Addition of Two Complex Numbers

Addition of Two Complex Numbers

When you add two complex numbers, you simply add their real parts together and their imaginary parts together. This is the most natural extension of addition from real numbers.

Take any two complex numbers $z_1 = a + ib$ and $z_2 = c + id$, where $a, b, c, d$ are real numbers. Their sum is defined as:

The result is always another complex number. For instance, if $z_1 = 2 + i3$ and $z_2 = -6 + i5$, then:

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Properties of Addition

Addition of complex numbers obeys five fundamental laws. Each one mirrors a property you already know from real numbers, but we verify them explicitly for complex numbers.

(I) Closure Law

Statement: The sum of any two complex numbers is always a complex number.

Proof: Let $z_1 = a + ib$ and $z_2 = c + id$ be any two complex numbers. By definition, $z_1 + z_2 = (a + c) + i(b + d)$. Since $a, b, c, d$ are real numbers, $a + c$ and $b + d$ are also real numbers. Therefore $(a + c) + i(b + d)$ is of the form $x + iy$ with $x, y \in \mathbb{R}$, which is precisely a complex number. So the sum stays within the set of complex numbers.

(II) Commutative Law

Statement: For any two complex numbers $z_1$ and $z_2$, $z_1 + z_2 = z_2 + z_1$.

Proof: Let $z_1 = a + ib$ and $z_2 = c + id$.

Since addition of real numbers is commutative, $a + c = c + a$ and $b + d = d + b$. Therefore the two sums are identical. The order in which you add two complex numbers does not matter.

(III) Associative Law

Statement: For any three complex numbers $z_1, z_2, z_3$, $(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$.

Proof: Let $z_1 = a + ib$, $z_2 = c + id$, and $z_3 = e + if$.

First compute $(z_1 + z_2) + z_3$:

Now compute $z_1 + (z_2 + z_3)$:

Because addition of real numbers is associative, $(a + c) + e = a + (c + e)$ and $(b + d) + f = b + (d + f)$. Hence the two expressions are equal. When adding three or more complex numbers, you can group them in any way you like.

(IV) Existence of Additive Identity

Statement: There exists a complex number $0 + i0$ (denoted simply as $0$) such that for every complex number $z$, $z + 0 = z$.

Proof: Let $z = a + ib$ be any complex number. The zero complex number is $0 = 0 + i0$.

The number $0 + i0$ acts exactly like the real zero: adding it to any complex number leaves the number unchanged. This is called the additive identity.

(V) Existence of Additive Inverse …