Properties of Real Numbers

Mike McGarry
Lesson by Mike McGarry
Magoosh Expert
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Arithmetic and Fractions, Properties of Real Numbers. Let me begin by saying this video really starts at square one. It starts at the very basic properties of all of mathematics. And if you're reviewing math, this is a wonderful place to start. We'll cover everything from the basics. But if you are already familiar with math, if math is something you are reasonably good at, then you probably don't need to watch all these introductory videos.

What I strongly suggest, is just skip ahead to the summary. Read the summary. If there is something in the summary that makes absolutely no sense, then go back to the video and watch the video. But don't force yourself to watch the video if it's stuff that's going to be way too easy for you.

It's very important to know yourself and pace yourself in this process. For folks who really want to learn everything from the beginning onward, let's start at the beginning. When the test says number, it always means a real number. What's meant by a real number? Well, a real number is any number on the number line.

This includes round numbers as well as fractions and decimals. So, all the numbers pictured here as well as all the numbers in all the spaces in between. Numbers like one third, or pi, or the square root of 7. All of these are numbers. They're all real numbers.

When the test says number, this number could be any number on the number line. It could positive, negative, or zero. It could be a whole number or a fraction, or a decimal. This is one of the way that the test loves to trap people, loves to begin a math problem. X is a number, and then it goes on.

And people assume that, since x is a number, it can only be, 1, 2, 3, 4, 5. In other words, that it has to be a counting number. People forget about negatives. They forget about fractions and decimals. It's very important not to fall into that crap. If you see the word number you have to be thinking about all these categories altogether.

We will discuss fractions and decimals in later videos. Right now, notice that zero is the only number that is neither positive nor negative. So we have these two very large categories, positive and negative. Zero's the only number on the number line that doesn't fall into either one of those categories.

One special type of number is the integer. Integers include all positive and negative whole numbers. If the question asks about integers, it could be any number in this set. It could be the positives or the negatives. If the question asks about positive integers, then that's ordinary counting numbers 1, 2, 3, 4, 5, 6, etc.

Very important to keep these three sets straight. Sometimes questions will ask about numbers, sometimes they'll ask about integers, sometimes they'll ask about positive integers. Those are three different sets. It's very important not to confuse them. Back to all the real numbers.

There are a few vocab words you need to know. These concern the four fundamental operations. You may want to pause the video here just to verify to yourself that you know these four terms. The result of addition is called a sum. The result of subtraction is called a difference.

The result of of multiplication is called a product. The result of division is called quotient. These are four vocab words you need to know. There are also some fundamental arithmetic properties common to all real numbers. To discuss these, I will use a, b, and c to represent real numbers. These are terms you don't need to know, you don't need to know the names of the properties, all you need to know are the properties themselves.

The first one is the commutative property, the ability to switch the order. So for example, we can switch the order of addition a+b = b+a. Doesn't matter which order we do it in, we get the same answer. Similarly, with multiplication we can swap the order around a*b = b*a. Again we can swap the order around, we get the same answer. Addition and multiplication are commutative.

Division and subtraction, in general, are not commutative. If we swap the order around to subtraction, we get different answers. 7 minus 4 does not equal 4 minus 7. But keep this in mind, if we rewrite the subtraction as the addition of a negative, that's a very sophisticated move, then it would be commutative. So 7- 4, we can express that as 7 + -4, and then we could swap the order, -4 + 7.

So we could actually perform that swap if we need it. So all these are properties you need to know. You need to know what you can do with these numbers but you don't need to remember the word commutative, that word will not be on the test. The second property is the associative property, the ability to group things. So if we're adding three numbers, we can group them we can add b + c first or we could add a + b first, either way of grouping leads to the same answer.

Here we can add the 3 and the 5 first, or we could add the 2 and the 3 first. Either way, we get the same answer. Similarly with multiplication, we can group them b*c first or a*b first. If we multiply the 3 and the 5 first or we multiply the 2 and the 3 first, either way, we get the same answer. Addition and multiplication are associative.

Subtraction and division, in general, are not associative. Again, you need to know what the numbers do. You don't need to know this word, associative. That's not a word that the test will test you on. The distributive property, this one is a big one. First of all, multiplication distributes over addition and it distributes over subtraction.

So we could do the addition and subtraction first, and then multiply, or we could multiply times each piece, and then perform the addition and subtraction. So for example, we can add 2+3 first, and then multiply. Or, we can do 6 times 2, 6 times 3 first, and then add those two products separately. Either way, we get the same answer.

Similarly, with subtraction we could subtract the 3 and the 7 first, get the 4, multiply that by 12. Or, we could multiply the 12 times 7 and separately multiply the 12 times 3 and then subtract those two products. Either way, we get the same answer. There are other properties that concern the special numbers 1 and 0.

The first one is multiplying and dividing by 1, very simple. We don't change a number when we multiply or divide by 1. So we can represent that symbolically. This way, a*1 = a, a/1 = a. Absolutely no change when we multiply or divide by 1. Multiplying by zero, anything times zero equals zero.

Again, very simple. So a*0 = 0*a = 0. So we're perfectly allowed to multiply by 0, and we always get 0. We're not allowed to divide by 0. That's a very important point. The zero product property, if the product of two numbers is zero, one of the factors must be zero.

So if a*b = 0, then a = 0 or b = 0. And notice that the word or there, that's an essential piece of mathematical equipment right there. That's actually part of the math. That's not decoration. That creates the logical connection between the two equations.

The zero product property is very important in factoring and in solving quadratics, so you'll run into it there. Those are in the algebra module videos. Dividing a number by itself. When any non-zero number is divided by itself, the quotient has to equal 1. We can express this as a/a, or a/a = 1.

Again, very straightforward. So in summary, the test is gonna use the word number. And this always means any real number, not just positive integers. It's very important to keep numbers versus integers versus positive integers straight. Those are three different sets.

Know those four terms, sum, difference, product and quotient. Those are actual vocab words you need to know. And know the properties. Again, we don't need to know the names of the properties, we just need to know the properties themselves. Here we have the commutative property and the associative property.

Here we have the distributive property which is a really important one. And then all the properties concerning the special numbers, 1 and 0.

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Arithmetic and Fractions