the distributing ability of operation over another mathematical operation within a bracket. It can be either distributive property of multiplication over addition or distributive property of multiplication over subtraction. Here integers are added or subtracted first and then multiplied or multiply first with each number within the bracket and then added or subtracted. This can be represented for any integers x, y and z as:
⇒ x × (y + z) = x × y + x × z
⇒ x × (y − z) = x × y − x × z
Example 8: −5 (2 + 1) = −15 = (−5 × 2) + (−5 × 1)
Property 5: Identity Property
Among the various properties of integers, additive identity property states that when any integer is added to zero it will give the same number. Zero is called additive identity. For any integer x,
x + 0 = x = 0 + x
The multiplicative identity property for integers says that whenever a number is multiplied by the number 1 it will give the integer itself as the product. Therefore, the integer 1 is called the multiplicative identity for a number. For any integer x,
x × 1 = x = 1 × x
If any integer multiplied by 0, the product will be zero:
x × 0 = 0 =0 × x
If any integer multiplied by -1, the product will be opposite of the number:
x × (−1) = −x = (−1) × x