Absolute Value of 100 - What Is |100|?

One rule, zero exceptions

The absolute value of a number is its distance from zero on the number line. Positive numbers stay as they are. Negative numbers drop the minus sign. Zero stays zero. That is the entire rule - and the calculator applies it for you, showing both the result and the reasoning behind it.

Quick start

Type -7 and click "Calculate |x|". Result: |x| = 7. Explanation: negative number changes sign to positive. The distance from -7 to 0 on the number line is 7 units.

Absolute value on the number line

Think of the number line as a ruler centered at zero. The absolute value tells you how far a number sits from the center, regardless of direction. Both -5 and +5 are exactly 5 steps away from zero, so |{-5}| = |5| = 5.

The formal definition uses a piecewise function:

|x| = x if x >= 0
|x| = -x if x < 0

The second line looks odd at first - "negative of a negative" - but -(-5) = 5, which is exactly what we want. The minus sign in the formula cancels the minus sign in the number.

Quick reference table

Input (x)	\|x\|	Why
-100	100	Negative - drop the sign
-7	7	Negative - drop the sign
-0.5	0.5	Works with decimals too
0	0	Zero is already at distance zero
3	3	Positive - no change
42	42	Positive - no change
3.14	3.14	Irrational approximations work the same way

Practical examples

Temperature difference: Yesterday was -3 degrees, today is 8 degrees. The difference is 8 - (-3) = 11, but you could also compute |8 - (-3)| = |11| = 11 degrees. Absolute value makes the order of subtraction irrelevant: |-3 - 8| = |-11| = 11.
|a - b| always gives a non-negative difference

Stock price change: A stock dropped from $142 to $137. The change is 137 - 142 = -5. The absolute change is |-5| = $5. Reports often state "the stock moved $5" without specifying direction - that is absolute value in action.
Financial reporting uses absolute values for magnitude

Manufacturing tolerance: A bolt must be 12.00 mm in diameter, with a tolerance of 0.05 mm. If the actual measurement is 11.97 mm, the error is |12.00 - 11.97| = |0.03| = 0.03 mm. Since 0.03 < 0.05, the bolt passes quality control.
Tolerances are defined as absolute deviations

Elevation from sea level: A submarine is at -250 meters (below sea level). Its depth is |-250| = 250 m. A mountain peak at +250 meters is equally far from sea level but in the opposite direction.
Depth and altitude are both distances from the zero reference

Signal strength: An audio signal oscillates between -0.8 and +0.8 volts. The peak amplitude is |0.8| = |-0.8| = 0.8 V. Signal processing uses absolute values to measure amplitude regardless of polarity.
Peak-to-peak voltage = 2 x |peak amplitude|

Absolute value in equations and inequalities

Equations with absolute value often produce two solutions, because the expression inside the bars can be either positive or negative.

Problem	Solution	Interpretation
\|x\| = 5	x = 5 or x = -5	Two numbers are 5 units from zero
\|x - 3\| = 7	x = 10 or x = -4	Two numbers are 7 units from 3
\|2x + 1\| = 9	x = 4 or x = -5	Solve 2x + 1 = 9 and 2x + 1 = -9
\|x\| < 3	-3 < x < 3	All numbers closer than 3 units to zero
\|x\| > 4	x < -4 or x > 4	All numbers farther than 4 units from zero

Where absolute value appears in real life

Field	How absolute value is used
Engineering tolerances	A part is acceptable if \|measured - target\| < tolerance. The direction of the error does not matter, only its magnitude.
Statistics (MAD)	Mean Absolute Deviation = average of \|xi - mean\|. Less sensitive to outliers than standard deviation because it does not square the differences.
Financial reporting	"The index moved 150 points" uses absolute change. Percentage change also often uses \|new - old\| / old for magnitude.
Signal processing	Rectifiers convert AC to DC by taking the absolute value of the voltage waveform. \|sin(t)\| is a full-wave rectified sine.
Computer science	The abs() function is one of the most commonly used math functions in any programming language. In Python: abs(-7) returns 7.
Navigation	Distance between two coordinates along a single axis is the absolute value of their difference: \|lat2 - lat1\| gives the latitude gap.

FAQ - Frequently Asked Questions

What is the absolute value of zero?

|0| = 0. Zero is neither positive nor negative. Its distance from itself on the number line is zero. It is the only number whose absolute value is itself AND whose negative is also itself (since -0 = 0).

Is absolute value the same as modulus?

For real numbers, yes - the terms are interchangeable. The notation |x| is called the modulus of x. In complex numbers, the modulus |a + bi| = sqrt(a^2 + b^2), which generalizes the concept to two dimensions. In programming, modulus sometimes refers to the remainder operator (%), which is a different concept entirely.

Can absolute value be negative?

Never. By definition, |x| >= 0 for all real numbers x. If someone writes |x| = -3, the equation has no solution. This property (non-negativity) is one of the four axioms that define a norm in mathematics.

What is the triangle inequality?

The triangle inequality states that |a + b| <= |a| + |b| for all real numbers a and b. In words: the absolute value of a sum is at most the sum of the absolute values. Example: |3 + (-5)| = |-2| = 2, while |3| + |-5| = 3 + 5 = 8. Indeed 2 <= 8. This property is fundamental in analysis and metric spaces.

How does abs() work in programming?

Most languages have a built-in absolute value function: Python: abs(-7) returns 7. JavaScript: Math.abs(-7) returns 7. C/C++: abs(-7) for integers, fabs(-7.0) for doubles. Java: Math.abs(-7). Internally, these typically check the sign bit and flip it if negative - a single CPU instruction on modern hardware.

How do you solve |x - 3| = 5?

Split into two cases: (1) x - 3 = 5, so x = 8. (2) x - 3 = -5, so x = -2. Check: |8 - 3| = |5| = 5. |(-2) - 3| = |-5| = 5. Both solutions are valid. The equation asks "which numbers are exactly 5 units away from 3 on the number line?" - and the answers are 8 (five to the right) and -2 (five to the left).

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|100| - Absolute Value

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