absolute value

(noun)

The magnitude of a real number without regard to its sign; formally, -1 times a number if the number is negative, and a number unmodified if it is zero or positive.

Related Terms

  • piecewise function
  • subdomain
  • number line
  • inequality
  • modulus

(noun)

For a real number, its numerical value without regard to its sign; formally, −1-1−1 times the number if the number is negative, and the number unmodified if it is zero or positive.

Related Terms

  • piecewise function
  • subdomain
  • number line
  • inequality
  • modulus

(noun)

The magnitude (i.e., non-negative value) of a number without regard to its sign; a number's distance from zero.

Related Terms

  • piecewise function
  • subdomain
  • number line
  • inequality
  • modulus

(noun)

The distance of a real number from 000 along the real number line.

Related Terms

  • piecewise function
  • subdomain
  • number line
  • inequality
  • modulus

Examples of absolute value in the following topics:

  • Absolute Value

    • Absolute value can be thought of as the distance of a real number from zero.
    • For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5, because both numbers are the same distance from 0.
    • The term "absolute value" has been used in this sense since at least 1806 in French and 1857 in English.
    • Other names for absolute value include "numerical value," "modulus," and "magnitude."
    • The absolute values of 5 and -5 shown on a number line.
  • Equations with Absolute Value

    • To solve an equation with an absolute value, first isolate the absolute value, and then solve for the positive and negative cases.
    • At face value, nothing could be simpler: absolute value simply means the distance a number is from zero.  
    • The absolute value of −5-5−5 is 555, and the absolute value of 555 is also 555, since both −5-5−5 and 555 are 555 units away from 000.
    • Recall that absolute value is a measure of distance, so it can never be a negative value.
    • The following steps describe how to solve an absolute value equation:
  • Inequalities with Absolute Value

    • Inequalities with absolute values can be solved by thinking about absolute value as a number's distance from 0 on the number line.
    • More complicated absolute value problems should be approached in the same way as equations with absolute values: algebraically isolate the absolute value, and then algebraically solve for xxx.
    • It is difficult to immediately visualize the meaning of this absolute value, let alone the value of xxx itself.
    • Now think: the absolute value of the expression is greater than –3.
    • Absolute values are always positive, so the absolute value of anything is greater than –3!
  • The Leading-Term Test

    • All polynomial functions of first or higher order either increase or decrease indefinitely as xxx values grow larger and smaller.
    • The properties of the leading term and leading coefficient indicate whether f(x)f(x)f(x) increases or decreases continually as the xxx-values approach positive and negative infinity:
    • Intuitively, one can see why we need to look at the leading coefficient to see how a polynomial behaves at infinity: When xxx is very big (in absolute value), then the highest degree term will be much bigger (in absolute value) than the other terms combined.
    • and the absolute value of xxx is bigger than MnKMnKMnK, where MMM is the absolute value of the largest coefficient divided by the leading coefficient, nnn is the degree of the polynomial and KKK is a big number, then the absolute value of anxna_nx^na​n​​x​n​​ will be bigger than nKnKnK times the absolute value of any other term, and bigger than KKK times the other terms combined!
  • Piecewise Functions

    • Example 1: Consider the piecewise definition of the absolute value function:
    • For all xxx-values less than zero, the first function (−x)(-x)(−x) is used, which negates the sign of the input value, making the output values positive.
    • For all values of xxx greater than or equal to zero, the second function (x)(x)(x) is used, making the output values equal to the input values.  
    • After finding and plotting some ordered pairs for all parts ("pieces") of the function the result is the V-shaped curve of the absolute value function below.
    • The piecewise function, ∣x∣={−x,if x<0x,if x≥0\left | x \right |= \left\{\begin{matrix} -x, & if\ x<0\\ x, & if\ x\geq0 \end{matrix}\right.∣x∣={​−x,​x,​​​if x<0​if x≥0​​, is  the graph of the absolute value function.  
  • One-to-One Functions

    • To see this, note that the points of intersection have the same y-value, because they lie on the line, but different x values, which by definition means the function cannot be one-to-one.
    • Another way to determine if the function is one-to-one is to make a table of values and check to see if every element of the range corresponds to exactly one element of the domain.  
    • This is an absolute value function, which is graphed below.
  • Complex Conjugates

    • The modulus symbol looks just like the absolute value symbol, which is okay because whenever b=0b=0b=0 so that z=a+bi=az=a+bi=az=a+bi=a is a real number, we have that the conjugate is a−bi=aa-bi=aa−bi=a.
    • So the symbol is consistent with the use of the absolute value symbol.
  • Basics of Graphing Polynomial Functions

    • As x34\frac {x^3}{4}​4​​x​3​​​​ tends to be much larger (in absolute value) than 3x24−3x2−2\frac {3x^2}{4} - \frac {3x}{2} - 2​4​​3x​2​​​​−​2​​3x​​−2 when xxx tends to positive or negative infinity, we see that yyy goes, like x34\frac {x^3}{4}​4​​x​3​​​​, to negative infinity when xxx goes to negative infinity, and to positive infinity when xxx goes to positive infinity.
    • Another easy point to draw is the intersection with the yyy-axis, as this equals the function value in the point zero, which equals the constant term of the polynomial.
    • In general, the more function values we compute, the more points of the graph we know, and the more accurate our graph will be.
    • Conversely, we can easily read the constant term of the polynomial by looking at its intersection with the yyy-axis if its graph is given (and indeed, we can readily read any function value if the graph is given).
    • The graph of a non-constant (univariate) polynomial always tends to infinity when the variable increases indefinitely (in absolute value).
  • Slope

    • The steepness, or incline, of a line is measured by the absolute value of the slope.
    • A slope with a greater absolute value indicates a steeper line.
    • Given two points (x1,y1)(x_1, y_1)(x​1​​,y​1​​) and (x2,y2)(x_2, y_2)(x​2​​,y​2​​), take a look at the graph below and note how the "rise" of slope is given by the difference in the yyy-values of the two points, and the "run" is given by the difference in the xxx-values.
  • Relative Minima and Maxima

    • Relative minima and maxima are points of the smallest and greatest values in their neighborhoods respectively.
    • The absolute maximum is the y-coordinate which is 161616.
    • The absolute minimum is the y-coordinate which is −10-10−10.
    • The local minimum is at the yyy-value of−16 and it occurs when x=2x=2x=2.
    • For the function pictured above, the absolute maximum occurs twice at y=16y=16y=16 and the absolute minimum is at (3,−10)(3,-10)(3,−10).
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