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Introduction to Real Numbers - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

When analyzing data, graphing equations and performing computations, we are most often working with real numbers. Real numbers are the set of all numbers that can be expressed as a decimal or that are on the number line. Real numbers have certain properties and different classifications, including natural, whole, integers, rational and irrational.

What we're going to talk about now is just different classifications of numbers. Okay? And more specifically what we're going to be talking about real numbers which are basically any number that can be written on your number line, okay?
So if your number line is just a spectrum of numbers, you have the numbers like 0, 1 on there but you also have a number of [IB] numbers in between there okay? okay and so we're going to talk about the different breakdown and all the classifications of those numbers.
So where I want to start is on natural and counting numbers and basically, what those are are the numbers 1, 2, 3 so on and so forth, 842 any sort of number that is a whole number larger than zero. Okay. Whole numbers include 0. So basically a whole number is a counting number plus zero and they continue upward as well.
Integers include the negative spectrum. So you're going you know up to say -2, -1, 0, 1, 2 so on and so forth, okay? So we have our counting numbers only positive, whole numbers throw in zero, integers throw in negatives.
Next part is what we call rational numbers. A rational number is any number that can be written as a fraction. Okay? So this can get a little bit tricky because a number like any integer can be written as a fraction. -2 can be written as -2 over 1. Therefore it can be a rational number. That also includes things like one fourth, or 0.23 because 0.23 can be written as 23 over 100, okay? Basically any number can be written as a fraction.
The last thing are what we call "irrational numbers" which can't be written as a fraction. And examples of those are say like pi, remember pi from Geometry or something like the square root of 3. Anything like that okay?
So whenever we're dealing with different types of numbers, this is just some language that we are dealing with, okay? And one thing to be aware of is that we can often simplify things up to put them into a category they may not appear.
Like we're used to saying okay radicals are irrational square roots. But something like the square root of 36 we know what that is. So therefore that square root doesn't have to be there because the square root of 36 is actually 6 which would then put it into all three of these categories, okay? Say like two fourths, you can simplify it to one half it's going to be a fractional number that actually didn't change too much so never mind that example. But something like eight halves you can simplify it to four. So it's not just a rational number it can also be a integer, whole number, accounting number as well okay?
So just sort of looking at your classifications always make sure you simplify it before you jump to a conclusion as to what type of number it is.

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