Define rational numbers and the properties. Explain
Answers
Properties of Rational Numbers. ... Rational numbers are the numbers which can be represented in the form of p/q, where q is not equal to 0. Basically, the rational numbers are the integers which can be represented in the number line. Let us go through all the properties here.
Numbers that are not rational are called irrational numbers. ... Any rational number is trivially also an algebraic number. Examples of rational numbers include. , 0, 1, 1/2, 22/7, 12345/67, and so on.
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Answer:
Rational numbers:
The numbers which can be expressed as ratio of integers are known as rational numbers. Examples: 1/4, 2/7, - 3/10, 34/7.
These are the numbers which can be expressed in x/y form; where y ≠ 0.
Properties of Rational Numbers:
(i) Closure Property:
When any operation is performed between two or more rational numbers and their result is also a rational number then we say that the rational numbers follow the closure property for that operation.
Addition
a) 5/3 + 3/2 = 19/6 (Rational No);
We can observe that addition of two rational numbers x and y, i.e. x + y is always a rational number.
Hence, rational numbers are closed under addition.
Subtraction
a) 5/3 - 3/2 = 1/6 (Rational No);
We can observe that subtraction of two rational numbers x and y, i.e. x - y is always a rational number.
Hence, rational numbers are closed under subtraction.
Multiplication a) 5/ 3 × 3/2 = 5/2 (Rational No);
We can observe that multiplication of two rational numbers x and y, i.e. x × y is always a rational number.
Hence, rational numbers are closed under multiplication.
Division a) 5/3 ÷ 3/2 = 10/9 (Rational No);
Hence, rational numbers are not closed under division.
(ii) Commutative Property:
When two rational numbers are swapped between one operator and still their result does not change then we say that the rational numbers follow the commutative property for that operation.
Addition
(a) 5/3 + 3/2 = 19/6;
3/2 + 5/3 = 19/6
We can observe that addition of two rational numbers x and y when inter changed yields the same answer, i.e. x + y = y + x.
Hence, rational numbers are commutative under addition.
Subtraction (a) 5/3 - 3/2 = 1/6;
3/2 – 5/3 = -1/6
We can observe that subtraction of two rational numbers x and y when inter changed does not yield the same answer, i.e. x - y ≠ y - x.
Hence, rational numbers are not commutative under subtraction.
Multiplication
(a) 5/3 × 3/2 = 5/2;
3/2 × 5/3 = 5/2
We can observe that multiplication of two rational numbers x and y when inter changed yields the same answer, i.e. x × y = y × x.
Hence, rational numbers are commutative under multiplication.
Division (a) 5/3 ÷ 3/2 = 10/9;
3/2 ÷ 5/3 = 9/10
Here, both answers are different(b) 12/3 ÷ 0 = ∞ ;
0 ÷ 12/3 = 0 ;
Hence, rational numbers are not commutative under division.
(iii) Associative Property:
When rational numbers are rearranged among two or more same operations and still their result does not change then we say that the rational numbers follow the associative property for that operation.
Addition
(a) 5/3 + (3/2 + 1/3) = 7/2;
(5/3 + 3/2) + 1/3 = 7/2
We can observe that addition of rational numbers x, y, and z in any order yields the same answer, i.e. x + (y + z) = (x + y) + z.
Hence, rational numbers are associative under addition.
Subtraction (a) 5/3 – (3/2 – 1/3) = 3/2;
(5/3 – 3/2) – 1/3 = -1/2
We can observe that subtraction of rational numbers x, y, and z in any order does not yields the same answer, i.e. x - (y - z) ≠ (x - y) - z.
Hence, rational numbers are not associative under subtraction.
Multiplication
(a) 5/ 3 × (3/2 × 2/3) = 5/3;
(5/ 3 × 3/2) × 2/3 = 5/3
We can observe that multiplication of rational numbers x, y, and z in any order yields the same answer, i.e. x × (y × z) = (x × y) × z.
Hence, rational numbers are associative under multiplication.
Division:(a) 5/3 ÷ (3/2 ÷ 1/4) = 5/18;
(5/3 ÷ 3/2) ÷ 1/4 = 40/9
We can observe that division of rational numbers x, y, and z in any order does not yields the same answer, i.e. x ÷ (y ÷ z) ≠ (x ÷ y) ÷ z.
Hence, rational numbers are not associative under division.
(iv) The role of zero:
When any rational number is added to zero (0), then it will result in the same rational number i.e. x/y + 0 = x/y.
Zero is called the identity for the addition of rational numbers.
Example: 2/3 + 0 = 2/3; -5/7 + 0 = -5/7; etc.
(v) The role of one :
When any rational number is multiplied with one (1), then it will result in the same rational number i.e. x/y × 1 = x/y.
One is called the multiplicative identity for rational numbers.
Example: 2/7 × 1 = 2/7; -8/3 × 1 = -8/3; etc.
(vi) Negative of a number:
When a rational number is added to the negative or additive inverse of its own, result will be zero (0) i.e. x/y + (-y/x) = 0.
Example: 2/7 + (-7/2) = 0; etc.
(vii) Reciprocal of a number:
When a rational number is multiplied with the reciprocal or multiplicative inverse of its own, result will be one (1) i.e. x/y × y/x = 1.
Example: 2/7 × 7/2 = 1; etc.
(viii) Distributive property of multiplication over addition:
If the rational numbers a, b, and c obey property of a × (b + c) = ab + ac, then it is said to follow Distributive property of multiplication over addition.
Example: 1/3 × (2/3 + 1/4) = 1/3 × 11/12 = 11/36
(1/3 × 2/3) + (1/3 × 1/4) = 2/9 + 1/12 = 11/36
Here, answer are same.
Hence, rational numbers follow distributive property of multiplication over addition.