Thanks for the post it was really helpful. I thought you might be instrested in another take on the intuition it is a little hard to comprehend buy very inuitive. https://youtu.be/1rVHLZm5Aho He also has a video on Euler’s formula.
Thanks Frederico! Just updated. For this phrase:
Thinking we’ve “figured out” a topic like numbers is what keeps us in Roman Numeral land.
I meant if we believe we’ve found the “final” form of numbers, we would have stuck with Roman numerals. We have to be open to a newer, better way to represent numbers. (Some people fight the notion of imaginary numbers, just like some Romans might have fought the notion of decimal numbers.)
Hi, Kalid!
I’m translating this post (to www.energiaeletrica.net, as I did with your Calculus Course). What do you mean by “Thinking we’ve “figured out” a topic like numbers is what keeps us in Roman Numeral land.”? I couldn’t catch the idea.
By the way: could you update the translation page? I’ve translated http://betterexplained.com/articles/rethinking-arithmetic-a-visual-guide/ in: http://www.energiaeletrica.net/blog/repensando-a-aritmetica-um-guia-visual/
Thanks.
Although, I learnt about complex numbers in my high school. This post just refreshed my lost memory instantly. Could you please also do the same on quaternions?
Being a Berkeleyan I can give an example of the ‘tangibleness’ of a negative quantity. Suppose you are used to drinking a cup of milk every morning. Now suppose that a shortage in milk made it necessary for you to drink only 1/2 cup of milk instead. The negative 1/2 cup that you can’t drink is just as tangible as the 1/2 cup that you do drink. You miss that 1/2 cup. The point is that everything is relative to your perception. And negative amounts are just as tangible as positive amounts. Only that which is perceptible is real. String theory is a delusion because it deals with dimensions that are not perceptible. IMHO 
Sir I have a question For U .
Which Family is bigger Real Numbers family Or imaginary Numbers family(Dont be confuse to take imaginary as complex)
Dear Khalid,
Thanks for the wonderful post. I never came across such an intuitive explanation before!
Could you kindly elaborate more on how 1 + i is 45 degrees??? I just didn’t get it…and it did not seem to connect from the earlier discussion too. May be I am too naive…
Thanks in anticipation,
HK
Hi Haris, you can consider 1 + i as making the sides of a triangle (1 unit “East” and 1 unit “North”). This triangle makes the diagonal of a square and is 45-degrees (half of a 90-degree angle). For other numbers (1 unit East and 3 units North) the angle has to be worked out with trigonometry. It might help to draw it out to see. Hope this helps!
Thank you! I have a teenage daughter in High School Algebra II wondering why she has to learn this stuff. I have a degree in Physics and Electrical Engineering and was really having a hard time finding a non college level physics or engineering example as to why she should know this other than making it easy to derive trigonometric identities via Euler’s formula!
Well done Sir!
@Swoorup: Glad it helped – yep, quaternions are a planned topic :).
@John: That’s great to hear, I struggled for ages to find a more accessible use case for imaginary numbers than phasors, voltage/current, etc. Making trig identities easier is a topic I’m working on now actually. Appreciate the note!
Thank you so much for that wonderful article explaining the confusion my mind was in!!! I googled and went through many sites but only yours helped me really understand about complex numbers. Why aren’t you my maths teacher!!! Keep posting. Great effort and work. Sharing this with my whole class!!
Wow what a CLEAR and BRILLIANT post!!! I admire your patience in answering the many questions you receive.
Here’s one more question: Why are the ‘real’ and ‘imaginary’ parts of complex numbers attached with a PLUS SIGN?? What does this plus sign represent? Where does addition come in???
Thank you for your clarity, time, and patience.
Can I steal your brain? I promise I won’t make bad use of it 
Hi Kalid,
I can’t begin to describe how much your articles have helped me. I am a college senior and I always knew that I didn’t really get what I learned in my math and physics classes. I just mechanically solved problems and took the tests…but that doesn’t fly in the electromagnetism class I’m taking now. But I think with following your tips on learning via analogies I have started to actually learn to retain this material that is really challenging for me. Thank you so very much for your amazing teaching.
Hi,
I am still waiting for a reply to my question (#400):
Why are the ‘real’ and ‘imaginary’ parts of complex numbers attached with a PLUS SIGN?? What does this plus sign represent? Where does addition come in???
Thank you for your clarity, time, and patience.
@Mary Wiltz.
The real part is the parallell component of a force/object/ influence etc.
And the imaginary part is the perpendicular component of the force /object / influence etc.
Everything in nature usually is slightly horizontal (parallel to the ground) and slightly vertical (perpendicular to the ground)
Cos(angle) = parallel component of the object/force etc.
i.Sin(angle) = perpendicular component of the object/force etc
You can also see it this way.
i^(0) = 1 …means no rotation
i^(90) = i …means rotation by 90 degree.
But what if an object rotates by only 60 degree.
Lets say the length/force of the object involved is 5 metres.
So the horizontal influence(real part ) of this stick = 5cos(60)
Its vertical influence (imaginary part) of this stick = i.sin(60)
Together the position of the stick = 5cos(60) + 5.i.Sin(60)
5cos (60) = top view of the stick.
i.5.sin(60) = side-view of the stick.
I wrote some articles on this on my blog
One is as below.
REAL AND IMAGINARY PART OF COMPLEX NUMBERS EXPLAINED
http://visualizingmathsandphysics.blogspot.in/2015/07/the-real-and-imaginary-part-of-complex.html?m=1
HOPE THIS HELPS
BINNOY
visualizingmathsandphysics.blogspot.in
If the question is about complex numbers being written as “a+bi”, where a and b are “real numbers”, it actually is addition.
However, the addition and multiplication symbols are also used in alternative systems of mathematics in what is called “a field”. A “field” is any set of “numbers” in which operations -similar- to addition and multiplication - and division - can be done.
One approach to inventing complex numbers - contrary to the topic of this article - is to say, “What if negative numbers do have a square root? For instance, i is the square root of minus 1”, and, it turns out that what happens is a two-dimensional geometry of numbers.
Of course you can add number together, including these ones, so it is quite legitimate to write “2+3i”, which is merely the number which is 2 real plus (3 real x i). Some more work can be done to establish that this is a definite number.
Another plan which is mainly this reader’s own, is to get away from the paradox of “square root of a negative number” which is clearly nonsense, and to think of a complex number “n” as being an ordered pair or “tuple” of two real numbers (a, b). You may see where this is going if I call them East and North - like geometric coordinates. For instance, (a, b) isn’t the same number as (b, a), and you can have (a, a) as an ordered pair.
Along with these tuples are rules of addition and multiplication: (a, b) + (c, d) = (a+c, b+d), i.e. you “add” two tuples by separately adding the first parts and adding the second parts.
(a, b) x (c, d) = (ac - bd, ad + bc). If I’ve got that right (which is uncertain), it looks goofy, but, take it from me, these rules are going to work.
If you look at the tuples with the second part 0, these rules are saying that (a, 0) + (b, 0) = (a+b, 0), and (a, 0) x (c, 0) = (ac, 0). That is, the tuples containing any real number followed by 0 behave exactly like the real numbers on their own. “Adding” the tuples is the same as adding the numbers.
The tuple (0, 1), on the other hand, is i - because (0, 1) x (0, 1) = (-1, 0).
And that’s how to make complex numbers without believing that -1 really has a square root. 
For the addition part of this, as you may already see, you can also think about “vectors”. What? Well, for instance, the location of the nearest pharmacist to your home is a vector - so far east (or west), and so far north (or south). Or, of course, a straight line direction, and a distance. Then, from the pharmacist to the post office, another vector. But what if you want to go straight from your home to the post office? You add the vectors together. That means adding the two east-west components, and adding the two north-south components. (It’s a lot harder in the “polar coordinate” system.)
Although strictly this works better on a flat map, than on a round planet.
Thank you for such a wonderful article. I really like your approach to learning and teaching. I am really interested in science, so I also want to learn as much about maths (the “language of the universe”) as possible, so I can learn more about the mechanisms behind various phenomena, and your article really helped by explaining something that was quite counterintuitive to me!
Hi Eunice, that’s great! I started the site to help other students and it’s great knowing when it’s making a difference. Learning via analogies and finding the real meaning behind the concepts is awesome!
Very helpful article, I like your methods & analogy in explaining mathmatical concepts
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