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Understanding Imaginary Numbers

By Kathleen Cantor, 03 Apr 2021

The term "imaginary number" describes any number that, when squared, gives a negative result. When you consider that man invented all numbers, you can also consider working with imaginary numbers. It's acceptable to invent new numbers as long as it works within the bounds of the rules that are already in place.

Simply put, an imaginary number is the square root of a negative number and does not have a tangible value. Although imaginary numbers are not real numbers and you cannot quantify them on a number line, these numbers are "real." We use them all the time in advanced Mathematics classes.

Solving Problems Involving Imaginary Numbers

For a time, the belief held that you can't get the square root of a negative number. This resulted from the "non-existence" of numbers that were negative after you squared them. It was impossible to work backward by taking the square root since every number was positive after you squared them.

So you couldn't square root a negative number and expect to come up with anything practical. But you don't have to worry about working out the square roots of negative numbers. To solve this problem, you can use a new number.

This new number was invented during the Reformation period. During this time, nobody believed that you could use this number for any "real world" use. It was strictly for making it easy to work out the computations in solving certain equations. As such, the new number was generally viewed as "a pretend number" invented only for convenience.

The new number we are talking about is called "i," denoting "imaginary." People believed that this number—the square root of a negative number—wasn't real. You write this imaginary number as:

  • i = √-1
  • Thus: i² = (√-1)² = -1
  • And: i² = (√-1)² = √(-1)² = √1=1

To determine the square root of a negative number in terms of the imaginary unit "i," use the following property where represents any non-negative real number:

√-a = √-1.a =√-1.√a= I√a

With this information, we can write:

√-9 = √-1.9 = √-1.√9= i.3 =3i

We would expect that 3i squared equals -9. So (3i)² =9i² =9(-1) = -9. You can write the square root of any negative number in terms of the imaginary unit. Such numbers are called imaginary numbers.

Solving Imaginary Numbers Involving Radicals

Since multiplication is commutative, the imaginary numbers are equivalent and are often misinterpreted as part of the radicand. To deal with this confusion, place the imaginary number in front of the radical, then solve the problem. Let's consider the complex number 21-20i.

Example 1

Solve the equation 21-20i.

Solution

  • From the definition of a square root, this number satisfies the equation: 21-20i=x2
  • Now express x as {a}<{e}{m}>+{b}{i} where a and b are real numbers: 21-20i=(a+bi)²
  • Then multiple out the term on the right-hand side: 21-20i=a²+(2ab)i+(b2)i²
  • As i²=-1 by definition of "i", rearrange the equation to give: 21-20i=(a²-b²)+(2ab)i.

Now that both sides of the equation are in the same form, compare the coefficients to obtain two equations in a and b. You have a²-b²=21 (call this equation 1). Next, compare the imaginary parts of the equation (the coefficients of i). You have 2ab=-20 (call this equation 2). You'll now have two equations with two unknowns. You can solve the simultaneous equations for a and b.

  • First, can make b the subject of equation 2 by dividing both sides by 2a like this: b=-10/a.
  • You can then substitute this expression for b into equation 1 like this: a²-(-10/a)²=21.
  • Simplify and factorize this equation to get the following: (a²+4)(a²-25)=0
  • The resultant equation is quadratic in disguise. Therefore: a²=-4 or a²=25

Remember the assumption that a and b are real numbers. a²=-4 has no solutions of interest to you. This means your solutions are a=5 and a=-5. Now substitute each a value into your earlier expression for b. This means that when a=5, b=-2 and when a=-5, b=2. Lastly, put a and b into the context of the question and get the solution as 5-2i and -5+2i.

Solving Imaginary Numbers With A Single Radical

If you have an equation with a single radical, follow the procedure below:

  1. Isolate the radical to the left side of the equation and leave everything else on the equation's left.
  2. Square both sides of the equation.
  3. Get the value of the unknown.
  4. Substitute the value of the unknown in the original equation to verify.

Example

Solve the equation √7x -2 = 5

Solution

  • √7x + 4 = 7
  • 7x +4 = 7²
  • 7x + 4 = 49
  • 7x = 45
  • x = 45/7

You can now substitute x = 45/7 into the original equation √7x -2 =5. Thus: 7-2 = 5

Solving Equations of Imaginary Numbers Involving Division

To divide imaginary numbers, you multiply the numerator and denominator by the complex conjugate a - bi. In this case, assuming a - bi is a complex number, then you will have:

(a + bi) (a - bi) = a²+ b²

Example

Divide

x = (5 - 3i) / 4 + 2i

Solution

Multiply top and bottom by 4 - 2i

  • (5 - 3i)(4 - 2i) / (4 + 2i)(4 - 2i)
  • (20-10i-12i+6i²) / (16 - 8i + 8i - 4i2)
  • (20 - 22i + 6i²) / 16 - 4i2
  • (14 - 22i) / 20

Understanding the Practical Application of the Concept of Imaginary Numbers

Also called complex numbers, imaginary numbers are applicable in real life. For instance, in quadratic planes, these numbers show up in equations that do not touch the x-axis. Imaginary Numbers are especially very useful in advanced calculus.

Imaginary Numbers are also very essential in electricity, especially in alternating current (AC) electronic devices. Here, the AC electricity alternates between positive and negative in a sine wave. As such, combining AC currents can be extremely challenging. So using imaginary currents and real numbers has helped solve this problem by making it possible to do the calculations to avoid electrocution.

Lastly, imaginary numbers are essential in signal processing. This is especially so if what you're measuring relies on cosine or sine wave. Signal processing is vital in cellular and wireless technologies as well as radar and brain waves.

Imaginary Doesn't Mean Impossible

Initially, imaginary numbers were considered impossible to solve. From this discussion, however, it's evident that they're not as complex as they seem. You can actually solve problems involving these types of numbers. Knowledge of imaginary numbers has deep significance and profound importance to the understanding of Physics and Mathematics.

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