Numbers fall into several broad categories: natural numbers, integers, rational numbers, irrational numbers, and even complex numbers. How many of these categories are you familiar with? Why is it important to have categories of numbers and to know which categories you are dealing with in a given expression?
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It can often be important to look for solutions to problems that fit certain parameters such as being a whole number or a rational number, or even a real number.
Equations with integer solutions are so important that they even have a name - Diophantine equations. Whole numbers are used when counting discrete or indivisible objects.
Solutions with rational numbers are used when looking for items that can be divided up into fractions, but for which dividing into non-rational parts is hard or impossible. The rationals are solutions to linear equations such as y = mx + b that come up in things such as calculating the position of a moving object that isn't accelerating, calculating sales tax, and dividing a commodity amongst several components - such as a halving a recipe, or splitting winnings among several participants.
Algebraic numbers are used when you want to find the solution to an equation of a polynomial. Polynomials have very nice properties, and they're easy to calculate. They come up in problems dealing with acceleration and bridge construction.
Real numbers are used when you need to find the solution that has a concrete existence - again, they often come up as the solution to polynomials, but only some polynomials have real solutions. They're also frequently solutions to exponentials such as half-life decay times (e.g. the natural log of 2). Others are complex numbers. They don't have to be rational or algebraic.
Complex numbers are used when you need to find a solution to a polynomial that does NOT have a real solution, or if you need to find a complex solution to an exponential problem. These come up very often in electrical circuits where the imaginary component is not part of the current-voltage relationship in the circuit (it may be stored in a capacitor as an electric field, or in an inductor as a magnetic field). They also occur in solutions to oscillating masses and dampening vibrations in bridges, airplane, and car engineering.
Why is it important to have categories of numbers and to know which categories you are dealing with in a given expression?
There's thousands of uses for each of these types of numbers, but I tried to highlight some useful and applicable answers here.
above from here:http://math.stackexchange.com/questions/442080/what-is-the-need-for-classifying-numbers-like-integer-whole-number-etc