You’ve heard of counting numbers, natural numbers, whole numbers, rational numbers, and real numbers. Now get ready for imaginary numbers! It’s not that Imaginary Numbers they don’t exist: rather, they defy the laws of mathematics. Find the square root of 4, now 9, now 16. Most people would find the answers to be 2, 3, and 4, respectively, but they aren’t completely correct. Both 22 and -22 will give you 4. Think of if it this way—what 2 numbers when multiplied by itself would give you four? (Hint: One of them is 2, and they are additive inverses). After all, a negative times a negative gives you a positive. That is why you might see teachers or mathematicians write ±2 as the answer for √ 2. Now here’s the dilemma: what do you get from √-4? √-9? Or √-16? When multiplied, two positive numbers and two negative numbers always give you a positive, but if a square root can’t be one negative number multiplied by a different positive number make that number. Confusing, right? That’s where ⅈ comes into play. ⅈ is a term used by mathematicians to denote √-1. Thus, you can separate √-4 into √4*√-1. That can then be simplified down to 2ⅈ. In other words, ⅈ can be used to turn really confusing terms such as √-2704 to simpler answers (52ⅈ). Just think of all the instances when you can take out a negative radical to make it simpler to solve. The best way to think of it when doing operations with imaginary numbers is to think of i as a variable. Just like you can raise numbers and variables to a power, you can do the same to imaginary numbers. Imaginary numbers are like a mix of numbers and variables. When you raise X to a power, it remains that power. If you raise 2 to a power, it becomes repeat multiplication of 2 for that many times. ⅈ has properties like that. The following chart explains how powers of ⅈ function: Thus, If you have ⅈn, you simply divide n by 4, and the remainder will be either 0-4, and you use that remainder to determine what power of ⅈ you need to use in that situation.
Now comes the topic of complex numbers. Don’t worry, it’s elementary ⅈ dear Watson! You might have heard teachers and mathematicians say that every number has three invisible 1s: 1x^11. Well guess what? It has another invisible section, but this doesn’t necessarily have to be a 1. Complex numbers are a mixture of both real and imaginary numbers: a+bⅈ. Now, the rules for complex numbers are just like those of binomial operations. For addition, you take (a+bⅈ)+ (c+dⅈ), which can finally be simplified out to (a+c)+(bⅈ+dⅈ)For subtraction, it’s the same exact thing, but with subtraction signs. (a+bⅈ)- (c+dⅈ) turns into (a-c)+(bⅈ-dⅈ). For multiplication, it’s like binomial multiplication. You can use FOIL or any other method to multiply it out. (a+bⅈ)(c+dⅈ) turns to ac+adi+cbi+bdi^2. Go back up to the imaginary number exponent rules, and that simplifies to ac+adi+cbi+(-bd). Now onto my final point, the imaginary grid. You know that real-number graphs have an x-axis and a y-axis. Imaginary number grids have an x-axis with real numbers (the real part of the complex number (a in a+bⅈ.)), and the y-axis with imaginary numbers. So for real number grids, there is an X and Y coordinate but for imaginary grids, it’s graphed as a+bⅈ with (a, bⅈ). Hope you enjoyed this quick intro to imaginary numbers!
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By Mohammad FaisaanMohammad is our Resident Technical Director and Math Whiz, whose wide interests involve martial arts, trading card games, and Rubiks Cubes. ArchivesCategories |