WEBVTT mathematics/algebra-2/eaton 00:00:00.000 --> 00:00:01.900 Welcome to Educator.com. 00:00:01.900 --> 00:00:07.300 In today's lesson, we will be discussing properties of real numbers. 00:00:07.300 --> 00:00:15.700 A real number corresponds to a point on the number line, and real numbers may be either rational or irrational. 00:00:15.700 --> 00:00:31.900 So first, just talking about the number line: some examples of real numbers would be 0, 1, 1/2, 00:00:31.900 --> 00:00:41.700 or (expressing a number as a decimal) it can be 2.8; negative numbers--maybe -2.18 or -4. 00:00:41.700 --> 00:00:50.900 OK, these are all real numbers, and they are rational; so let's talk about the difference between rational and irrational numbers. 00:00:50.900 --> 00:01:02.300 Rational numbers can be expressed as a fraction. 00:01:02.300 --> 00:01:15.200 So, rational numbers are expressed in the form a/b, where a and b are integers, and b does not equal 0, 00:01:15.200 --> 00:01:23.900 because as you will recall, we cannot have 0 in the denominator, because that would result in an undefined expression. 00:01:23.900 --> 00:01:37.400 When a rational number is expressed as a decimal, it will either be terminating or repeating. 00:01:37.400 --> 00:01:48.000 Let me explain what I mean by this: if I have an example, such as 1/2, and I convert that to decimal form, it is going to terminate; it is .5. 00:01:48.000 --> 00:02:01.900 I might have another number, 1/3, which is also rational; and it is going to end up being .3333, and on indefinitely. 00:02:01.900 --> 00:02:06.400 This could also be written as .3 with a bar over it, indicating it is repeating. 00:02:06.400 --> 00:02:19.300 So, this is also rational, because it repeats; the repeating pattern could be longer--it could be 2.387387387, so this is repeating. 00:02:19.300 --> 00:02:25.100 The point is that they either terminate or repeat when expressed in decimal form. 00:02:25.100 --> 00:02:36.300 Irrational numbers cannot be expressed in the form a/b, so they are expressed as decimal form. 00:02:36.300 --> 00:02:54.100 But they neither terminate nor repeat in decimal form; they just go on indefinitely. 00:02:54.100 --> 00:03:08.900 For example, I could have 4.871469837246, and on and on and on, with no pattern--no repeating and no ending. 00:03:08.900 --> 00:03:21.600 Another example would be π; we often express π as 3.14, but it actually goes on and on indefinitely; it is just approximately equal to 3.14. 00:03:21.600 --> 00:03:29.500 Therefore, we can add π up here (that is an irrational number) to represent some irrational numbers up here. 00:03:29.500 --> 00:03:37.100 In addition, the square root of numbers, other than the square root of numbers that are perfect squares, are irrational numbers. 00:03:37.100 --> 00:03:41.500 So, the square root of 3 is, or the square root of 2. 00:03:41.500 --> 00:03:46.800 If it is a perfect square, that means it is the result of multiplying an integer by itself. 00:03:46.800 --> 00:03:54.500 For example, 2 times 2 is 4, so that is a perfect square; and the square root of 4 is rational--in fact, it just equals 2. 00:03:54.500 --> 00:04:04.600 All other square roots are irrational numbers; so again, both rational and irrational numbers are real numbers, but they have different properties. 00:04:04.600 --> 00:04:12.200 Sometimes, we express the relationship between the various types of real numbers using a Venn diagram. 00:04:12.200 --> 00:04:21.400 So, I am going to go ahead and break down the real number system into its subsets, and then show you how this works as a Venn diagram. 00:04:21.400 --> 00:04:42.100 We have the real number system, and we have irrational numbers, and we also have rational numbers. 00:04:42.100 --> 00:04:54.700 And a Venn diagram is a visual way of understanding the relationship between these and their various subsets. 00:04:54.700 --> 00:05:00.600 You could use circles; I am using rectangles and squares...whichever. 00:05:00.600 --> 00:05:12.900 Irrational numbers: this is sometimes just known as Q with a line over it, and some examples that I just discussed-- 00:05:12.900 --> 00:05:18.800 the square root of 2, π, the square root of 5--these are irrational numbers. 00:05:18.800 --> 00:05:33.200 And then, I have rational numbers, sometimes expressed as Q; OK, this is the real number system. 00:05:33.200 --> 00:05:43.900 Within rational numbers are some subsets; the first one is the natural numbers. 00:05:43.900 --> 00:05:53.800 And the natural numbers are the numbers that we use to count things (1, 2, 3, and on); they are the numbers we use in counting. 00:05:53.800 --> 00:06:02.100 There is a slightly bigger set of numbers known as the whole numbers. 00:06:02.100 --> 00:06:08.500 The whole numbers include the natural numbers (this square is around the natural numbers, because it includes all of those). 00:06:08.500 --> 00:06:19.600 And it also includes 0; so, we add 0 to this set; and it does not include negative numbers. 00:06:19.600 --> 00:06:38.800 Next are the integers: the integers include natural numbers, whole numbers, and also negative numbers. 00:06:38.800 --> 00:06:42.900 So, 0 is included, and on. 00:06:42.900 --> 00:06:51.400 OK, so then, you can come out to just rational numbers (that are not whole numbers, natural numbers, or integers). 00:06:51.400 --> 00:07:02.600 And you can get fractions included, like -1/2, 0, 1, 2, 3/2, and on. 00:07:02.600 --> 00:07:06.400 The real number system is broken down into rational and irrational; and rational numbers 00:07:06.400 --> 00:07:16.300 are further broken down into the natural numbers, the whole numbers, and the integers. 00:07:16.300 --> 00:07:21.400 In algebra, it is important to understand the properties of real numbers--what it is 00:07:21.400 --> 00:07:25.700 that you are allowed to do, and not allowed to do, when working with real numbers. 00:07:25.700 --> 00:07:32.600 So, we call the various properties commutative, associative, identity, inverse, and distributive. 00:07:32.600 --> 00:07:42.700 Reviewing those: the commutative property applies both to addition and to multiplication. 00:07:42.700 --> 00:07:50.900 And what the commutative property tells us is that two terms can be added in either order, or multiplied in either order. 00:07:50.900 --> 00:07:58.300 So, if I have two numbers, a+b, I can change that order, b+a, and it is not going to change my result. 00:07:58.300 --> 00:08:07.400 For multiplication, the commutative property under multiplication, I could say a times b equals b times a. 00:08:07.400 --> 00:08:15.400 The associative property also applies to both addition and multiplication. 00:08:15.400 --> 00:08:23.900 The associative property tells you that, when you are adding or multiplying, the terms can be grouped in any way, and the result will be the same. 00:08:23.900 --> 00:08:29.800 Remember that we sometimes use grouping symbols with expressions and equations. 00:08:29.800 --> 00:08:42.500 So, looking at this, I could group it as (a+b)+c, or I could group it as a+(b+c)--group those together. 00:08:42.500 --> 00:08:46.500 Either way, they are equivalent; it is not going to change my result. 00:08:46.500 --> 00:08:55.700 The same holds true for multiplication: if I have (ab)c, that equals a(bc), and it doesn't matter 00:08:55.700 --> 00:09:04.200 if I decide to group it like this or like this--if I multiply these first or these first. 00:09:04.200 --> 00:09:13.200 The identity property: when I think of this, I just remember that the identity property tells me that the number maintains its identity. 00:09:13.200 --> 00:09:25.100 It doesn't change; so, for addition, what this says is that the sum of any number and 0 is the original number. 00:09:25.100 --> 00:09:35.300 So, a+0 is still a; so, the identity of the number does not change just because you add 0 to it. 00:09:35.300 --> 00:09:46.100 For multiplication, the product of any number and 1 is the original number; that is the identity property under multiplication. 00:09:46.100 --> 00:09:53.200 So, a times 1 is still the original number, a. 00:09:53.200 --> 00:10:07.000 The inverse property, as applied to addition, says that, when you add the same number, 00:10:07.000 --> 00:10:17.400 but with the opposite sign (a + -a, or say, 3 + -3), the result is 0. 00:10:17.400 --> 00:10:25.100 A number plus its additive inverse is equal to 0. 00:10:25.100 --> 00:10:40.200 This property can also be applied to multiplication; and this only applies to real numbers other than 0, when applied to multiplication. 00:10:40.200 --> 00:10:41.300 And you will see why. 00:10:41.300 --> 00:10:49.700 What this says is that, if you multiply a number by its reciprocal with the same sign, the result will be 1. 00:10:49.700 --> 00:10:57.200 We can't apply 0 here, because that would give us a 0 in the denominator, which is an undefined expression. 00:10:57.200 --> 00:11:03.600 So, a number times its reciprocal gives you 1. 00:11:03.600 --> 00:11:08.800 Finally, the distributive property: you can review more about this in the Algebra I lectures. 00:11:08.800 --> 00:11:16.700 This is a very important property when working with equations, solving equations, and working with algebraic expressions. 00:11:16.700 --> 00:11:24.700 Recall that a(b+c) equals ab + ac. 00:11:24.700 --> 00:11:30.900 So, we go forward to multiply; and when we go in the reverse direction, recall that that is factoring. 00:11:30.900 --> 00:11:44.100 This property also applies to multiplying a number by terms that are subtracting (ab-ac). 00:11:44.100 --> 00:11:55.700 If you put the numbers you are adding first, it doesn't change the property--it still applies; you get ab + ac. 00:11:55.700 --> 00:12:02.100 The same for subtraction. 00:12:02.100 --> 00:12:05.300 And finally, recall that this property can be applied to several numbers. 00:12:05.300 --> 00:12:12.100 You can have more than two numbers in the parentheses: this could be a(b + c + d). 00:12:12.100 --> 00:12:16.800 And then, you just multiply each one out: ab + ac + ad. 00:12:16.800 --> 00:12:21.500 And again, this is something we are going to be using a lot throughout the course. 00:12:21.500 --> 00:12:30.000 All right, applying some of these concepts to the examples: Example 1: What sets of numbers do these belong to (starting with 6)? 00:12:30.000 --> 00:12:40.100 Well, 6 is a real number; it is also a rational number (I can easily express this as a fraction: 6/1). 00:12:40.100 --> 00:12:47.400 And so, it is a rational number; then, I think about the subsets. 00:12:47.400 --> 00:12:56.300 Is it a natural number? Yes, it is a number that can be used in counting (1, 2, 3, 4, etc.), so it is a natural number. 00:12:56.300 --> 00:13:05.800 And it is therefore also a whole number; the whole numbers encompass the natural numbers; and it is an integer. 00:13:05.800 --> 00:13:09.000 6 belongs to all of these categories. 00:13:09.000 --> 00:13:18.900 The square root of 20 is a real number; however, recall that, unless you are talking about the square root of a perfect square, it is irrational. 00:13:18.900 --> 00:13:28.600 Square roots of perfect squares are rational; other square roots are irrational. 00:13:28.600 --> 00:13:40.900 -4/5: this is expressed as a fraction, so...well, it is a real number; and it is also rational, because it can be expressed as a fraction. 00:13:40.900 --> 00:13:46.300 It is not a natural number, because it is negative, and it is a fraction. 00:13:46.300 --> 00:13:50.300 The same thing: it is not a whole number, and it is also not an integer. 00:13:50.300 --> 00:13:57.700 So, this belongs to the two groups real and rational. 00:13:57.700 --> 00:14:09.100 OK, Example 2 asks what properties are used: so, there is an expression here--a mathematical expression--and various steps are taken. 00:14:09.100 --> 00:14:14.800 We need to determine which properties were used that allowed those steps to be taken. 00:14:14.800 --> 00:14:22.800 Well, looking at what happened between here and here, we started out with 2, times 4 plus 3 plus 7. 00:14:22.800 --> 00:14:32.300 The order of these was switched: 3+7 is still grouped together in parentheses, but it was put before the 4. 00:14:32.300 --> 00:14:36.800 Remember that the commutative property is the property that says that, when adding, 00:14:36.800 --> 00:14:43.900 you can change the order that you are adding terms in, and still get the same result; so this is the commutative property. 00:14:43.900 --> 00:14:51.300 OK, the next step: I look at what happened, and this big set of parentheses is gone. 00:14:51.300 --> 00:14:54.200 And the way it was removed is by use of the distributive property. 00:14:54.200 --> 00:15:04.500 Recall that the distributive property says that a, times (b + c), equals ab + ac; and that is what was done here. 00:15:04.500 --> 00:15:19.100 2 times the whole expression (3+7), and then 2 times 4; this is the distributive property. 00:15:19.100 --> 00:15:26.600 In the next step, the order of these two numbers was changed; so that is commutative. 00:15:26.600 --> 00:15:31.100 And a 0 was also added to 4; and remember that, according to the identity property, 00:15:31.100 --> 00:15:37.500 you can add 0 to a number, and the original number is unchanged (4 + 0 is 4). 00:15:37.500 --> 00:15:40.200 So, this is commutative and identity. 00:15:40.200 --> 00:15:51.600 Finally, it is getting rid of the rest of these parentheses by using the distributive property: 2 times 7, plus 2 times 3, plus 2 times 4, plus 2 times 0. 00:15:51.600 --> 00:16:00.900 So again, we are using the distributive property. 00:16:00.900 --> 00:16:06.900 Example 3 asks what is the multiplicative inverse of -6 and 7/8. 00:16:06.900 --> 00:16:19.100 Recall: multiplicative inverse, the definition, is a number times the reciprocal of that number; and recall that that equals 1. 00:16:19.100 --> 00:16:26.800 OK, so the multiplicative inverse--I have -6 and 7/8, so I am going to change this from a mixed number to a fraction. 00:16:26.800 --> 00:16:36.300 6 times 8 is 48, plus 7 is 55; and this is negative, so that is -55/8. 00:16:36.300 --> 00:16:46.800 So, looking at it as a fraction makes it much simpler; and I just need to take the multiplicative inverse of that. 00:16:46.800 --> 00:16:51.800 And so, I would change that to -8/55. 00:16:51.800 --> 00:17:00.500 And it does satisfy this formula right here, because if I took -55/8 (which is my original number), 00:17:00.500 --> 00:17:08.300 and I multiplied it by -8/55, two negatives (a negative times a negative) gives me a positive; 00:17:08.300 --> 00:17:13.300 the 8's cancel; then, the 55's cancel to give me 1. 00:17:13.300 --> 00:17:18.000 So, I was able to check my work by seeing that it satisfies this equation. 00:17:18.000 --> 00:17:24.400 OK, in Example 4, we are asked to simplify and state the property used for each step of simplification. 00:17:24.400 --> 00:17:30.700 First, I want to get rid of the parentheses; so I am going to use the distributive property. 00:17:30.700 --> 00:17:39.000 Multiplying out, recall that the distributive property is: a(b+c)=ab+ac. 00:17:39.000 --> 00:17:48.000 OK, this is 2(6x) + 2(3y + 4z). 00:17:48.000 --> 00:17:53.700 So, right now, I am just removing these outer parentheses; I am keeping these intact--that will take a second round. 00:17:53.700 --> 00:18:04.500 So, this is plus -3, times the entire expression in the parentheses, plus -3, times z. 00:18:04.500 --> 00:18:15.300 OK, I am going to apply the distributive property again, in order to remove the remaining parentheses. 00:18:15.300 --> 00:18:38.700 This gives me: 2(3y), plus 2(4z), plus -3(3x), plus -3(-y), plus -3(z). 00:18:38.700 --> 00:19:04.900 Now, I am going to multiply these out: this is 12x + 6y + 8z - 9x + (a negative and a negative is a positive, so that gives me) 3y - (this is -3z). 00:19:04.900 --> 00:19:16.800 OK, now I am going to group together like terms: and that is using the commutative property--I can change the order of these terms. 00:19:16.800 --> 00:19:38.700 I have my x's, 12x-9x; I have my y's, and that is 6y and 3y; and then finally, z's: 8z and -3z. 00:19:38.700 --> 00:19:52.800 All that is left to do is add like terms; so, 12x-9x is 3x; 6y and 3y gives me 9y; and 8z-3z is 5z. 00:19:52.800 --> 00:19:58.400 So, we are simplifying this, using first the distributive property (to remove the outer parentheses), 00:19:58.400 --> 00:20:03.200 then the distributive property to remove these other sets of parentheses, 00:20:03.200 --> 00:20:10.700 and the commutative property to re-order this to group like terms, and then simply adding or subtracting. 00:20:10.700 --> 00:20:15.000 That concludes this lesson from Educator.com; see you next lesson.