Complex Absolute Value

If $a+bi$ is a Gaussian Integer, then...

$|a+bi|=\sqrt{a^2+b^2}=5 ~~~~~ \Longrightarrow ~~~ a^2+b^2=5^2=25$

Since $a$ and $b$ are integers, we first look for Pythagorean triples. We have, $3^2+4^2=4^2+3^2=5^2$ . Since $a$ and $b$ are not necessarily natural numbers, we must also consider the negative versions of them. Considering possible arrangements, we have $2\times 2^2=8$ solutions.

Note that, $0^2+5^2=5^2+0^2=5^2$ is also a solution. Again considering negative cases, we get $4$ more solutions.

In total, we obtain: $8+4=\fbox{12}$ solutions.

For the magnitude of the given complex number to be 5, there are following combinations possible in the format of (a,b):

(5,0), (-5,0), (0,5), (0,-5), (3,4), (3,-4), (-3,4), (-3,-4), (4,3), (4,-3), (-4,3), (-4,-3).

These are the 12 Solutions possible.

Of course the problem has not been stated correctly :) $a$ and $b$ are assumed to be integers (otherwise answer would be infinite!).

We want: $\sqrt{a^{2}+b^{2}}=5$

Thus,

$a^{2}+b^{2}=25$ .

There only only two possibilities (without ordering!) in positive integers: $(0,5)$ and $(3,4)$ for $a$ and $b$ ..

Now we note that for a given pair, we can attach different signs to $a$ and $b$ and also we can permute the resulting pairs. This way, it is easy to see that there are total $12$ possibilities!

2 points (3,4), (4,3) in QI, similarly by changing signs, 2 more in each quadrant, giving 2*4 = 8. Two points (0,5), (0,-5) on x-axis. Similarly, 2 on y-axis for 4 more. So answer is 8+4 = 12.

this are that integers 1)4+3i 2)3+4i 3)5 4)0+5i 5)-3+4i 6)-4+3i 7)-5 8)-3-4i 9)-4-3i 10)0-5i 11)3-4i 12)4-3i

The absolute value of a complex number is of the form $a+bi$ is the same as $sqrt{a^2+b^2}$ . $a^2+b^2$ must be equal to 25 we can see that the only solutions are $(\pm5,0)$ , $(0,\pm5)$ , $(\pm3,\pm4)$ . Thus, there are 12 solutions.