The idea of absolute value and the term itself have been around for centuries, but yet the concept initially seems to elude many. A part of the broadly defined discipline of Algebra, absolute value pertains to many different mathematical concepts, one of which being complex numbers. Read on to learn the basics of the absolute value complex values beginning with the history, then moving on to the concepts. To learn more, you should check out this course, Matrix Algebra.
The Origins of the Idea behind Absolute Value
The ideas that became the concept of absolute value were first explored in the 17th century in France by a talented amateur mathematician named Jean-Robert Argand. Not first termed as absolute value, Argand’s work explored the idea of units of measure for absolute value, then described as “module.” The term and associated concepts have been a part of French mathematical thinking since that time, but really didn’t enter into the English mathematical vocabulary until more than fifty years later.
Of even greater interest is the fact that the concept at the core of absolute value in complex numbers has been around much longer, arguably before 500 BC. While this concept itself has been around much longer, it could not be applied to that of absolute value until absolute value became a concept in and of itself. When we look at absolute value in complex numbers, the nexus between the two becomes quite clear.
Basic Concept of Absolute Value
While absolute value seems quite complex on its face, it is considered to be a relatively easy topic to comprehend in terms of learning Algebra. In fact, current US mathematical curriculum shows absolute value as being introduced to students as early as the middle school years. Other titles used synonymously with absolute value include magnitude and numerical value; all three of these are used to describe the symbol for absolute value, seen below, which was introduced by a German mathematician in the 1840’s and is still used today.
Absolute value = │x│
In the most simple terms, the definition of absolute value of a number is the linear distance of that number from zero. If the numbers 0 and 5 are on a number line, the absolute value of the number 5 is 5 because it is 5 units away from 0.
It should also be noted that by this same definition, if the numbers -5 and 0 are on a number line, the absolute value of the number -5 is also 5 because it is 5 units away from 0.
The absolute value in its simplest form will always be positive because the distance between the two numbers will always be positive.
Absolute Value and Complex Numbers
There is one main distinction when looking at absolute value in basic terms versus an absolute value of a complex number, but it has nothing to do with finding the absolute value. Rather, the difference lies in the plane that complex numbers lie in from a mathematical perspective. Simple numbers are a linear distance from zero whereas complex numbers exist on a complex and two- dimensional plane, making the mathematical equation get to the absolute value slightly more challenging.
Calculating the absolute value of a complex number is actually one application of everyone’s favorite Algebraic concept, Pythagorean theorem. To refresh your memory, Pythagorean theorem involves a three sided right triangle and allows us to calculate a third side based on the two other known sides. Or as your high school math teacher likely drilled into your memory, the hypotenuse squared (hypotenuse being the side of the triangle that is opposite the right angle) is equal to summing the squares of the other two sides of the right triangle.
a² + b²= c²
Why the digression to high school Algebra? Complex numbers exist on the complex plane in a direction from the origin that when adding a y axis, creates the hypotenuse of a right angle. This is to say that Pythagorean theorem can be used to determine the absolute value of a complex number because of its relation to 0 on a complex plane.
Relating back to Pythagorean theorem above, if c is the absolute value, or │c│ and we use the above equation to get to the value of the hypotenuse, than we can get the absolute value of the complex number, or c by calculating the square root of the sum of a both sides squared. The equation below puts this in context:
│c│ = √a²+b²
As with the basic calculation of absolute value, it is also true for complex numbers that even if the complex number itself is negative, or to the left of the Y axis, the absolute value is still positive. As with the basic calculation, this is true for complex numbers because a distance from the complex number to 0 will always be positive. Also, keep in mind that squaring any number whether positive or negative will make it a positive number.
As logic would dictate, the idea of absolute value is closely tied in with the concept of distance. Always remember that distance will always either be positive or zero; as a result, absolute value will not be a difficult concept for you to remember, even as it relates to complex numbers.
For many people, the simple idea of math is so daunting, they do not make the effort to try to learn even the basics. With just a little consistent effort and willingness to learn, math can very quickly change from being your Achilles heel to being a point of pride, even a source of confidence, and even a skill you can pass on to others. In addition, you should build upon your understanding of Algebra by taking these Udemy courses, Beginning Algebra: Building a Foundation and Advanced Algebra: Strategies for Success. Moreover, if you need help grasping some of the concepts, check out this blog, Problem Solving Skills: Training Your Brain to Think Effectively. Once you get started, on finding the absolute value of complex numbers, you will wonder why you waited so long!