Is zero (0) the only complex number that is both purely imaginary and purely real, simultaneously?

Question

Oh come on people... think a little harder!

a complex number is in the form a+bi

If b = 0 then its purely real, if the a = 0 then its purely imaginary.

If a and b both = 0 then its 0+0i, or simply 0.

0 is the only numerical value on the complex number plane that exists on each of the pure axes.

Answer 1

It is "purely" a question of definition:

A complex number has the form a + bi where a and b are real numbers and i is such that i^2 = -1.

A "purely" real number is a complex number where b=0.
A "purely" imaginary number is one where a=0.

Since 0 = 0 + 0i, then 0 meets both definitions.

In real numbers, there is only one "zero". Therefore there are no other ways to write a + bi with a=0 and b=0. This confirms that the complex 0 is the only element that meets both requirements simultaneously.

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However, there are such things as complex cyclic groups. For example, the group
{a + bi : a and b belong to Z(7)}
where each of a and b are elements of the Z(7) group. Therefore, IN THAT GROUP, the complex element 14 + 7i is congruent to the complex 0 and is simultaneously purely imaginary and purely real.

Answer 2

As I understand it, the only imaginary number is "i" which is used as a function of the square root of a negative number.

Answer 3

yes