Confusion in representation of complex numbers .

Question

To begin, in the above image, $z$ has been taken to to be $x+iy$, allowing a multiplication by $i$ to result in: $iz=-y+ix$. In the diagram, they have actually shown the 2nd quadrant than the 4th quadrant. If the point given by $z$ was in the 4th quadrant, then wouldn't the angle be $-90^{\circ}$?

Finally, when we have $x+iy$, then it means that the y axis is the imaginary axis. Only, here we see the x axis as said imaginary axis. How is this possible?

Answer 1

The notation is misleading. It would be easier to understand, if they used following notation:

• Coordinates of the point A: z1 = x1 + iy1
• Coordinates of the point B: z2 = iz1 = x2 + iy2 = -y1 + ix2

Answer 2

The coefficient of $i$ gives us the vertical position.

$x+iy$ corresponds to the coordinate $(x,y)$. When $x, y $ are both positive, it is a point in the first quadrant.

$iz=ix-y = -y+ ix$ corresponds to the point $(-y, x)$. If $x, y$ are both positive, the first coordinate $-y$ is negative and the second coordinate $x$ is positive and hence we get a point in the second quadrant.

In general $iz$ rotate $z$ by $90^\circ$ anticlockwise.

Answer 3

You have two axes: the axis in the "$1$" direction and the axis in the "$\mathrm{i}$" direction. Any point of the plane $(a,b)$ can be written as having coordinate $a$ in the $1$ direction and coordinate $b$ in the $\mathrm{i}$ direction: $$ 1 \cdot a + \mathrm{i} \cdot b \text{.} $$ Of course, no one ever writes down the "$1\cdot {}$", so we actually write and see $$ a + \mathrm{i} b \text{.} $$

There is a potential ambiguity in the uses of $x$ and $y$. $x$ is being used as both the label for the real axis and for the first coordinate of the point $A$. $y$ is being used as the label for the imaginary axis and for the second coordinate of the point $A$. It would have been better to not use the axis labels as coordinates of the point $A$, but it was done. You may be better served (mentally) replacing the label on the horizontal axis with "Re" and the label on the vertical axis with "Im".

So you start with point $A$ having coordinates $(x,y)$, corresponding to the complex number $x + \mathrm{i} y$. Notice that $x$ and $y$ are positive real numbers, so $A$ is in the first quadrant. Then you multiply by $\mathrm{i}$, obtaining $\mathrm{i} x - y$. That is, you have $$ 1 \cdot (-y) + \mathrm{i} \cdot x \text{.} $$ This is the point with coordinates $(-y,x)$ which has negative first coordinate (left half-plane) and positive second coordinate (upper half-plane), so is in quadrant II, as labelled by the point $B$.

Answer 4

You’re letting the letters $x$ and $y$ confuse you. Let’s start with a complex number $u+iv$ instead, where $u$ and $v$ are positive real numbers. This complex number has real part $u$ and imaginary part $v$. (Remember, the imaginary part is the coefficient of $i$.) It is plotted at the point $\langle u,v\rangle$ in the first quadrant, because we normally plot the real part on the $x$-axis and the imaginary part on the $y$-axis.

When you multiply the number by $i$, you get $ui-v$, or $-v+ui$. Now the real part, i.e., the part without the factor of $i$, is $-v$, and the imaginary part is $u$, so we plot the complex number at the point $\langle -v,u\rangle$, which is in the second quadrant.

It’s fine to use the terms $x$-coordinate and $y$-coordinate instead of real coordinate and imaginary coordinate, or abscissa and ordinate, but only if you remember that the $x$ and $y$ in these terms have nothing to do with the symbols used to denote the actual coordinate numbers. This is true even when you’re just dealing with points in the plane: the $x$-coordinate of the point $\langle y,x\rangle$ is $y$, not $x$.