# Multiplication of complex number by $~i~/-i~$

Multiplication of non-zero complex number with $$~-i~$$ rotates the point about origin through a right angle in anticlockwise direction. True or false?

My attempt :- actually I don't know how to proceed in general but I did it with an example of $$~1+i~$$ and by multiplying it with $$~-i~$$ the point turns $$~ 90~$$ in clockwise direction. Also if we multiple the above complex number by $$~i~$$ it rotates by $$~90~$$ in anti-clockwise direction. Is the above result is true in general of just for particular example.

Also general proof will be appreciated. Thanks

• OK, not everybody can be bothered to switch to upper case when typing the word I. But surely when you are talking about multiplication by the square root of $-1$, you can seen the sense in making the effort? Commented Jul 27, 2019 at 13:22
• Yes, sorry i forget about it Commented Jul 28, 2019 at 11:16
• It seems you are incorrigible :-) Commented Jul 28, 2019 at 11:18

I assume that by “anticlockwise” you mean “with the same verse of rotation by which the positive $$x$$-semiaxis reaches the positive $$y$$-axis by the least angle”; “clockwise” is the opposite verse of rotation.

Then multiplication by $$-i$$ induces a clockwise rotation by a right angle, because $$1\cdot(-i)=-i$$, which in the standard representation of complex numbers has a negative $$y$$-coordinate.

As you see, there are many implicit assumptions in the statement of the problem.

If you decide to represent the $$x$$-axis in the usual fashion (horizontal, positive verse from left to right) and the $$y$$-axis upside down with respect to the usual way, then, with the intuitive understanding of clockwise and anticlockwise, multiplication by $$-i$$ induces an anticlockwise rotation by a right angle. That's why I initially gave a definition that's independent on the graphical representation.

Another case: suppose your teacher is drawing on a side of a transparent board and you're on the opposite side; you and the teacher will see different verses of rotation, under the common understanding of clockwise ad anticlockwise. Not if you stick to the definition I gave.

By the way, the “anticlockwise” verse I defined at the top is usually and better called positive.

It's easy to see in general, by drawing in the coordinate system. The complex number $$a+bi$$ is represented as (the vector from the origin to) the point with coordinates $$(a, b)$$.

We have $$(a+bi)\cdot i=-b+ai\\ (a+bi)\cdot(-i)=b-ai$$

• So then what can we conclude now? Commented Jul 28, 2019 at 11:22
• That multiplying by $\pm i$ rotates the vector corresponding to the complex number by $\pm 90^\circ$. Commented Jul 28, 2019 at 13:56

This is wrong. Usually one puts the unit $$i$$ on the vertical coordinate axis in upright position, so that to go from $$1$$ to $$i$$ is an anti-clockwise rotation by $$90°$$. Then $$-i$$ is in the downward position and multiplication by $$-i$$ rotates by $$90°$$ in clockwise direction. Remember that $$1=1+0i$$ is at the 3:00 position on the clock, $$i$$ at the 12:00 position and $$-i$$ at the 6:00 position.

• According to you also multiplying by '-i' rotate the complex number in clockwise direction, then why so said this is wrong! Commented Jul 28, 2019 at 11:44
• Because in your question, the first claim is that the rotation by $-i$ is anti-clockwise. Commented Jul 28, 2019 at 13:09