In algebra and in vectors we come across numbers in negative directions although the magnitudes of such quantities have no signs or supposed to be positive. Let us take a closer study.
What is Absolute Value in Math
Let us consider the numbers 5 and -5. Algebraically, they are not same. But when you plot these numbers on a number line, they are no doubt on the opposite sides of the origin 0. But they are at the same distance from the origin. That is -5 is also 5 units away from the origin on a number line. Hence considering only the fact of ‘how far’, we can say that the absolute value 0f -5 is 5.
Thus we can define that an absolute value of a number is only its physical value and hence it is always referred as positive. Symbolically the absolute value of a variable ‘x’ is denoted as lxl and it is always equal to + x.
Finding Absolute Value
As explained earlier an absolute value of a number cannot be negative. Thus, the easiest method of finding absolute value of a number is just consider the number only, ignoring the sign before it. That is for finding absolute value of -10, just consider only the number 10. That is l-10l = 10.
Limit of Absolute Value
Since the absolute value of a variable is always positive, the upper limit of the absolute value is infinity. However, because an absolute value can never be negative, it cannot cross below 0. Therefore, the lower limit of absolute value is 0. Thus the limit of absolute value of a variable can be expressed as [0, ∞).
Absolute Value Practice
The concept of absolute value is important. For example, A is situated 10 miles from his office and B is situated 5 miles from the same office but exactly in the opposite direction. Now to calculate the distance between A and B, we cannot algebraically say 10 + (-5) = 5 miles or 5 + (-10) = -5 miles. Here you need to apply the absolute value practice and say the distance is l10l + l5l = 10 + 5 = 15 miles.
Properties of Absolute Values
The important properties of absolute values are,
lxl = x for x ≥ 0, but = -x for x < 0.
If lf(x)l = a then there can be two cases. That is f(x) = a and f(x) = -a.
Suppose lf(x)l + a = b and if a > b, then there is no real solution to the equation.
