Algebra 2 Properties of Real Numbers

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In today's lesson, we will be discussing properties of real numbers.0002

A real number corresponds to a point on the number line, and real numbers may be either rational or irrational.0007

So first, just talking about the number line: some examples of real numbers would be 0, 1, 1/2,0016

or (expressing a number as a decimal) it can be 2.8; negative numbers--maybe -2.18 or -4.0032

OK, these are all real numbers, and they are rational; so let's talk about the difference between rational and irrational numbers.0042

Rational numbers can be expressed as a fraction.0051

So, rational numbers are expressed in the form a/b, where a and b are integers, and b does not equal 0,0062

because as you will recall, we cannot have 0 in the denominator, because that would result in an undefined expression.0075

When a rational number is expressed as a decimal, it will either be terminating or repeating.0084

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 .5.0097

I might have another number, 1/3, which is also rational; and it is going to end up being .3333, and on indefinitely.0108

This could also be written as .3 with a bar over it, indicating it is repeating.0122

So, this is also rational, because it repeats; the repeating pattern could be longer--it could be 2.387387387, so this is repeating.0126

The point is that they either terminate or repeat when expressed in decimal form.0139

Irrational numbers cannot be expressed in the form a/b, so they are expressed as decimal form.0145

But they neither terminate nor repeat in decimal form; they just go on indefinitely.0156

For example, I could have 4.871469837246, and on and on and on, with no pattern--no repeating and no ending.0174

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 3.14.0189

Therefore, we can add π up here (that is an irrational number) to represent some irrational numbers up here.0201

In addition, the square root of numbers, other than the square root of numbers that are perfect squares, are irrational numbers.0209

So, the square root of 3 is, or the square root of 2.0217

If it is a perfect square, that means it is the result of multiplying an integer by itself.0221

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 2.0227

All other square roots are irrational numbers; so again, both rational and irrational numbers are real numbers, but they have different properties.0234

Sometimes, we express the relationship between the various types of real numbers using a Venn diagram.0244

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.0252

We have the real number system, and we have irrational numbers, and we also have rational numbers.0261

And a Venn diagram is a visual way of understanding the relationship between these and their various subsets.0282

You could use circles; I am using rectangles and squares...whichever.0295

Irrational numbers: this is sometimes just known as Q with a line over it, and some examples that I just discussed--0300

the square root of 2, π, the square root of 5--these are irrational numbers.0313

And then, I have rational numbers, sometimes expressed as Q; OK, this is the real number system.0319

Within rational numbers are some subsets; the first one is the natural numbers.0333

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.0344

There is a slightly bigger set of numbers known as the whole numbers.0354

The whole numbers include the natural numbers (this square is around the natural numbers, because it includes all of those).0362

And it also includes 0; so, we add 0 to this set; and it does not include negative numbers.0368

Next are the integers: the integers include natural numbers, whole numbers, and also negative numbers.0379

So, 0 is included, and on.0399

OK, so then, you can come out to just rational numbers (that are not whole numbers, natural numbers, or integers).0403

And you can get fractions included, like -1/2, 0, 1, 2, 3/2, and on.0411

The real number system is broken down into rational and irrational; and rational numbers0422

are further broken down into the natural numbers, the whole numbers, and the integers.0426

In algebra, it is important to understand the properties of real numbers--what it is0436

that you are allowed to do, and not allowed to do, when working with real numbers.0441

So, we call the various properties commutative, associative, identity, inverse, and distributive.0446

Reviewing those: the commutative property applies both to addition and to multiplication.0452

And what the commutative property tells us is that two terms can be added in either order, or multiplied in either order.0463

So, if I have two numbers, a+b, I can change that order, b+a, and it is not going to change my result.0471

For multiplication, the commutative property under multiplication, I could say a times b equals b times a.0478

The associative property also applies to both addition and multiplication.0487

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.0495

Remember that we sometimes use grouping symbols with expressions and equations.0504

So, looking at this, I could group it as (a+b)+c, or I could group it as a+(b+c)--group those together.0510

Either way, they are equivalent; it is not going to change my result.0522

The same holds true for multiplication: if I have (ab)c, that equals a(bc), and it doesn't matter0526

if I decide to group it like this or like this--if I multiply these first or these first.0536

The identity property: when I think of this, I just remember that the identity property tells me that the number maintains its identity.0544

It doesn't change; so, for addition, what this says is that the sum of any number and 0 is the original number.0553

So, a+0 is still a; so, the identity of the number does not change just because you add 0 to it.0565

For multiplication, the product of any number and 1 is the original number; that is the identity property under multiplication.0575

So, a times 1 is still the original number, a.0586

The inverse property, as applied to addition, says that, when you add the same number,0593

but with the opposite sign (a + -a, or say, 3 + -3), the result is 0.0607

A number plus its additive inverse is equal to 0.0617

This property can also be applied to multiplication; and this only applies to real numbers other than 0, when applied to multiplication.0625

And you will see why.0640

What this says is that, if you multiply a number by its reciprocal with the same sign, the result will be 1.0641

We can't apply 0 here, because that would give us a 0 in the denominator, which is an undefined expression.0650

So, a number times its reciprocal gives you 1.0657

Finally, the distributive property: you can review more about this in the Algebra I lectures.0663

This is a very important property when working with equations, solving equations, and working with algebraic expressions.0669

Recall that a(b+c) equals ab + ac.0677

So, we go forward to multiply; and when we go in the reverse direction, recall that that is factoring.0685

This property also applies to multiplying a number by terms that are subtracting (ab-ac).0691

If you put the numbers you are adding first, it doesn't change the property--it still applies; you get ab + ac.0704

The same for subtraction.0716

And finally, recall that this property can be applied to several numbers.0722

You can have more than two numbers in the parentheses: this could be a(b + c + d).0725

And then, you just multiply each one out: ab + ac + ad.0732

And again, this is something we are going to be using a lot throughout the course.0737

All right, applying some of these concepts to the examples: Example 1: What sets of numbers do these belong to (starting with 6)?0741

Well, 6 is a real number; it is also a rational number (I can easily express this as a fraction: 6/1).0750

And so, it is a rational number; then, I think about the subsets.0760

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.0767

And it is therefore also a whole number; the whole numbers encompass the natural numbers; and it is an integer.0776

6 belongs to all of these categories.0786

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.0789

Square roots of perfect squares are rational; other square roots are irrational.0799

-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.0808

It is not a natural number, because it is negative, and it is a fraction.0821

The same thing: it is not a whole number, and it is also not an integer.0826

So, this belongs to the two groups real and rational.0830

OK, Example 2 asks what properties are used: so, there is an expression here--a mathematical expression--and various steps are taken.0838

We need to determine which properties were used that allowed those steps to be taken.0849

Well, looking at what happened between here and here, we started out with 2, times 4 plus 3 plus 7.0855

The order of these was switched: 3+7 is still grouped together in parentheses, but it was put before the 4.0863

Remember that the commutative property is the property that says that, when adding,0872

you can change the order that you are adding terms in, and still get the same result; so this is the commutative property.0877

OK, the next step: I look at what happened, and this big set of parentheses is gone.0884

And the way it was removed is by use of the distributive property.0891

Recall that the distributive property says that a, times (b + c), equals ab + ac; and that is what was done here.0894

2 times the whole expression (3+7), and then 2 times 4; this is the distributive property.0904

In the next step, the order of these two numbers was changed; so that is commutative.0919

And a 0 was also added to 4; and remember that, according to the identity property,0926

you can add 0 to a number, and the original number is unchanged (4 + 0 is 4).0931

So, this is commutative and identity.0937

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 0.0940

So again, we are using the distributive property.0951

Example 3 asks what is the multiplicative inverse of -6 and 7/8.0961

Recall: multiplicative inverse, the definition, is a number times the reciprocal of that number; and recall that that equals 1.0967

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.0979

6 times 8 is 48, plus 7 is 55; and this is negative, so that is -55/8.0987

So, looking at it as a fraction makes it much simpler; and I just need to take the multiplicative inverse of that.0996

And so, I would change that to -8/55.1007

And it does satisfy this formula right here, because if I took -55/8 (which is my original number),1012

and I multiplied it by -8/55, two negatives (a negative times a negative) gives me a positive;1020

the 8's cancel; then, the 55's cancel to give me 1.1028

So, I was able to check my work by seeing that it satisfies this equation.1033

OK, in Example 4, we are asked to simplify and state the property used for each step of simplification.1038

First, I want to get rid of the parentheses; so I am going to use the distributive property.1044

Multiplying out, recall that the distributive property is: a(b+c)=ab+ac.1051

OK, this is 2(6x) + 2(3y + 4z).1059

So, right now, I am just removing these outer parentheses; I am keeping these intact--that will take a second round.1068

So, this is plus -3, times the entire expression in the parentheses, plus -3, times z.1074

OK, I am going to apply the distributive property again, in order to remove the remaining parentheses.1084

This gives me: 2(3y), plus 2(4z), plus -3(3x), plus -3(-y), plus -3(z).1095

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).1119

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.1145

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.1157

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.1179

So, we are simplifying this, using first the distributive property (to remove the outer parentheses),1193

then the distributive property to remove these other sets of parentheses,1198

and the commutative property to re-order this to group like terms, and then simply adding or subtracting.1203

That concludes this lesson from Educator.com; see you next lesson.1211