# Thread: How do you find the argument of 8i?

1. ## How do you find the argument of 8i?

The modulus is $\displaystyle \sqrt {x^2 + y^2}$ which here is just $\displaystyle \sqrt{64}$ = $\displaystyle 8$

And as the argument is defined by $\displaystyle tan^{-1} ({|\frac{y}{x}|)}$, and $\displaystyle x$ in this case is zero, you would be basically calculating the inverse tan of $\displaystyle \frac{8}{0}$, and it's impossible to divide through by zero! How, then, is it possible to find the argument of $\displaystyle 8i$?

2. Originally Posted by db5vry
The modulus is $\displaystyle \sqrt {x^2 + y^2}$ which here is just $\displaystyle \sqrt{64}$ = $\displaystyle 8$

And as the argument is defined by $\displaystyle tan^{-1} {|\frac{y}{x}|}$, and $\displaystyle x$ in this case is zero, you would be basically calculating the inverse tan of $\displaystyle \frac{8}{0}$, and it's impossible to divide through by zero! How, then, is it possible to find the argument of $\displaystyle 8i$?
What you're actually (kind of) trying to do is find the inverse tan of infinity. What is the value of $\displaystyle \tan{\frac{\pi}{2}}$ ?

Also, an easier way to see it is to just draw it on an argand diagram. 8i is on the imaginary axis, what angles does the imaginary axis make with the positive real axis?

3. Originally Posted by db5vry
The modulus is $\displaystyle \sqrt {x^2 + y^2}$ which here is just $\displaystyle \sqrt{64}$ = $\displaystyle 8$

And as the argument is defined by $\displaystyle tan^{-1} ({|\frac{y}{x}|)}$, and $\displaystyle x$ in this case is zero, you would be basically calculating the inverse tan of $\displaystyle \frac{8}{0}$, and it's impossible to divide through by zero! How, then, is it possible to find the argument of $\displaystyle 8i$?
The number $\displaystyle 8i$ is located on the positive imaginary axis.
That makes a right angle with the positive real axis.
Therefore $\displaystyle \arg(8i)=\frac{\pi}{2}$.

4. Originally Posted by pomp
What you're actually (kind of) trying to do is find the inverse tan of infinity. What is the value of $\displaystyle \tan{\frac{\pi}{2}}$ ?

Also, an easier way to see it is to just draw it on an argand diagram. 8i is on the imaginary axis, what angles does the imaginary axis make with the positive real axis?

I drew 8i on the argand diagram, and plotted the value of 8 on the imaginary axis but then that's the thing - there is no value to plot on the real axis because it is the only term so it doesn't make any angle at all to me, it looks just like a line and now I'm very confused

The value of $\displaystyle \tan{\frac{\pi}{2}}$ is....an error, according to my calculator. Does this mean infinity, then?
Obviously, this can't be found on a calculator, or to me on an argand diagram. Thanks for the help so far. It's just probably one of many simple things in mathematics that I haven't learned yet, and this example seems difficult. Is there any way you might be able to explain further?

EDIT: I see it now! It's just clicked. Thanks very much to you and Plato for this.

5. You're welcome. Glad we could help.