This equation shows the cartesian form of any given complex number, given a real part (\( \htmlClass{sdt-0000000032}{\Re} \)) which here is assigned to the variable \(a\), an imaginary part (\( \htmlClass{sdt-0000000129}{j} \)) which here is assigned to the variable \(b\), and the standard imaginary unit (\( \htmlClass{sdt-0000000129}{j} \)).
| \( j \) | This symbol represents the imaginary unit, which is defined as the square root of \(-1\). \( j = \sqrt{-1}\). It is the most fundamental unit in the field of complex numbers, allowing for the expression of numbers that cannot be represented on the real number line. |
| \( Z \) | This symbol represents any given complex number on the complex plane. |
Consider the definition of the real part and imaginary part of a complex number:
The symbol \(\Re\) represents a function that takes in a complex number, and outputs the real part of that complex number. For example, \(\Re(a + b\htmlClass{sdt-0000000129}{j}) = a\). To emphasize this, the real part of a complex number is always a real number itself.
The symbol \(\Im\) represents a function that takes in a complex number, and outputs the imaginary part of that complex number. For example, \(\Im(a + b\htmlClass{sdt-0000000129}{j}) = b\). It is worth noting, however, that the output is still a real number as the fact that it is multiplied by \(\htmlClass{sdt-0000000129}{j}\) is implied.
From here it follows that the form is:
\[ \htmlClass{sdt-0000000006}{Z} = a + b \htmlClass{sdt-0000000129}{j} \]
as required...
Consider some complex number \(\htmlClass{sdt-0000000006}{Z}\) that is has a real part of 5, and an imaginary part of 10. What is it in it's cartesian form?
From here it follows that, in cartesian form, the complex number is
\[ \htmlClass{sdt-0000000006}{Z} = 5 + 10\htmlClass{sdt-0000000129}{j} \]