The identity element of complex numbers under addition is 0.
In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation that leaves other elements unchanged when combined with them. For an operation denoted by *, an element e is an identity element if for every element a in the set, a * e = a and e * a = a.
Identity Element for Complex Numbers
For complex numbers, which are typically represented in the form $a + bi$ (where $a$ and $b$ are real numbers, and $i$ is the imaginary unit), the operation of addition is defined as combining the real parts and the imaginary parts separately.
Let $z$ be any complex number, $z = x + yi$.
The identity element for addition, denoted as $e$, must satisfy:
$z + e = z$
$(x + yi) + e = x + yi$
For this to hold true, the identity element $e$ must be $0 + 0i$, which simplifies to 0.
When you add $0$ to any complex number, the complex number remains unchanged:
- $(x + yi) + 0 = (x + yi) + (0 + 0i) = (x+0) + (y+0)i = x + yi$
Examples from Reference
As highlighted in the provided reference from Wikipedia's "Identity element" page, the identity element for complex numbers under addition is explicitly stated:
| Set | Operation | Identity |
|---|---|---|
| Complex numbers | + (addition) | 0 |
| Complex numbers | ยท (multiplication) | 1 |
This table clearly demonstrates that for the set of complex numbers, the element that acts as the identity under the operation of addition is 0.
Practical Significance
Understanding the identity element is fundamental in algebra and number theory, as it defines the "neutral" state for a given operation. For addition, 0 represents the absence of change, allowing for concepts like additive inverses (e.g., for any complex number $z$, its additive inverse is $-z$, such that $z + (-z) = 0$).
