Understanding Complex Numbers: A Visual Explanation

In summary, complex numbers represent an extension of the one-dimensional number line to the two-dimensional complex plane. They consist of a real part and an imaginary part, and can be visualized as a number line representing the real number a, with a y-axis extending to represent i. Multiplication by i^2 represents rotation by 180 degrees, while multiplication by i represents rotation by 90 degrees. A YouTube video has demonstrated that complex numbers can be understood as representing rotation.
  • #1
QuarkCharmer
1,051
3
Complex Numbers, I just don't understand what they are supposed to represent. I understand how they are used, to some degree in solving equations etc. but the definitions leave me clueless. The idea that they "extend the idea of the one-dimensional number line to the two-dimensional complex plane" sort of makes sense. If each number consists of a real part and an imaginary part (a+bi etc.), then starting from a number-line representing the real number a, you can extend a y-axis to represent i. But what does that really mean?? What would be a good visual explanation of this concept?
 
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  • #2
I have been struggling with the same difficulty for several years. Finally, there was a YouTube video a few months ago which gave a reasonably logical demonstration that complex numbers represent rotation. Think of a number represented in just one direction on a number line. Multiplication by i^2 is rotation by 180 degrees, and multiplication instead by just i is rotation by just 90 degrees.
 

Related to Understanding Complex Numbers: A Visual Explanation

1. What are complex numbers?

A complex number is a number that comprises both a real and imaginary part. It is usually written in the form a + bi, where a is the real part and bi is the imaginary part, with i being the imaginary unit, equal to the square root of -1.

2. How are complex numbers represented visually?

Complex numbers can be represented visually on a complex plane, with the real part plotted on the horizontal axis and the imaginary part plotted on the vertical axis. The point where these two axes intersect is known as the origin, and it represents the complex number 0.

3. What is the significance of the modulus and argument of a complex number?

The modulus of a complex number is its distance from the origin on the complex plane, and it is equal to the square root of the sum of the squares of its real and imaginary parts. The argument of a complex number is the angle between the positive real axis and the line connecting the origin to the complex number, measured in a counterclockwise direction. These two values help to fully describe a complex number and are useful in various mathematical operations.

4. How do I perform arithmetic operations with complex numbers?

To add or subtract complex numbers, simply combine the real parts and the imaginary parts separately. To multiply complex numbers, use the FOIL method (First, Outer, Inner, Last) and remember that i2 = -1. To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator, which is the same as the original denominator but with the sign of the imaginary part flipped.

5. What is the geometric interpretation of complex number operations?

The geometric interpretation of complex number operations can be understood through the concept of rotation and scaling on the complex plane. Multiplying a complex number by another complex number results in a rotation and scaling of the original number. Addition and subtraction can also be visualized as translations on the complex plane. This geometric interpretation provides a deeper understanding of complex numbers and their operations.

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