(1-i).jpeg "(1+i)(1-i)")
Simplifying (1 + i)(1 - i)
This article explores the simplification of the expression (1 + i)(1 - i), where 'i' represents the imaginary unit.
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit. The imaginary unit 'i' is defined as the square root of -1, i.e., i² = -1.
Simplifying the Expression
To simplify (1 + i)(1 - i), we can use the distributive property of multiplication (also known as FOIL method):
(1 + i)(1 - i) = 1(1 - i) + i(1 - i)
Expanding the terms:
= 1 - i + i - i²
Since i² = -1, we can substitute:
= 1 - i + i - (-1)
Simplifying further:
= 1 - i + i + 1
= 2
Therefore, the simplified form of (1 + i)(1 - i) is 2.
Significance of the Result
The simplification of (1 + i)(1 - i) to a real number (2) highlights a key concept in complex numbers:
- Conjugate Pairs: (1 + i) and (1 - i) are complex conjugates of each other. A complex conjugate is formed by changing the sign of the imaginary part of a complex number.
- Product of Conjugates: The product of a complex number and its conjugate always results in a real number. This is a useful property for simplifying complex expressions.
In conclusion, the expression (1 + i)(1 - i) simplifies to 2, demonstrating the concept of complex conjugates and their impact on simplifying complex number expressions.