Find the modulus of \(\frac{1+i}{1+i}-\frac{1-i}{1+i}.\)
Solution and Explanation
\(\frac{1+i}{1+i}-\frac{1-i}{1+i}=\frac{(1+i)^2-(1-i)^2}{(1-i)(1+i)}\)
\(=\frac{1+i^2+2i-1-i^2+2i}{1^2+1^2}\)
\(=\frac{4i}{2}=2i\)
\(∴|\frac{1+i}{1-i}-\frac{1-i}{1+i}|=|2i|=\sqrt2^2=2\)
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.