Problem comprehending the representation of complex numbers by a unit circle

I'm having a very basic problem with understanding the representation of complex numbers using the unit circle.

An example of what I'm struggling with is shown in the initial diagram on this page (top right):

https://en.wikipedia.org/wiki/Euler%27s_formula

The circle is shown centred at the origin of axes whereby 'x' represents the real number component and "y" represents the imaginary number component.

The unit circle crosses the x axis at +1 and -1. It crosses the y axis at +i and -i.

My problem with comprehending this representation is that it suggests to me that "1" on the x axis is the same magnitude as "i" on the y axis. This condition would conform with the concept of the unit circle.

In my (obviously mistaken) way of looking at this I would have thought that "1" and "i" cannot be considered to have the same magnitude - and thus the representation is not circular but something else such as elliptical.

I'd appreciate any suggestions to help achieve the necessary correction to my faulty way of thinking!

Many thanks, Peter.

I think a helpful approach for me might be as follows:

The complex number 'a + bi' is plotted in a complex plane where the horizontal axis is to be thought of as the real axis and the vertical axis is to be thought of as the imaginary axis.

In the form 'a + bi' both a and b are real numbers.

The magnitude or absolute value of the complex number is then thought of its distance from the origin of these axes.

This can be written as |a + bi| and is equal to be square root of the sum of a squared and b squared.

In this context the unit circle is the set of complex numbers whose magnitude is one. On the complex plane they form a circle centred at the origin with a radius of one. It includes the value of 1 on the right extreme, the value 1i at the top extreme, the value -1 at the left extreme, and the value -1i at the bottom extreme.

Thus in the visualisation of complex numbers by a unit circle the axes are intersected where |a| = 1 and |b| = 1.

This would allow me to be satisfied with the representation as specifically circular rather than more generally elliptical.

In order for me to completely happy with this I wonder if it is fair to say that a complete representation of a complex number is actually 'a1 + bi' - as this makes it totally clear that a and b are conceptually the same - i.e. coordinate values. So in the initial diagram that I refered to above, my difficulty in comprehension arose from the labelling of the intersection of the vertical axis as 'i' when strictly it might better be shown as '1' (of imaginary units of which this axis is formed).

• It might help to understand why you think $1$ and $i$ cannot have the same magnitude? Note that your intuition for magnitude from real numbers may not apply to complex numbers. – Dustan Levenstein Nov 3 '17 at 13:50
• Thanks very much for your comment. My difficulty was not in thinking that 1 and i cannot have the same magnitude - but that it seemed to me like an unfounded assumption that we CAN show them as having the same magnitude in the unit circle representation. However RoyPJ's answer below suggests that the scale chosen for the axes is actually completely arbitrary, which I need to try and assimilate into my way of thinking about the representation. – P Tetlow Nov 4 '17 at 9:28
• Your thoughts are absolutely correct! The absolute value calculation is derived from the Pythagoras formula by the way, but I guess that's clear. I have nothing more to add :) – RoyPJ Nov 9 '17 at 12:07

I think your thought that $1$ cannot be considered to have the same magnitude comes from $i = \sqrt{-1}$ and therefore you think that $i$ should somehow be 'smaller' than $1$.
So first of all make yourself free from any correlation of the real part of a number and its imaginary part. Just consider $a = 1+i$ as a different notation of a point $a$ in a two dimensional grid at $(1|1)$. Hence $Re(a)$ is the $x$ coordinate and $Im(a)$ the $y$ coordinate (on a $x,y$-scale of course). How big or small the you choose the scales of the two axis is completely arbitrary, so visually you are right, the unit circle could as well be elliptical if you choose the scale differently.