Question

First simplify each of the following numbers to the \(x+i y\) form or to the \(r e^{i \theta}\) form. Then plot the number in the complex plane. $$\frac{1}{i-1}$$

Short answer

Simplified form: \(-\frac{1}{2} - \frac{1}{2}i\). Plot: move \(-\frac{1}{2}\) units left and \(-\frac{1}{2}\) units down.

Step-by-step solution

Write the expression

The given expression is \(\frac{1}{i-1}\).

Multiply by the complex conjugate

To remove the complex number from the denominator, multiply both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of \(i-1\) is \(i+1\). So, \[ \frac{1}{i-1} \times \frac{i+1}{i+1} = \frac{i+1}{(i-1)(i+1)} \].

Simplify the denominator

Simplify the denominator using the difference of squares: \[ (i-1)(i+1) = i^2 - 1^2 = -1 - 1 = -2 \].

Simplify the numerator

After simplifying the denominator, write the simplified fraction: \[ \frac{i+1}{-2} = -\frac{1}{2} - \frac{i}{2} \].

Express in the form \(x + iy\)

Rewrite the simplified form in terms of \(x + iy\): \[ -\frac{1}{2} - \frac{i}{2} \]. So the complex number is \(-\frac{1}{2} - \frac{1}{2}i\).

Plot the number in the complex plane

To plot \(-\frac{1}{2} - \frac{1}{2}i\) in the complex plane, move \(-\frac{1}{2}\) units left on the real axis and \(-\frac{1}{2}\) units down on the imaginary axis.

Key concepts

Complex Conjugate

When you work with complex numbers, sometimes you need to get rid of the imaginary part in the denominator. That's where a complex conjugate comes in. If you have a complex number like \( z = a + bi \), its conjugate is \( \overline{z} = a - bi \). Simply, you flip the sign of the imaginary part. This trick helps you simplify fractions with complex numbers. For example, to simplify \( \frac{1}{i-1} \), multiply by the conjugate \( i+1 \) to get a simpler form.

Complex Plane

The complex plane is similar to the regular coordinate plane but with a twist. It has two axes: the real axis (horizontal) and the imaginary axis (vertical). You can represent any complex number \( z = x + yi \) as a point \( (x, y) \) in this plane. Moving along the real axis shows the real part, while moving along the imaginary axis shows the imaginary part. So, plotting \( -\frac{1}{2} - \frac{1}{2}i \) means moving left \( -\frac{1}{2} \) units on the real axis and down \( -\frac{1}{2} \) units on the imaginary axis.

Difference of Squares

The difference of squares is a handy algebraic identity. It states that \( (a - b)(a + b) = a^2 - b^2 \). This is really useful for simplifying complex numbers. For example, when simplifying \( \frac{1}{i-1} \), multiplying by the conjugate \( i+1 \) gives us \( (i-1)(i+1) \). Applying the difference of squares, it simplifies to \( i^2 - 1^2 = -1 - 1 = -2 \). This turns the denominator into a simple real number.

Imaginary Unit

The imaginary unit is represented by \( i \), which is defined as \( \sqrt{-1} \). Essentially, \( i^2 = -1 \). This might seem strange, but it is fundamental in dealing with complex numbers. In our example, \( (i-1)(i+1) \) simplifies using \( i^2 = -1 \). So when you see \( i \) in any expression, remember it's just a tool to handle numbers that would otherwise be impossible to work with.

Real and Imaginary Parts

Every complex number \( z = x + yi \) has a real part \( x \) and an imaginary part \( yi \). These parts help us to break down and understand complex numbers better. For example, in \( -\frac{1}{2} - \frac{1}{2}i \), \( -\frac{1}{2} \) is the real part and \( -\frac{1}{2}i \) is the imaginary part. Understanding the real and imaginary parts makes plotting on the complex plane straightforward. Just treat the real and imaginary parts as coordinates on the plane.