Adding Complex Numbers
In this detailed exploration, we will delve into the process of adding two complex numbers.
When tasked with adding two complex numbers, denoted as (a,b) and (c,d), you will find that the operation boils down to simply adding the corresponding elements of each pair, which results in the new pair, (a+c, b+d). In equation form, we can represent it as follows: $$ (a,b)+(c,d)=(a+c,b+d) $$
In essence, this operation involves adding the real component to the imaginary component of the complex numbers.
To provide a better understanding, let's consider a concrete example.
Let's choose two complex numbers arbitrarily:
$$ z_1 = (2,3) = 2+3i $$
$$ z_2 = (4,2) = 4+2i $$
These numbers exist on the complex plane.
Now, let's proceed to add the real and the imaginary parts of the complex numbers together
$$ z_1 + z_2 = (2,3) + (4,2) = (2+4,3+2) = (6,5) $$
This operation yields a new complex number, (6,5) or 6+5i.
From a geometric perspective, this process is equivalent to vector addition on a plane.
Therefore, the sum of two complex numbers can be visually represented and calculated using either the parallelogram method or the tip-to-tail method.
Furthermore, complex numbers can still be added when they are expressed in their algebraic form
$$ z_1+z_2 = (2+3i)+(4+2i) $$
The operation remains straightforward - simply combine and add the real and imaginary parts of the complex numbers:
$$ z_1+z_2 = (2+3i)+(4+2i) = (2+4)+(3i+2i) = 6+5i $$
As we can see, the result remains consistent.
It is worth noting that if the imaginary parts of the complex numbers are zero, the sum simplifies to the addition of two real numbers: $$ z_1+z_2=(a,0)+(b,0)=(a+b,0+0) = (a+b,0) = a+b $$ For instance, let's consider two complex numbers that are purely real: $$ z_1=(4,0) \\ z_2=(2,0) $$ The sum is also a purely real complex number: $$ z_1+z_2=(4,0)+(2,0)=(4+2,0+0) = (6,0) $$ This simply reflects the sum of real numbers. Both the complex numbers z1 and z2, and their sum z1+z2, reside on the real axis of the complex plane.
Properties of Complex Numbers
Lastly, the addition of complex numbers is a well-behaved operation. It demonstrates key mathematical properties such as commutativity and associativity:
- Commutativity property
Changing the order of the numbers doesn't affect the sum. $$ z_1 + z_2 = z_2 + z_1 $$ - Associativity property
The grouping of numbers doesn't affect the sum. $$ (z_1 + z_2) + z_3 = z_1 + (z_2+z_3) = z_2 + (z_1+z_3) $$ - Identity element
The identity (or neutral) element for addition in the set of complex numbers is the number zero, represented as (0,0) $$ z_1 + 0 = (a,b) + (0,0) = (a+0,b+0) = (a,b) $$
If you find this exposition from Nigiara helpful, we encourage you to stay tuned for more in-depth explanations.
