That’s amazing!!! I just sat back in wonder when i read your explanation of how complex numbers were rotations…and then again when you explained how to use them instead of finding angles. I really like the way you explain math, it’s all very visual, and even the parts that don’t have pictures really conjure up understandable mental images in the reader.
I only wish high schools taught math like this. Most teachers don’t even introduce the unit circle when teaching trig (which is horrible!) and I know that my fellow ib nerds would love to be taught this way.
Hi Ian, great question. When multiplying two complex numbers, we need to combine all the parts (more here: http://betterexplained.com/articles/how-to-understand-combinations-using-multiplication/). Imagine multiplying 13 x 14:
(10 + 3)(10 + 4) = 1010 + 310 + 410 + 3*4 = 100 + 30 + 40 + 12 = 182
We split 13 into “10 + 3” and 14 into “10 + 4” then combined each possibility: 10 & 10, 3 & 10, 4 & 10, 3 & 4. In a similar way, multiplying the parts of the complex number means we combine each part. When combining, if we ever get i*i, we can simplify that to -1.
Thank you very much. Very clear explanation of imaginary numbers! Helpful to understanding imaginary time.
Hi Cindy, thanks for the note! Hope it helps with your explanations :).
Ah-ha! Many thanks for your reply Kalid 
I’m fully with you now! If you can teach complex numbers to someone as dumb as me then you’re definitely doing something right! lol. Thanks again
Great explanation. I came across this trying to refresh my memory and this did it.
Thanks very much.
Really nice explanation. Thanks a lot!
Love the article; it made things so much clearer. The thing about turning 1 to -1 intrigued me, even though the concept seems so simple… Are you saying that all multiplication sort of starts with 1? Likewise, does all arithmetic start with 0? I wouldn’t have thought this idea was all that important before reading this article, but now it does seem like it has some significance; the concept of getting from 1 to -1 wouldn’t make sense if 1 was not a starting point. This is because 1 is the multiplicative identity (if that’s the right term…), right?
Thank you! The link was wonderful. Keep up the great work, it’s a lot of fun to read your pages. -Tom
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My teacher wants me to make a power point on “The real life application of imaginary numbers” hurrray. Now i get to research. Thanks for the article and helping explaint to me what imaginary and complex numbers are all about.
Thanks Tyler, glad it helped.
Thanks Tom, glad you enjoyed it!
No problem, you’re more than welcome. And it’s awesome that you’re helping encourage kids to keep going with math! 
- Great question, let me try to clarify.
- A “unit” is a measure of distance (like a mile)
- An “axis” is a measure of direction (like North/South or East/West)
Imagine a street map. There are 2 axes (North/South and East/West), and both are measured in units of a “mile” (let’s say). If we want to move somewhere, however, we might say “1 Mile North” or “3 miles East” – i.e., you need to specify the distance and direction.
In the case of imaginary and real numbers, we have two axes (East/West = real (positive and negative), and North/South = imaginary (positive and negative)). Distance traveled on each axis is measured in the same “unit” (there’s no specific unit like miles, we just call them “units”).
It’s true that you could have different measures of distance (i.e., the real axis uses “feet” and the imaginary axis uses “miles”) but in math, we assume the units are the same size on each dimension. (And if they weren’t, in some given application, you’d probably convert them first… i.e., a mile would be 5280 feet, and you’d write that down on the “feet” axis).
So, the assumption is that when dealing with the real & imaginary axis, the distance measures are using the exact same units.
- Awesome questions.
This question of “how” almost assumes that the essence of the two numbers can take on an interaction between each other. Do you believe your description can extend to concepts beyond numbers?
Yep, I think the concept of “multiplication” can be used on lots of other ideas in math. In calculus, we “integrate” functions, which is a beefed-up form of multiplication (or alternatively, multiplication is a special-case of integration). The essence of “applying” one concept to another shows up in many places, but we don’t always use the term “multiply”.
Speaking of extension, to extend this conversation even further, how would you then make a similar abstraction for Addition? What would you say are the distinguishing characteristics between the two abstractions such that they deserve their own term?
You may like this article:
http://betterexplained.com/articles/rethinking-arithmetic-a-visual-guide/
To me, addition can mean “accumulate, slide or combine” depending on what is being added. In the case of chemistry, yep, I’d say addition corresponds to combining. (2 Hydrogen + 1 Oxygen = Water + Excess Energy).
With nature, yes, a lot of it depends on the metaphors we’re using. Individual families may add new members, specials as a whole may multiply exponentially. I think part of it is stepping back and looking at what is happening, vs. what we named it :). We shouldn’t confuse the sign on the road for the road itself (very Zen!). Basically, we’ve given words some meaning (addition, multiplication, etc.) but we need to see what’s actually happening to the number, and whether a better word could apply (“Hrm, I’m saying add here, but do I mean combine? I’m saying multiply, but do I mean scale?”).
For addition vs. multiplication, my intuitive meanings are basically:
- Addition: accumulate, slide, combine
- Multiplication: repeated counting, scale, apply
Multiplication can look like “faster addition” if you’re talking about repeated counting vs. accumulating one at a time. But that’s only one use of the term, and other uses (like scaling something to make it larger or smaller) don’t have a direct analogue to addition.
Hi Kalid, glad to hear that the units are assumed to be the same. This is consistent with my new understanding of what complex numbers are and how they are used. Nice !
Ready for some more curious (but familiar) questions ?
In your response to number 2, your wrote:
“Basically, we’ve given words some meaning (addition, multiplication, etc.) but we need to see what’s actually happening to the number, and whether a better word could apply (“Hrm, I’m saying add here, but do I mean combine? I’m saying multiply, but do I mean scale?”).”
So, what seems to happen is that depending on the area of math one is studying, you may be given a different definition and interpretation for operations such as Multiplication (I think even within the same area of Mathematics there will sometimes exist varying definitions). I find it odd that one can create a set of axioms for numbers and then say “Here is how we will define multiplication for this number system…”. And then, the next number system being stated will say: “And - here is how WE will define multiplication…” And of course, the two definitions do not always seem consistent as you try to interpret the implications/interpretations/intuitions. One intuition says scale, one says repeat, the other says rotate - all derived from the use of the same term. So, I have to ask…with all of these different definitions and interpretation of these operations that float around, do they all have to map back to some set of axioms or laws in order for a mathematical system to invoke its definition of multiplication ? (Sounds similar to my first question, but differs b/c now it is less about intuition and more about requirements)
Or is it the case that the terms gets used in various systems of mathematics with no minimum criteria that must be met. That is, could I create a set of axioms and then use those axioms to define Multiplication as whatever I see fit - no relationship at all required with the previous use of term? Or is it that mathematicians use an agreed upon intuitive principle to define “how” Multiplication should look like in your system when trying to impose the properties of one mathematical object on another ?
If you tell me, nope it is whatever you want to define it as (just depends on context, just like in the English language) and is just a tragedy (or perhaps a blessing in some unexpected way) in the field of mathematics - it would mean something to me. If you told me, Mathematicians use the high level principle which maps back to “how” you believe the essence should be applied b/w 2 numbers - it would mean something to me. If you say, Alonzo there are a set of axioms that are required to be true or properties that must hold before you can use the term multiplication and here they are - it would mean something to me. No matter what the answer, clearing it up would mean something to me. Although if you say I have no idea, that actually would mean something to me also because it tells me I am not the only one with this question.
And maybe you have already said it in one of your previous posting to me and I missed it. For example, you wrote, “I think we’re still discovering what multiplication in it’s purest form”…so is this support of the idea that we don’t really know what it is or has become - so folks just define it in a way that is useful for their system? But then, wouldn’t this force the use of the term multiplication to become meaningless overtime. Why, why, why so many different definitions for the same term? How are we allowed to do this in such an precise field of study ? It seems like math is consistent once a term is defined for a particular system, but the field of study is not consistent with the use of its terms among the various areas. I am rambling now - so I’ll shut myself up.
Hi Alonzo, glad if some things are coming together :). You’re raising very good, very deep math questions which are nudging up to the edge of my formal math knowledge!
Intuitively, we’ve developed better and better understanding of what “numbers” are. We first thought numbers were for counting rocks (integers). But wait, we have fractions! (Rational numbers). And there are numbers between fractions! (Real numbers). And negatives (only discovered in the 1700s!). And complex numbers.
You’re correct: each new set of numbers required us to redefine what “multiplication” meant (“Hrm, what does -1 x -3 mean? It can’t be repeated counting…”). It often happened that at the “lower” levels (integers), multiplication was a special case of what happened at the higher levels (repeated counting is a special case of scaling, when you’re dealing with whole units).
Mathematically, the definitions of numbers and their operations is called an “algebra”. (“Elementary algebra” is what we think of as solving x^2 + 3x = 5; “Algebra” as a class is about the very definitions of math). In an algebra, you can define what a number is, what types of operations can be done (add / subtract / multiply / divide), whether there are any special elements (anything times 0 is 0), and so on.
Wikipedia has an article (http://en.wikipedia.org/wiki/Algebra), but like many Wiki articles, you already need to be an expert to understand it. But there is a cool chart midway down that shows different types of numbers (Natural numbers, Integers, etc.) and the different operators and properties they have. Phew!
Yes, for laymen, I think “multiplication” is changing over time, just as “number” is changing over time (500 years ago, numbers could only be positive; 2000 years ago, numbers could only be fractions… in 500 years, who knows what “number” will refer to?).
In everyday discussions, I’d say “number” and “multiplication” refer to the most popular systems of the day, so “number” means real number (since people are comfortable with decimals, but not imaginary numbers) and “multiplication” means scaling (since they are comfortable with decimals, multiplication can stretch or shrink a number, but not rotate it).
In more rigorous math discussions, however, you would need to mention what type of numbers you’re describing (one of the common types, like Integers or Complex numbers, or your own type). And if it’s your own type, you need to describe the behavior of the “x” symbol (which you may call Multiplication).
[…] Don’t forget, we thought systems like x^2 + 1 were “non-zeroable” until imaginary numbers came […]
@Jyeisha: Glad it was helpful. To me, when I see a number like -1, the negative sign encompasses the “opposite-ness” of the number. -1 is an “opposite 1”.
When you see “i”, you can consider it encompassing the “rotation-ness” of a number. i (really, 1*i) is a “rotated 1”. The key trick is allowing numbers to exist in 2 dimensions (and why not? Numbers are ideas used to keep track of things).