Imaginary Numbers Essay, Research Paper

The History of Imaginary Numbers

The origin of imaginary numbers dates back to the ancient Greeks. Although, at

one time they believed that all numbers were rational numbers. Through the years

mathematicians would not accept the fact that equations could have solutions that were

less than zero. Those type of numbers are what we refer to today as negative numbers.

Unfortunately, because of the lack of knowledge of negative numbers, many equations

over the centuries seemed to be unsolvable. So, from the new found knowledge of

negative numbers mathematicians discovered imaginary numbers.

Around 1545 Girolamo Cardano, an Italian mathematician, solved what seemed to

be an impossible cubic equation. By solving this equation he attributed to the acceptance

of imaginary numbers. Imaginary numbers were known by the early mathematicians in

such forms as the simple equation used today x = +/- ^-1. However, they were seen as

useless. By 1572 Rafael Bombeli showed in his dissertation ?Algebra,? that roots of

negative numbers can be utilized.

To solve for certain types of equations such as, the square root of a negative

number ( ^-5), a new number needed to be invented. They called this number ?i.? The

square of ?i? is -1. These early mathematicians learned that multiplying positive and

negative numbers by ?i? a new set of numbers can be formed. These numbers were then

called imaginary numbers. They were called this, because mathematicians still were

unsure of the legitimacy. So, for lack of a better word they temporarily called them

imaginary. Over the centuries the letter ?i? was still used in equations therefore, the name

stuck. The original positive and negative numbers were then aptly named real numbers.

What are Imaginary Numbers?

An imaginary number is a number that can be shown as a real number times ?i.?

Real numbers are all positive numbers, negative numbers and zero. The square of any

imaginary number is a negative number, except for zero. The most accepted use of

imaginary numbers is to represent the roots of a polynomial equation (the adding and

subtracting of many variables) in one variable. Imaginary numbers belong to the complex

number system. All numbers of the equation a + bi, where a and b are real numbers are a

part of the complex number system.

Imaginary Numbers at Work

Imaginary numbers are used in a variety of fields and holds many uses. Without

imaginary numbers you wouldn?t be able to listen to the radio or talk on your cellular

phone. These type of devices work by receiving and transmitting radio waves. Capacitors

and inductors are used to make circuits that are used to make radio waves. In order to

determine the right values of capacitors and inductors to use in the circuits, designers

need to use imaginary numbers.

Another use of imaginary and complex numbers is in physics, quantum mechanics

to be exact. In quantum mechanics a big problem is to find the position of a particle.

Unfortunately, only the probability distribution of it?s position is possible to find. The

only way to calculate this is to use imaginary and complex variables.

Lastly, electrical engineers use imaginary numbers. However, instead of using ?i?

in their equations they use ?j.? This is because in the equations they commonly use, ?i?

means current, so to represent imaginary numbers they use ?j.?

Four Most Familiar Number Concepts

There are four of the most common numbers that we, the common person, know

about and can understand why they exist. At one point or another you might have used

one of these four concepts in your math classes. The first concept are Natural Numbers,

which are abstract numbers that answer questions, like ?how many.? They are able to

describe sizes and sets. The second concept are Integers, they describe the relative sizes

between two sets. They answer questions, like ?how many more does A have than B??

Rational numbers are what describes ratios and fractions. For example you might tell

Karen that you ate 3/4 of an apple pie. This will let Karen know you ate three quarters out

of a four quarter pie. A real number is a number that will describe a measurement like

weight, length and fluid. However, in none of the four concept can you see the square

root of -1 fall into place. There exist a fifth concept which is referred to as a complex

number. As mentioned earlier a complex number equation = a+bi. It is a real number with

an imaginary number.

Quadratic Formula and Imaginary numbers

Throughout our lifetime, teachers have informed students that negative numbers

cannot be squared. With imaginary numbers we are able to do so. With a very simple

example it can be shown how this is true. With an equation like y = x^2+ 4x+29, we can

get the x intercept by using the quadratic equation. By following all appropriate steps you

will find out where the x intercepts are at. Roots are all places that a graph will touch the

x intercept. The quadratic equation = -b+- square root of b^2-4(a)(c)/ 2(a). Therefore,

x^2=a , 4x= b and 29 = c.

-4+- square root of 4^2 – 4(1)(29)

2(1)

-4+- square root of 16-116

2

-4+- square root of -100

2

-4 +-10i

2

=-2+-5i

The answer -2 +-5i, lets you know that it is a complex root, meaning that it does

not touch the x intercept. By graphing the equation y=x^2+4x+29, you will see the

parabolas location. This parabola will not touch the x intercept. This table will show you

how:

i^1=i i^4=1

i^2=-1 i^5=i

i^3=-i i^6=i

Complex Root and Complex Conjugate Root

This is an ongoing cycle that will help you solve problems that deal with i^n

power. Another amazing technique you can use is when you are given a complex root and

the complex conjugate root and you need to derive the equation by the root given and

complex root. A complex conjugate root is that exact opposite of a complex root. For

example if you are given one complex root of 2-5i, and you are asked to find the equation

you simply multiply 2-5i by the conjugate root of 2+5i. By using foil method you will

find out the equation. For example:

(2-5i) (2+5i) = 4+10i-10i-25i

elimination will give you

=-25i^2+4

you know i^2= -1

therefore

=-25(-1)+4

a negative times a negative equals a positive

=25+4=29

You then add the complex root (2-5i) with it?s complex conjugate root (2+5i).

2+5i

=2-5i

=4

This will let you know that your equation is y=x^2+4x+29 and you are

able to graph and see how and where the complex roots are located on the graph.

Making An imaginary Number A Real Number

You can multiply, add, divide , subtract and even take the square root of a

negative number. Like mentioned in the History of Imaginary Numbers, negative

numbers were not believed to be a valid answer. However, we know that a negative

number does have meaning and is a valid answer. A negative number will let us

determine many different things. We see them in our check books, when graphing, and

even when finding the expected number of a roulette game. Complex numbers can be

added to show you how they can become real numbers. For example:

5i^2 + 4i^2= 9i^4

9(1)=9

The answer is a real number that we obtained after adding it to imaginary numbers. You

can refer to the imaginary number cycle. It is known that i^4=1, nine is then multiplied by

1 to get a positive nine. Weather you get a negative or positive number they are real

numbers.

Conclusion

Imaginary numbers are in fact very real. They have common uses and very

intricate uses. Little does the average person know the imaginary number is one of the

oldest and greatest discoveries ever found.

Dr. Anthony, ?Ask Dr. Math? The Math Forum, 1994-2001.

http://forum.swarthmore.edu/dr.math/faq/faq.imag.num.html

(2001-March-15).

Mathews, John. Howell. ?Complex Analysis,? 2000.

www.ecs.fullerton.edu/~mathews/c2000/c01/Links/c01_lnk_3.html

(2001-March-15).

Nahin, Paul. (1998). ?An Imaginary Tale.? New Jersey: Princeton University Press.

Ross, Kelley. Ph.D.. ?Imaginary Numbers,? 2000.

www.friesian.com/imagine.htm

(2001-April-4).

Snyder, Bill. Personal Interview. 20 April. 2001.

Spencer, Philip. ?Do Imaginary Numbers Really Exhist?? University of Toronto

Mathematics Network, 1997.

www.math.toronto.edu/mathnet/answers/imaginary.html

(2001-April-4)

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