Addition of Imaginary Numbers

Adding imaginary numbers is straightforward. Just add their imaginary parts together. In ordered-pair notation, the rule is:

$$ (0,a) + (0,b) = (0,a+b) $$

In algebraic notation, where the ordered pair (0,b) is written as bi, the same operation can be expressed as:

$$ ai + bi = (a+b)i $$

Example

Consider the following imaginary numbers:

$$ (0,2)=2i $$

$$ (0,3)=3i $$

To find their sum, add the imaginary components:

$$ (0,2)+(0,3)=(0,2+3)=(0,5) $$

The same calculation in algebraic notation gives:

$$ 2i+3i=(2+3)i=5i $$

So, the sum of 2i and 3i is 5i.

Note: The imaginary number 5i is simply another way of writing the ordered pair (0,5). Since the imaginary unit is defined as i=(0,1), we can write:

$$ 5i = 5 \cdot (0,1) = (5 \cdot 0,\; 5 \cdot 1) = (0,5) $$

Therefore, the ordered pair (0,5) and the imaginary number 5i are equivalent representations of the same imaginary number.

In general, adding imaginary numbers works just like adding like terms in algebra: add the coefficients and keep the imaginary unit i unchanged.