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.