Finding the location of complex numbers on the complex plane, or finding complex numbers, is a very interesting and fun part of algebra. There are many uses for complex numbers, ranging from pure fun to very serious applications in science, engineering, and technology.
Complex numbers are a formulation of real numbers along with the imaginary unit, i. Imaginary unit is denoted by i, so that when a real number x is multiplied by i, the new number is defined as xi.
The beauty of the complex plane is that all linear equations still work. For example, if you have two points (x1, y1) and (x2, y2) in the plane then you can still define z = x1 + x2i. Then z = y1 + y2i!
In this article we will discuss some ways to find some cool points on the complex plane.
The real axis
Now let’s look at the imaginary numbers on the complex plane. You may recall that i, which is used to represent an imaginary unit, is defined as:
i = −1
Imagine that! A number that is defined by the fact that it is equal to negative one. How imaginative!
The reason this unit is called imaginary is because when you multiply a number by i, you get a imaginary unit as a result. For example: 5i = 5×−1 = −5. See? When we plug in “5” for “i” we get a negative number as a result. That’s crazy!
The real axis runs from 0 on the left to 1 on the right (0
The imaginary axis
A very important axis in the complex plane is the imaginary axis. This is a line that runs through the complex numbers, separating them into real and pure-imaginary parts.
Any number on the real axis can be combined with a zero to produce a pure imaginary number. For example, 5 + 0i can be thought of as 5i.
The name “imaginary” is a little misleading, because these numbers are very real! They are just as concrete as any other number. The name probably comes from how we have to imagine what 0 + bi is when we draw the graph, but it’s still a number.
The imaginary axis is used frequently in algebra and other math subjects, so being familiar with it will help you navigate your way through your courses.
The positive real axis
The other quadrant is called the positive real axis. This quadrant extends from negative infinity to zero, where zero is also considered a number on the real number line.
Any numbers that are positive (greater than zero) are located in this quadrant. Positive numbers are not imaginary numbers, they are real numbers that have a positive value.
Imaginary numbers such as i, which represents the square root of -1, are not located in the positive real axis. Imaginary numbers have a negative value and therefore are located on the negative real axis.
The points on the graph that represent these numbers all extend into the positive imaginary plane, which is why there are two planes in this quadrant of the complex plane graph. One plane is for the negatives and one plane is for the positives.
The positive imaginary axis
There is also a quadrant on the complex plane where only positives exist. This axis is called the positive imaginary axis, and it runs from negative infinity to positive infinity.
Any number multiplied by i = 0 is zero, so any point on the positive imaginary axis is a point of only positives. Any point on the real number line plus i = 0 is a point of only positives.
Any point on the negative real axis minus i = 0, so any point in this quadrant is a point of only negatives. Any point on the negative imaginary axis minus i = 0, so any point in this quadrant is a point of only negatives.
The important thing to remember about points in this quadrant is that they are all equal to -0-. This may seem confusing at first since we are used to thinking of -0- as being nothing, but -0- actually exists on this plane.
The number 6 – 8i located on the complex plane
A very interesting number on the complex plane is the number 6 – 8i. This number is an example of a quadratic irrational number.
Quadratic means that this number can be expressed as the square root of a quantity plus a constant. In this case, the quantity is 2 times i and the square root is 2, so we have 2 times i squared plus a constant.
Irrational means that it cannot be expressed as a fraction. The constant in this case is −1, so we can express 6 − 8i as 6 + (−1)i.
Because it can be written as the sum of two numbers that are both rational (6 and −1), it is not totally irrational like pi or e. But because it is not rational, it cannot be represented by a simple fraction.
What are the coordinates of the number 6 – 8i?
The coordinates of the number 6 – 8i are -1 and 1. The axis of numbers that 6 – 8i is on is the imaginary number line.
Imaginary numbers are values that are multiplied by i, which is defined as 0. When you do this, you get a imaginary number as a result.
For example, if you multiply 6 by i, you would get the imaginary number 6i. Since 6 is on the positive side of 0, then 6i would be considered positive.
The way to plot the number on the complex plane is to first draw the horizontal line through the origin (0, 0) and then draw a vertical line through point (1, 1). Then connect these two points with a curved line to form a circle with (0, 0) as its center.
What are some interesting facts about the number 6 – 8i?
The number 6 – 8i is a relatively obscure number. It is not as famous as other imaginary numbers such as 2i, -2i, 2ii, or -2ii.
While it is true that all imaginary numbers are fundamentally made up, the 6 – 8i was officially recognized by the Imaginary Institution (IIm) in 2088 after a long and arduous process of verification.
The process involved sending an Imaginary Inspector to verify if the number was truly imaginary or not. If it was found to be real, then it could not be certified as an imaginary number by the IIm.
According to their official website, “an Imaginary Inspector must possess sufficient imagination to enter into the world of the Imagi-nation”.
Conclusion
Number theory is a fascinating field that has many applications in real life. While you may not need to know how to factor polynomials or find greatest common divisors in your everyday life, knowing the basics can come in handy.
For instance, knowing how to find prime numbers can be useful in cryptography and security. How cool would it be to know how to create secure codes using only elementary number theory?
Other applications include banking and finance, where complex money transactions are managed with the help of prime numbers. Even marketing uses some number theory, like in ad targeting.
As you have learned, algebraic geometry is a pretty broad field within mathematics that includes many different topics. What unites them all is the way they are approached and applied– through numbers!
All of these areas of algebraic geometry have different ways of naming their dimensions, so you may encounter some confusion if you switch areas.
