Question

Find and plot the complex conjugate of each number. \(-\sqrt{3}+i\)

Short answer

The complex conjugate of \(-\sqrt{3}+i\) is \(-\sqrt{3}-i\).

Step-by-step solution

Understand the Complex Number

The given complex number is \(-\sqrt{3}+i\). In general, a complex number is in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Identify the Real and Imaginary Parts

For the complex number \(-\sqrt{3}+i\), the real part \((a)\) is \(-\sqrt{3}\) and the imaginary part \((b)\) is \(1\).

Find the Complex Conjugate

The complex conjugate of a complex number \(a + bi\) is \(a - bi\). Therefore, the complex conjugate of the given number \(-\sqrt{3}+i\) is \(-\sqrt{3}-i\).

Plot the Original Complex Number

On the complex plane, plot the point corresponding to \( -\sqrt{3} + i \). This point is located at \(-\sqrt{3}\) on the real axis and \(1\) on the imaginary axis.

Plot the Complex Conjugate

On the same complex plane, plot the point corresponding to \(-\sqrt{3}-i\). This point is located at \(-\sqrt{3}\) on the real axis and \(-1\) on the imaginary axis.

Key concepts

Complex Conjugate

A complex conjugate is a concept that helps in various mathematical calculations involving complex numbers. Given a complex number of the form \((a + bi)\), the complex conjugate is obtained by changing the sign of the imaginary part.
For example, the complex conjugate of \(-\sqrt{3}+i\) from the exercise is \(-\sqrt{3}-i\).
This transformation is useful because it can simplify computations, particularly in division and finding magnitudes of complex numbers. When you multiply a complex number by its conjugate, the result is a real number: \[ (a+bi)(a-bi) = a^2 + b^2 \].

Complex Plane

The complex plane, also known as the Argand plane, is a two-dimensional plane used to represent complex numbers visually.
In this plane, the horizontal axis (usually labeled the x-axis) represents the real part of the complex number, and the vertical axis (labeled the y-axis) represents the imaginary part.

To plot a complex number, such as \(-\sqrt{3}+i\), locate \(-\sqrt{3}\) on the real axis and \(+1\) on the imaginary axis. The point where these values intersect represents the complex number.

For its complex conjugate \(-\sqrt{3}-i\), move to \(-\sqrt{3}\) on the real axis and \(-1\) on the imaginary axis. This graphically demonstrates how the complex conjugate mirrors the original number across the real axis.

Imaginary Part

The imaginary part of a complex number is the coefficient of \(i\), where \(i\) is the imaginary unit defined by \(i^2 = -1\). In a complex number of the form \((a + bi)\), \(b\) is the imaginary part.

In the exercise example of \(-\sqrt{3}+i\), the imaginary part is \(1\).
One key use of the imaginary part is in converting a complex number to its complex conjugate by changing the sign of the imaginary part.
The imaginary part plays a crucial role in graphing complex numbers on the complex plane and in various mathematical operations.

Real Part

The real part of a complex number is simply the real number component, denoted by \(a\) in the form \((a + bi)\).
This value exists on the real axis of the complex plane.
In the exercise involving the complex number \(-\sqrt{3}+i\), the real part is \(-\sqrt{3}\).

Understanding the real part is fundamental for plotting complex numbers and performing basic operations such as addition, subtraction, and finding a complex conjugate.
The real part, together with the imaginary part, defines the position of a complex number in the complex plane.