Questions & Answers
Question
AnswersState whether true or false.
The absolute value of an integer is greater than the integer.
A. TRUE
B. FALSE
Answer
Verified128.4k+ views
Hint: Here we use the definition of absolute value of an integer and using the definition we compare both the values in magnitude. Absolute value is the modulus of a number.
Complete step-by-step answer:
We know the set of integers contains all the negative numbers, zero and all the positive numbers and can be written as the set \[Z = \{ .... - 3, - 2, - 1,0,1,2,3....\} \]
So, if we look at a real line, integers are extended on both sides.
Now we know that the absolute value of an integer is defined as the numeric value of the integer without considering the sign along with it. Absolute value of an integer \[x\] is denoted by a modulus sign \[\left| x \right|\].
Value of \[\left| x \right| = x\] if \[x\] is positive.
Value of \[\left| x \right| = - x\] if \[x\] is negative.
Absolute value of an integer is always non-negative as the negative sign along with the integer gets removed when we apply modulus to it.
So, the absolute values of integers are represented on a real line from zero to infinity, which means the whole real line except the negative numbers.
If we take an integer \[ - x\], then absolute value of the integer will be \[\left| { - x} \right| = - ( - x) = x\] which is positive. So, we can say the absolute value of an integer is greater than the integer.
If we take an integer \[x\], then absolute value of the integer will be \[\left| x \right| = x\] which is positive. So, we can say the absolute value of an integer is equal to the integer.
From both the above statements we conclude that the absolute value of an integer is greater than or equal to the value of the integer.
So, the statement given in the question is wrong.
So, option B is correct.
Note: Students many times make mistake while calculating the absolute value of a negative integer as they take only the numeric value in place of \[x\] in \[\left| x \right| = - x\] where they have to take the negative sign along with the numeric value then only the both negative signs will multiply to give positive value.
Complete step-by-step answer:
We know the set of integers contains all the negative numbers, zero and all the positive numbers and can be written as the set \[Z = \{ .... - 3, - 2, - 1,0,1,2,3....\} \]
So, if we look at a real line, integers are extended on both sides.
Now we know that the absolute value of an integer is defined as the numeric value of the integer without considering the sign along with it. Absolute value of an integer \[x\] is denoted by a modulus sign \[\left| x \right|\].
Value of \[\left| x \right| = x\] if \[x\] is positive.
Value of \[\left| x \right| = - x\] if \[x\] is negative.
Absolute value of an integer is always non-negative as the negative sign along with the integer gets removed when we apply modulus to it.
So, the absolute values of integers are represented on a real line from zero to infinity, which means the whole real line except the negative numbers.
If we take an integer \[ - x\], then absolute value of the integer will be \[\left| { - x} \right| = - ( - x) = x\] which is positive. So, we can say the absolute value of an integer is greater than the integer.
If we take an integer \[x\], then absolute value of the integer will be \[\left| x \right| = x\] which is positive. So, we can say the absolute value of an integer is equal to the integer.
From both the above statements we conclude that the absolute value of an integer is greater than or equal to the value of the integer.
So, the statement given in the question is wrong.
So, option B is correct.
Note: Students many times make mistake while calculating the absolute value of a negative integer as they take only the numeric value in place of \[x\] in \[\left| x \right| = - x\] where they have to take the negative sign along with the numeric value then only the both negative signs will multiply to give positive value.