# Properties of Real Numbers

BAM! Mr. Tarrou. In this lesson, we are going
to be looking at some basic properties of real numbers, associative and commutative
properties of multiplication and division, excuse me addition and multiplication, division
doesn’t have associative properties as we will talk about it or commutative properties
for that matter. we will be looking at identity properties of addition and multiplication
what can you add and multiply by and still get the same answer, zero product property,
what happens when you multiply by -1. we are going to simplify some algebra 1 type expressions
and finish off our video talking about counter examples in other words how can I prove a
statement is false by simply using a counter example and how does that compare with deductive
reasoning which is kind of basically how we prove that two algebraic expressions are equal
by walking through these properties we are going to learn and show that one expression
is equal to another, usually another expression in simplified terms. A relationship or relationships
that are always true for real numbers are called properties. The key word here is ‘always
true’. For something to be true in mathematics, it must be true an infinite number of times,
like 10 million, 10 billion times and never fail. If you can come up with one counterexample
like we will talk about at the end of the video, even if it fails one time out of just
an uncountable number of successes, then it is a false statement. So we use these properties
to re-write and compare expressions that compares expressions is what I am talking about with
deductive reasoning. So equivalent expressions are two algebraic expressions that are equal
for all values of their variables. So if you have, well we’ll go through some examples
later but basically I will give you an algebraic example or two of them and regardless of what
number you plug into, or numbers you plug into those expressions, you will get the same
numerical answers. Those are equivalent expressions. Properties of real numbers. Well we are going
to let a,b and c be any real numbers and we have the commutative properties of addition
which says that we have a+b equals b+a. In other words we can change the order of the
addition and still get equivalent expressions or equal answers and there I have the multiplication
which is a times b is equal to b times a so again we can turn that order around, commute,
move the numbers. Like if I want to commute to work, I move from where I am here at home
to work, I drive to work. So commuting is taking these values and moving them around,
basically just switching places in these expressions or equations as it may be.
So here we have some examples of some statements that are true and we are going to actually
validate whether they are true or not with a little bit of arithmetic. so we have 3+5
which is equal to 8 and we have 5+3 which is equal to 8 and hopefully we know that from,
obviously from our earlier years of learning math but when you add on a number line you
move to the right and when you subtract you move to the left or when you are adding negative
numbers, you move to the left so from zero we move right there and then right 5 more
times, that’s of course going to give us an answer of 8. We start from 0 and move to
the right 5 and then 3 more steps to the right because of the addition of these positive
numbers, we get 8 again and of course that’s a true statement .that’s an example of commutative
property of addition. I have taken these numbers, I have moved around the addition side and
I get equivalent answers. Now here I have got some negative numbers going on and hopefully
we are a little bit comfortable with positive and negative numbers combining them, if not
we will be talking about that in the next section or the next lesson for algebra. We
have 2 plus -3. And you can think about positive and negative numbers in a couple of ways.
You can talk about money. Like a positive number is money coming in and negative numbers
is money going out so if we have positive \$2 and we want to spend \$3, we are going to
be short or owe a dollar giving you an answer of -1.
Now another way of thinking about that is again with the number line and if you have
0 here and we move two places to the right because that’s a positive 2 so 1,2 and then
we are adding a -3 well we moved two spaces to the right and that -3 is going to move
us backwards so I am going to move back 1, back again and that’s -2 we are back to 0
and back to the left one more time stepping 3 spaces to the left on the number line again
from 0, positive 2, 2 to the right, -3, 3 to the left and we are at -1. 1 space to the
left of 0 and again that’s how we get -1. well you got to be careful with this example
here which I am including purposely even though I am going to talk about combing positive
and negative numbers in the next section because there is no commutative property with subtraction,
but I am taking the negative with the, that’s in front of the 3 and I am moving it with
the 3 so here I have +2 plus a -3 and I have moved those numbers around the addition sign
and moved the -3, moved the negative with the 3 and kept it out front so we have -3
plus 2 well that’s also going to be -1. And we have our true statements. So there’s
two examples here of the commutative property. Just one more time, if this, my hand is zero,
we have 3 movements to the left and two movements to the right, so here we go 1, 2, 3 and then
back 2, we are still going to land one space on the left of zero or that’s going to put
as at -1. Well here, see I made these very close on
purpose. I have 2 plus -3 and that is actually an equivalent statement with 2-3. Again 2
movements to the right, 3 units to the left. Or if you want to have, or you have \$2 and
you want to spend \$3, you are going to owe somebody or be short of a dollar and if I
come over here to the other side, you notice how I took those numbers and I changed their
places but here the 3 has a minus in front of it or a negative sign and over here the
three is positive. So I have just taken the positive 2 and the 3 ignoring the minus sign
and changed their places and I have tried to show what would like the commutative property
of subtraction. Well +3 minus 2, I mean if you have \$3 and you want to spend \$2, you
will still have 1 left over and -1 is not equal to positive 1 and this is an example
of why, a counterexample of why we don’t have the commutative property of subtraction.
Or again with the number line, if there’s a zero here, there’s positive 3 to the right,
3 steps to the right of -2 is going to back us up 2 and we are 1 space to the right of
zero as opposed to before when we were one space to the left.
Multiplication. 4 times 5 is equal to 20 and 5 times 4 is also equal to 20. So again we
have that commutative property of multiplication going on here. And one third of 15, well one
third of 15, when you start working with fractions, you really should just make everything look
like a fraction. So that you can see there’s a couple of ways you can simplify this but
one times 15 is equal 15 over 3 times 1 which is equal to 3 and on the right side we have
if we place that 15 over 1 to make sure our numerators and denominators are lining up.
And you notice I am not going through the process of finding common denominators because
you don’t need those when you are multiplying. So we have 15 times 1 is 15 over 1 times 3
which is equal to 3 and we have an equivalent statement here which is 15/3 is equal to 15/3.
Now I really kind of prefer you and my students to write 15 divided by 3 is equal to 5 so
really as a simplified form this should say 5 is equal to 5. I showed a counterexample
for subtraction showing you that you cannot just change the order, use the commutative
property with subtraction. The same is going to be true with division. There is no commutative
property with division. So if I clean this off real quick. And let’s say we have 20
divided by 4 well that is the same as 20 divided by 4. Either way you look at it, is equal
to 5. If I try and change this order around and write 4 divided by 20 that is going to
be 4 divided by 20. well now this time that doesn’t equal 5 again because it’s the
top number divided by the bottom and 20 doesn’t even go into 4 once much less 5 times. Now
4 and 20 are both divisible by 4. So I can reduce the fraction, dividing the numerator
and the denominator by 4 and getting an answer of 1/5ths and again we do not have a commutative
property of subtraction, we do not have a commutative property of division. I just gave
you a little counterexample for division. The associative property of addition and multiplication
that says, well now associative, to associate with people is to group with them, to make
friends basically. So the associative property of addition and multiplication is about regrouping
the numbers not moving them like we do with the commutative property. So our example here
says, or our rule, excuse me, (a+b)+c is equal to a+(b+c). And we basically have a very similar
statement for multiplication. Well if you remember your order of operation we just spoke
of recently or you learnt in your math class, you have to work inside the parentheses first
so when I see (7+2)+1, those parentheses are telling me that I need to add the 7 and 2
first so we get 9 +1 and over here we have 2 plus 1 which is 3. And 9 plus 1 is equal
to 10. Excuse me. And 7 plus 3 is also equal to 10 and we have equivalent statements. So
we were allowed to move the grouping symbols, if you will in this example and get equivalent
statements. So at least one example, not a proof. But it is at least one example of the
associative property working. Now why don’t we have associative property of subtraction?
Well if I come back up here and let’s change the plus sign between the 7 and the 2 into
a negative sign. Actually let’s just highlight it in a different color so you can know that
I changed the problem. 7 minus 2, working inside the parentheses first and well we got
to work in the parentheses first now, trying to apply the associative property to an expression
that includes subtraction and we have 2+1 which is 3 and 5 plus 1 is 6. 7 minus 3 is
equal to 4 and that of course is not true. So again we have an associative property of
addition like my original example right before I changed it, which worked. I included a little
bit of subtraction in there, now the associative property fails. That counterexample shows
that we do not have an associative property of subtraction. It works with multiplication,
I am not going to do a counterexample for this but we have 4 times 2 is 8. We have,
working the parentheses first 2 times 5 is 10 and 8 times 5 is 40. And 4 times 10 is
40 as well. If you are just getting into algebra, this is basically my third or fourth lesson,
I believe, in my Algebra 1 course. So you might be tempted still to use a little cross
for multiplication, well of course if algebra working with variables and a lot of times
those variables are x’s. The variable x, the letter x looks exactly like a multiplication
symbol of x so we avoid those at all costs. So don’t ever just use x for multiplication
is algebra. You are going to get confused with that multiplication symbol and sometimes
when it’s actually a variable. So use little dots or parentheses to indicate multiplication
not x’s. Next screen! Nanana. Identity property of addition and
multiplication. Addition. If you add by a zero, nothing changes. So that any number
plus zero, is equal to any number. Well if I have got \$4 in my pocket, you give me absolutely
nothing, I spent nothing. I still have \$4. Okay multiplication. Any number times 1 is
equal to itself. So 5 times 1 is 5. Oh actually I can put a number up here how about 7 plus
0 is still equal to 7. We have 7 times 1 is still equal to 7. Any number times 1 is equal
to itself. and by the way when you are setting up algebra, like adding like terms or even
multiplying if you want, but especially and subtracting like terms, it’s a really good
idea, actually if you are adding a variable and there’s no number out front of it, might
be a little much for these rules, but if you do a+5a and you keep forgetting to say its
6a, cause 1 plus 5 is equal to 6, then go ahead and remind yourself by literally writing
that 1 in there, that coefficient of 1 in front of your variable. Okay so, but these
identities are unique to that particular math operation. If I add by 0, nothing happens,
like 7+0 is still 7 but if I were to multiply by 0, a lot happens. Anything times zero,
the product of any number and zero is zero itself. Well this comes in very very handy.
Kind of seems like it’s obvious but, yeah 3 times 0 is equal to0. You don’t have to
tell me that. I have been doing arithmetic and using that for years now maybe and your
math history. But this idea in algebra that anything multiplied by 0 is equal to 0 is
extremely important because later on this year when you are in geometry and you are
reviewing it again in Algebra 2 and so on and so on, you are going to learn something
like factoring. you are going to solve equations by factoring them and this product or this
zero property of multiplication comes in very handy and is very very important so a times
zero is equal to zero. Anything times zero is equal to zero.
And when you multiply a number by -1, well when you multiply by 1, you get the exact
same answer back, but when you multiply by -1, that negative sign is going to change
the sign. So any number multiplied by -1, any number multiplied by -1 is going to be
that same number because we multiply it by 1, which is the identity for multiplication
but that negative is going to change the sign. So if we have 5 times -1, that’s going to
be equal to -5, yes, or if it was initially -5 and I multiply it by -1, we know that negative
times negative is positive and we get a positive 5 so maybe we want to think of multiplying
by negative 1 doesn’t give us a negative answer but actually changes the sign. Okay
so because a, this idea of ‘a’ could be any number, doesn’t have to be a positive number.
So -5 times -1 is +5. So when you take any number and multiply it by -1, you actually
get the opposite answer. You shouldn’t really say this and I think I just did so I am going
to correct myself now. You shouldn’t think of this as negative a, but the opposite of
a. so any number times -1 gives you or equals the opposite of what you originally started
with. So -5 times -1 is +5 cause 5 is the opposite of -5 and by the way with opposites,
you know the two numbers are opposites when you add them and you get zero. Any two numbers
opposites. And again if you started with that +5 and you multiplied by a -1, you get -5
which is the opposite of positive 5. Simplify the algebraic expressions or just, excuse
me, expressions in general. Even though these are all algebraic, they have variables in
them. So we are going to take this 3 times 7x.and if you are really strong in your algebra
skills, you are probably going to know the answer, but if you are using a lot of these
new rules and properties of real numbers that I gave you, you would actually recognize that
all of this is actually being multiplied together so you could apply the associative property
of multiplication to move the parentheses around the 7x and take them away from or move
them away from 7x and regroup the 3 times 7 instead and do that first and you can really
see here now, this is 3 times 7 or 21x and that is the simplified form of 3 times 7 x
done with the associative property of multiplication. Well here we have 2+5x+10 and this expression
has a couple of like terms that we can add together. Indeed those like terms are the
constants or the regular value, the term here that has no variable in it, just the two sitting
alone and then we have another term, by the way terms are separated by addition and subtraction
so we have 1..2.. 3 terms here. And the terms that can be added together, the like terms
are 2 and 10. Now eventually, maybe your teacher’s doing it right now, but just circling the
2 and 10 to show you what like terms can be added together but if you are just starting
to learn, like we are, the properties of real numbers, we can show that we can move these
numbers around the addition signs using the commutative property of addition. I am going
to write 2+10+5x. And now that we have the constant of 2 and the constant of 10 and I
am saying that because they don’t have a variable in them. These like terms can be
added together now that they are right next to each other. And positive 2 plus positive
10 or just 2+10 is equal to 12 plus 5x. And then down here, this is, we use a little bit
of properties but it kind of just seems like I am simplifying here. I got 42ab of 42 times
a times b, anytime a number or letter or letters, variables are touching, that’s multiplication.
So we have 42 times a times b divide by, fraction bar’s division, 6 times a. Well I can simplify
this and help the process along but, writing every sort of similar item with its own fraction
of these numbers. Well 42 divided by 6 a divided by a and then I need to show that the b is
over something. Well remember all this is being multiplied together and I want to show
a numerator and denominator. So what will I show in the denominator that has something
there but yet if I multiply this, it doesn’t change the value of 6a, the original denominator?
I am going to put a denominator of 1 and use that, basically sort of that multiplication
slash, in this case I can say, kind of like we have an identity really of multiplication,
multiplying by 1 gives us the same answer. Really, you can kind of think there’s a division
property as well. If you divide by 1, nothing changes. So we can say now that this 42 over
6 is equal to 7, we have anything divided by itself, what’s 2/2? Well 2 goes into itself
once. 3/3. 3 goes into itself once. 10/10. Anything divided by itself is equal to 1 so
I am just simplifying here and we get a divided by a which is just equal to 1. So 7 times
1 and then b divided by 1. Well anything divided by 1, just like when we were multiplying by
1, doesn’t change the answer. So we just simply have b and our final answer is 7b.
Let me, I am just going to step off here and write in the properties that I used a little
bit to simplify these expressions now and then we’ll be moving on to our last , I think
it’s going to be our last screen of notes. Board of notes. Our previous 3 examples I
just worked through showing them step and doing the associative property, commutative
property and ran a little bit down there as I was going along. Those are done with deductive
reasoning which is the process of reasoning logically from given facts to a conclusion.
It’s one of the greatest things about, that you should be learning from algebra is this
idea of deductive reasoning. Being given a complex situation, understanding that there’s
a set of rules that needs to be followed and then following those in a specific order so
that you logically get from the beginning of your problem to a solution. And then of
course if you go through more and more mathematics, you are going to start to see some good real
life applications of some of those mathematics but one of the most important things we are
teaching here in algebra is this idea of logically working your way through some complex problems
and you can apply those logic skills you build to other situations even outside of math.
prove a mathematical statement wrong. I have sort of written down this commutative property
of subtraction one more time but actually drew the number lines for you so you can see
the movement as opposed to just my hand moving around to validate that no, 2 is not equal
to -2 which is what we get from attempting to reverse the order of 7 minus 5 to become
5 minus 7. Or a minus b, any number minus another number is equal to that same second
number minus the first. Just doesn’t work. Okay? So we only have a commutative property
of addition and multiplication. Our example here looks very complex and I certainly would
not want to work this out by hand. 5 to the ninth minus 7 to the fourth. 5 to the 9th
is a very very large number. But thankfully I don’t need a calculator to evaluate this
expression if I understand again that multiplication rule of zero, that anything multiplied by
zero is itself equal to zero. Well that’s what 6 minus 6 is. We have 5 to the ninth
power minus 7 to the fourth times 0 and as soon as one of my factors, that’s a long
string of basically multiplication, pieces that are being multiplied together. Once one
of my factors is equal to zero, I know the entire expression is going to be zero and
I am done. A box of chalk holds 12 pieces of chalk. Just happen to have one right here!
Actually quite a few of them, you can imagine! A sleeve of chalk holds 9 boxes and a case
of chalk holds 10 sleeves. Honestly I made those numbers up but I just wanted to give
you a word problem that we have to think through and decide what is the best approach of solving
it. And so this box of chalk holds 12, this box of chalk holds 12, this box of chalk holds
12 and it’s like continuing this addition saying that, cause ultimately, what, I have
12 piece in a box. Now I want to do something about the fact that there’s 9 of those boxes.
Well I can add 12, every time I put one box down or I can simply multiply by 9. Repeated
addition is multiplication. And in the end I want to know how many pieces of chalk I
would have if I were to buy the entire case and these numbers right too, besides the fact
that 12 pieces aren’t a box of chalk. So one box is 12 pieces. Okay. Well sleeve is
9 of those boxes, each one containing 12, so it’s going to be 12 times 9 and okay
then we get an answer from that which we’ll do in a second. But then we have a whole case
which holds 9 of those sleeves, or excuse me, not nine a case which holds 10 of those
sleeves so we are going to do those 12 pieces of chalk in a single box, times 9 for the
sleeve and then there’s 10 of those sleeves in 1 case of chalk so we are going to multiply
that by 10 and we got here 9 times 12, that’s the tens place, 9 times 10 is 90 and 9 times
2 is 18. So that’s 90 and 18 so that’s a 108 and basically I am really just following
the order of operation working my multiplication from left to right or you can think of me
as wrapping that 12 times 9 in the set of parentheses and associating those first two
together and this is all just multiplication so I can basically multiply any pair of numbers
I like, re-associate those groups, associative property of multiplication. So again we have
12 times 9 is a 108 times 10 is going to be a 1080 pieces of chalk. So let’s see here…
kind of running out of room. But when you do a word problems, it’s always good to
summarize what does that number actually mean. As we get into solving more problems and you
yourself in your algebra class, and a lot of times I have students well there’s always
difficult to set up word problems but then sometimes I get students that set them up
correctly. I will work out the algebra required to solve those equation or simplify the expressions
or whatever and then I get a number at the end and I don’t know what it stands for!
So we are looking for, and I ran out of room actually, but we are looking for the number
of pieces or the pieces of chalk that are not in just a box, not in a sleeve but in
an entire case that holds 10 of those sleeves. And so 12 in a box times the 9 boxes that
are in the sleeve times the 10 sleeves which we said were in one case. So I am Mr. Tarrou!