Addition of Complex Numbers

Definition

Addition of complex numbers is the process of combining two or more complex numbers to create a new complex number. Each complex number is expressed in the form $$a + bi$$, where $$a$$ is the real part and $$b$$ is the imaginary part. This addition involves adding the real parts together and the imaginary parts together separately, resulting in a new complex number that retains the structure of the original numbers.

5 Must Know Facts For Your Next Test

1. To add two complex numbers, simply add their real parts and their imaginary parts: if you have $$z_1 = a + bi$$ and $$z_2 = c + di$$, then $$z_1 + z_2 = (a+c) + (b+d)i$$.
2. Graphically, complex numbers can be represented on the complex plane, where the x-axis corresponds to the real part and the y-axis corresponds to the imaginary part.
3. When adding complex numbers, commutativity holds true, meaning that $$z_1 + z_2 = z_2 + z_1$$ for any two complex numbers.
4. Associativity also applies, so when adding three or more complex numbers, the grouping does not affect the result: $$(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$$.
5. The result of adding a complex number and its conjugate will yield a real number since the imaginary parts cancel each other out.

Review Questions

• How do you perform the addition of two complex numbers? Give an example.
  • To add two complex numbers, you combine their real parts and their imaginary parts separately. For instance, if we take the complex numbers $$z_1 = 3 + 4i$$ and $$z_2 = 1 + 2i$$, we add them like this: first add the real parts: $$3 + 1 = 4$$; then add the imaginary parts: $$4i + 2i = 6i$$. Therefore, the sum is $$4 + 6i$$.
• Explain why addition of complex numbers is commutative and provide a mathematical demonstration.
  • The addition of complex numbers is commutative because rearranging the order of addition does not change the sum. For example, if we have two complex numbers $$z_1 = a + bi$$ and $$z_2 = c + di$$, when we calculate both forms of addition, we find: $$z_1 + z_2 = (a+c) + (b+d)i$$ and $$z_2 + z_1 = (c+a) + (d+b)i$$. Since addition of real numbers is commutative ($$a+c = c+a$$ and $$b+d = d+b$$), we see that both forms yield the same result.
• Analyze how the graphical representation of adding complex numbers reflects their algebraic addition.
  • When adding complex numbers graphically on the complex plane, each complex number is represented as a point or vector from the origin to its coordinates $(a,b)$. To add two complex numbers visually, you can use vector addition. For example, if you have vectors for both numbers starting at the origin, you can place them head-to-tail. The resultant vector from the origin to where you end up after placing both vectors represents their sum. This visual method shows that adding their corresponding real and imaginary parts directly corresponds to moving along those axes in a straightforward manner.

Related terms

Complex Number: A number that consists of a real part and an imaginary part, typically expressed as $$a + bi$$.

Real Part: The component of a complex number that does not involve the imaginary unit $$i$$, denoted as $$a$$ in the expression $$a + bi$$.

Imaginary Part: The component of a complex number that is multiplied by the imaginary unit $$i$$, denoted as $$b$$ in the expression $$a + bi$$.