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Properties of Real Numbers
Transcript
Arithmetic and Fractions. Properties of Real Numbers. Let me begin by saying this video really starts at square one, it starts with the very basic properties of all of mathematics. And if you are reviewing math, this is a wonderful place to start. We'll cover everything from the basics.
But, if you're already familiar with math, if math is something you're reasonably good at, then you probably don't need to watch all these introductory videos. What I'd strongly suggest is just skip ahead to the summary, read the summary. If there's 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 to 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 and spaces in between. Numbers like one-third, or pi, or the square root of seven, 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 be 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 one, two, three, four, five, or, in other words, that it has to be a counting number.
People forget about negatives. They forget about fractions and decimals. Very important not to fall into that trap. If we see the word number, you have to be thinking about all these categories all together. 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 is 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 multiplication is called a product. The result of division is called the quotient.
These are four vocab words that 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 is 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. It doesn't matter which order we do it in, we get the same answer. Similarly with multiplication, we get to swap the order around, a * b = b * a. Again we could 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 the subtraction, we get different answers. Seven minus four does not equal four minus seven. 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 needed. So all these are properties you need to know, you need to 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 the three and the five first, or we can add the two and the three first. Either way, we get the same answer. Similarly, with multiplication, we can group them b times c first or a times 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 the subtraction.
So for example, we can add two plus three first and then multiply, or we can do six times two, six times three first and then add those two products separately. Either way we get the same answer. Similarly with subtraction we could subtract the three and the seven first get the four, multiply that by twelve, or we could multiply the 12 * 7 and separately multiply the 12 * 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 one, very simple. We don't change a number when we multiply or divide by one. So we can represent that symbolically this way, a * 1 = a, a/1 = a. Absolutely no change if we multiply or divide by one.
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 zero and we always get zero, we're not allowed to divide by zero, 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 verses integers versus positive integers straight.
Those are three different sets. Know those four terms, sum, difference, product, and quotient. Those are actually 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, the associative property. Here we have the distributive property, which is a really important one, and then all the properties concerning the special numbers one and zero.
Read full transcriptBut, if you're already familiar with math, if math is something you're reasonably good at, then you probably don't need to watch all these introductory videos. What I'd strongly suggest is just skip ahead to the summary, read the summary. If there's 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 to 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 and spaces in between. Numbers like one-third, or pi, or the square root of seven, 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 be 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 one, two, three, four, five, or, in other words, that it has to be a counting number.
People forget about negatives. They forget about fractions and decimals. Very important not to fall into that trap. If we see the word number, you have to be thinking about all these categories all together. 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 is 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 multiplication is called a product. The result of division is called the quotient.
These are four vocab words that 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 is 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. It doesn't matter which order we do it in, we get the same answer. Similarly with multiplication, we get to swap the order around, a * b = b * a. Again we could 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 the subtraction, we get different answers. Seven minus four does not equal four minus seven. 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 needed. So all these are properties you need to know, you need to 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 the three and the five first, or we can add the two and the three first. Either way, we get the same answer. Similarly, with multiplication, we can group them b times c first or a times 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 the subtraction.
So for example, we can add two plus three first and then multiply, or we can do six times two, six times three first and then add those two products separately. Either way we get the same answer. Similarly with subtraction we could subtract the three and the seven first get the four, multiply that by twelve, or we could multiply the 12 * 7 and separately multiply the 12 * 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 one, very simple. We don't change a number when we multiply or divide by one. So we can represent that symbolically this way, a * 1 = a, a/1 = a. Absolutely no change if we multiply or divide by one.
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 zero and we always get zero, we're not allowed to divide by zero, 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 verses integers versus positive integers straight.
Those are three different sets. Know those four terms, sum, difference, product, and quotient. Those are actually 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, the associative property. Here we have the distributive property, which is a really important one, and then all the properties concerning the special numbers one and zero.
