state whether the following pairs of rational numbers are closed under the operation given or not.
a. 21 and 7 , addition.
b. 144/27 and 24/27 , subtraction
c. 100/23 and 67/12 , multiplication
d. 34 and 0 , division
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Answers
A rational number is a number that can be expressed as the quotient or fraction pqpq of two integers, a numerator p and a non-zero denominator q. Since, q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as ‘ the field of rationals’ is usually denoted by Q .
A set that is closed under an operation or collection of operations is said to satisfy a closure property. Often a closure property is introduced as an axiom, which is then usually called the axiom of closure.
For any two rational numbers, say x and y, the results of addition, subtraction and multiplication operations give a rational number. Division is not under closure property because division by zero is not defined.
For example, if we take two rational numbers 12 and 3412 and 34, then:
12 +34=1×2+3×14=2+34=5412 +34=1×2+3×14=2+34=54, which is a rational number.
Also, 12 -34=1×2−3×14=2−34=−1412 -34=1×2−3×14=2−34=−14, which is a rational number.
And, 12×34=3812×34=38, which is also a rational number.
Thus, we see that for addition, subtraction as well as multiplication, the result that we get is itself a rational number. This means that rational numbers are closed under addition, subtraction and multiplication.
Hence, option (d) is the correct answer.
Note: Students should remember the meaning of the closure property. One should also note that the denominator of a rational number can never be 0 otherwise it will not be defined.
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