Why do we have abstractions in Mathematics?

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Everybody will tell you that math is about numbers!

Counting numbers 1, 2, 3 …………………..
Whole numbers 0, 1, 2, 3……………….
Integers……….. -3,-2,-1, 0, 1, 2, 3……………….
Rational numbers 1/2, 1/3, -7/8

Real numbers …. √2 = 1.4142……
Complex numbers..... x + y √ -1

Where x and y are real numbers
Each subsequent number system includes the previous number system. So far example integers are included in rational numbers and rational numbers are included in real numbers and so on.

Each enlargement helps us to solve more difficult equations and something we could not solve earlier!

x + 2 = 5 (needs negative numbers)
4x = 1 (needs rational numbers)
x4 = 7  (needs irrational numbers)
x4 = - 7 (needs complex numbers)

With each of these enlargements we are able to solve harder problems but the math becomes more abstract and less intuitive


For example, we understand when we have 5 apples but what does it mean to have  √-5  apples?


The great mathematician Gauss said



“The true metaphysics of square root of -1 is elusive”

He was adept at using complex numbers and applying them to solve many problems but it still bothered him!

In fact, Kronecker famously said “God made integers all else is the work of man”

By studying these more abstract number systems we are getting more and more away from studying concrete things like integers but they are necessary to solve harder problems!

So next time when you are learning abstract algebra and struggling to have a good intuitive feel of what a commutative ring really is, don’t despair! In fact everybody is struggling! 

Comments

  1. UnknownJune 9, 2020 at 9:09 AM

    Yes, I agree that abstract mathematics does not seem an easy going concept at first. But it offers unparalleled practical applications. Very well explained the necessity of complex number system.

    ReplyDelete

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