The absolute value is one of the fundamental concepts in algebra and mathematical analysis. At first glance it may seem like nothing more than an operation that "strips the minus sign" from a number; in reality, its meaning runs deeper: the absolute value measures a distance.
This idea is essential. When we write \(|x|\), we are not merely changing the sign of \(x\) — we are expressing how far \(x\) lies from \(0\) on the real number line. For this reason, the absolute value is always a non-negative number.
Table of Contents
- Definition of absolute value
- Geometric meaning of the absolute value
- First examples
- Fundamental properties
- Absolute value of a product
- Absolute value of a quotient
- Absolute value and powers
- Distance between two real numbers
- Triangle inequality
- Common mistakes to avoid
Definition of absolute value
Let \(x\) be a real number. The absolute value of \(x\), denoted \(|x|\), is defined as follows:
\[ |x|= \begin{cases} x & \text{if } x\geq 0,\\ -x & \text{if } x<0. \end{cases} \]
This definition deserves careful reading. If \(x\) is positive or zero, its absolute value coincides with \(x\) itself. If \(x\) is negative, its absolute value is \(-x\).
The expression \(-x\) in the second case does not mean the result is negative. In fact, whenever \(x<0\), we have \(-x>0\). For instance, if \(x=-5\), then:
\[ -x=-(-5)=5. \]
Thus the absolute value always returns a number greater than or equal to zero.
Geometric meaning of the absolute value
The most important interpretation of the absolute value is geometric: \(|x|\) represents the distance from \(x\) to \(0\) on the real number line.
For example, the number \(4\) lies \(4\) units away from \(0\), so:
\[ |4|=4. \]
Likewise, the number \(-4\) also lies \(4\) units away from \(0\), so:
\[ |-4|=4. \]
This explains why two opposite numbers have the same absolute value: they are equidistant from \(0\), but on opposite sides of the real number line.
In general:
\[ |x|=|-x|. \]
First examples
Let us compute a few absolute values.
If \(x=7\), then \(x\) is positive, so:
\[ |7|=7. \]
If \(x=-7\), then \(x\) is negative, so:
\[ |-7|=-(-7)=7. \]
If \(x=0\), then:
\[ |0|=0. \]
The absolute value of \(0\) is \(0\), since \(0\) is at zero distance from itself.
Fundamental properties
Several fundamental properties follow directly from the definition.
For every real number \(x\):
\[ |x|\geq 0. \]
This property reflects the fact that a distance cannot be negative.
Moreover:
\[ |x|=0 \quad \Longleftrightarrow \quad x=0. \]
Indeed, the only number at zero distance from \(0\) is \(0\) itself.
Another important property is:
\[ |x|=|-x|. \]
The numbers \(x\) and \(-x\) are symmetric about the origin, so they lie at the same distance from \(0\).
Absolute value of a product
The absolute value of a product equals the product of the absolute values:
\[ |xy|=|x|\cdot |y|. \]
This property holds for every pair of real numbers \(x\) and \(y\).
For example:
\[ |-3\cdot 5|=|-15|=15. \]
On the other hand:
\[ |-3|\cdot |5|=3\cdot 5=15. \]
The two results agree.
The reason is that the absolute value disregards direction on the real number line and retains only the magnitude of the quantity. In a product, the signs may affect the sign of the result, but they leave its magnitude unchanged.
Absolute value of a quotient
If \(y\neq 0\), then:
\[ \left|\frac{x}{y}\right|=\frac{|x|}{|y|}. \]
The condition \(y\neq 0\) is necessary, since division by zero is undefined.
For example:
\[ \left|\frac{-6}{2}\right|=|-3|=3. \]
On the other hand:
\[ \frac{|-6|}{|2|}=\frac{6}{2}=3. \]
Again, the two results agree.
Absolute value and powers
A particularly useful property concerns the square:
\[ |x|^2=x^2. \]
Indeed, if \(x\geq 0\), then \(|x|=x\), so \(|x|^2=x^2\). If instead \(x<0\), then \(|x|=-x\), and therefore:
\[ |x|^2=(-x)^2=x^2. \]
From this property it also follows that:
\[ |x|=\sqrt{x^2}. \]
This formula is very important but must be interpreted with care. The principal square root is always non-negative, so \(\sqrt{x^2}\) is not equal to \(x\) for every \(x\); rather, it equals \(|x|\).
For example:
\[ \sqrt{(-3)^2}=\sqrt{9}=3, \]
whereas:
\[ -3\neq 3. \]
Therefore:
\[ \sqrt{x^2}=|x|, \]
not simply \(x\).
Distance between two real numbers
The absolute value provides a natural way to express the distance between two real numbers. If \(a\) and \(b\) are real numbers, the distance between \(a\) and \(b\) is:
\[ |a-b|. \]
Equivalently, one can write:
\[ |b-a|. \]
The two expressions are equal, since:
\[ |a-b|=|-(b-a)|=|b-a|. \]
For example, the distance between \(2\) and \(7\) is:
\[ |7-2|=|5|=5. \]
The distance between \(-3\) and \(4\) is:
\[ |4-(-3)|=|7|=7. \]
This interpretation is fundamental for understanding equations, inequalities, and functions involving the absolute value.
Triangle inequality
One of the most important properties of the absolute value is the triangle inequality:
\[ |x+y|\leq |x|+|y|. \]
This inequality states that the absolute value of a sum does not exceed the sum of the absolute values.
Geometrically, it means that the distance covered by travelling directly from one point to another cannot be greater than the distance covered by making an intermediate stop.
For example:
\[ |3+(-5)|=|-2|=2. \]
Whereas:
\[ |3|+|-5|=3+5=8. \]
Therefore:
\[ |3+(-5)|\leq |3|+|-5|. \]
In this case:
\[ 2\leq 8. \]
Equality holds when \(x\) and \(y\) have the same sign, or when at least one of them is zero.
Common mistakes to avoid
The first mistake is thinking that the absolute value always makes its argument positive. It is more precise to say that the absolute value returns a non-negative number.
In fact:
\[ |0|=0, \]
and \(0\) is not positive — it is zero.
The second mistake is writing:
\[ \sqrt{x^2}=x. \]
This equality does not hold for every real number. The correct form is:
\[ \sqrt{x^2}=|x|. \]
The third mistake is distributing the absolute value over addition. In general:
\[ |x+y|\neq |x|+|y|. \]
For example:
\[ |2+(-2)|=|0|=0, \]
whereas:
\[ |2|+|-2|=2+2=4. \]
Therefore:
\[ |2+(-2)|\neq |2|+|-2|. \]
The absolute value is far more than a rule for dropping minus signs. It is a tool for measuring distances, controlling magnitudes, and rigorously describing many properties of the real numbers.
Its piecewise definition shows that the behaviour of \(|x|\) depends on the sign of \(x\), while its geometric meaning makes clear why the result is always non-negative.
A solid understanding of the absolute value is indispensable for tackling equations and inequalities involving moduli, piecewise-defined functions, intervals on the real line, and many subsequent topics in algebra and mathematical analysis.