Simplifying Complex Fractions: Converting 1i/(1-i) to Rectangular Form
In the realm of mathematics, particularly in the domain of complex numbers, we often encounter expressions that require simplification. One such expression is the conversion of 1i/(1-i) into its rectangular form. This process involves several steps, each building on basic algebra and complex number properties. Let's break down this simplification step-by-step to understand the transformation fully.
Understanding the Expression
The given expression is a complex fraction:
1i / (1 - i)
Here, 1i (or i in standard notation) represents the imaginary unit, and the denominator (1 - i) is a complex number. Our goal is to simplify this fraction into a form that can be expressed in the standard rectangular form of a complex number: a bi, where a and b are real numbers.
Step-by-Step Simplification
Step 1: Multiply the Numerator and Denominator by the Complex Conjugate
To eliminate the imaginary unit in the denominator, we multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 1 - i is 1 i. So, we have:
1i / (1 - i) (1 i) / (1 i)
Step 2: Expand the Expression
When we multiply the numerator and denominator, we use the distributive property:
(1i 1 1i i) / (1 1 1 i - i 1 - i i)
This simplifies to:
(1i 1i^2) / (1 - i i - i^2)
Since i^2 -1, the expression further simplifies to:
(1i 1(-1)) / (1 - (-1))
Which is:
(1i - 1) / (1 1)
Step 3: Simplify the Fraction
Now, we combine the simplified terms in the numerator and denominator:
(-1 1i) / 2
This can be written as:
-1/2 1i/2
In standard rectangular form, this is:
-1/2 i/2
Final Expression
The simplified form of 1i/(1-i) in rectangular form is:
-1/2 i/2
Conclusion
This step-by-step guide on simplifying complex fractions into their rectangular form demonstrates the power of algebraic manipulation and the importance of understanding complex numbers. By following these steps, we can transform complex expressions into more understandable and useful forms, facilitating further operations and analyses.