Hello, in this video, I will briefly recall all the rules, the computational rules with real numbers
and why and the reason for this is that, well, you are going to perform computations on real numbers.
So it's good to repeat how you do this.
And the second reason is that when we will define operation arithmetical operations on matrices like
matrix additions or matrix multiplication, we will try to check if they if the same rules apply for
those computations.
So this is the second reason why it is good to repeat all those rules.
So let's start roles for computations with real numbers.
And you know that you have basically two kind of arithmetical operations with real numbers.
You have addition and multiplication.
Well, you could also say that you have subtraction and division, but actually subtraction is just
an addition of an opposite number.
And division is multiplication by by an inverse.
So basically, you have just two operations, a multiplication and addition.
And for both addition and multiplication, you have the so-called commutative law.
And this law tells you that addition and multiplication are not dependent on the order of the terms
or factors.
So these numbers here, in case of addition, are called terms.
And when you multiply numbers, then you call them factors.
So if you take like one plus two or two plus one, you arrive at the same result and the same thing
with multiplication.
Three times seven is the same as seven times three.
It should be obvious then the is associative low.
It tells you that you can well put parentheses as you like.
And what is what, what does it mean in practice?
This means that if you have to add three numbers to each other, it doesn't matter if you first add
that to second numbers and then the result is added to the first one.
Or if you add like two first numbers and add the result to the third one, you will arrive at the same
result and the same for multiplication.
Then you have the so-called the distributive law, which kind of connects multiplication with with addition.
And sometimes this rule is called a parenthesis multiplication.
So it tells you basically that if you have some factor and it has to be multiplied by a sum, then you
have to multiply this factor with each and every element in the parentheses.
So each term of this sum and if you have a hard time remembering this rule, then you can use the following
mental picture.
Well, it works, of course, only for A, B and C, which are positive numbers.
Then you can think of them of A, B and C as of length of some interval.
So this interval has length A, this one has length B and this one has legs C.
So from here to here and well, what is the left hand side of this equality?
It is the area of the rectangle with one side of length A and the second side of length B plus C, so
you know that the area is one length at times width.
So it is A times B plus C, but you can split this rectangle in two rectangles.
One of them has size a baby and the second one A by C.
So that area of this first one is eight times B and the area of the second one is eight times C.
So the total some of the areas of these two rectangles are A, B plus A C, but this the sum of the
two areas is exactly the same as the area of the total, a large rectangle.
So this means that indeed A times B plus C is equal to a B plus a C.
But this is just a geometrical interpretation, it's not a proof of this role because actually this
rule is just that all which which does not need any proof rules like this are called axioms.
These are just truth, which are in some way reasonable and which we follow in our computations.
But they don't need any and any justification.
But it is good to to have this mental picture, if you like.
Don't remember this from.
That next month, that neutral element of the addition is a zero, and this means that if you add zero
to anything, you will not to change this anything, it will just be like the same thing.
So addition of zero changes, nothing.
And in this way, a zero is a neutral element of the addition.
And the same way for multiplication with one one is, say, the neutral element of multiplication.
It means that if you multiply anything by one, you will not change anything.
It will still be the same number.
And the next one, the zero product property, the zero product property tells us that if a product
of two numbers is equal to zero, then it must.
That some of these numbers may be two of them, but at least one of these numbers must be equal to zero.
So only the product of two or more numbers is equal to zero.
Only if some of the factors, some of the numbers is equal to zero.
This is a small sign.
It is or it's a mathematical symbol for or and this implication, it means that from this follows this.
So if a B is equal to zero, then either A is equal to zero or B, C equal to zero, possibly both of
them, but at least one must be equal to zero.
Then we have this cancellation property.
And what does it tell us?
That if a B is equal to a C and this small guy is a mathematical symbol for end and a non zero is different
from zero, then B must be equal to C and why is it called a cancellation property?
Well, you can think of it that you can divide both sides of this equality by a and the equality will
be still preserved.
And this is why you have to demand that a non-zero because you cannot divide by zero is one reason.
And the second reason is that if A is equal to zero, then this equality is always true regardless of
B and C. So if A is equal to zero, you cannot you cannot have this conclusion that the baby must be
equal to C because well, zero times seven is equal to zero times thirty four, but seven is not equal
to 34.
Right.
And you use this cancellation property, for example, in cases like this, when you know the two X
is equal to two Y, then you can be sure that X is equal to Y.
So you have made this cancellation by two.
So in this case, A is equal to two.
It's obviously not zero and a B is equal to X and C is equal to Y, right.
So in this you apply this cancellation property in this way.
And the last property existence of inverse element, so each element A has its, well, kind of inverse
element.
But inversely, in this case, in the case of additions, we usually don't use the word invest.
We we we we would rather use the word opposite number.
So the opposite number is the number which added to the given number gives the neutral element zero.
So it's kind of minus a and each element has its well opposite number and this opposite number in case
of multiplication then it is really always called the inverse element and this is well one divided by
a.
So for example four for a equal to two.
It's a multiple multiplicative inverse is one half because two times one half is equal to one for a
equal to one third.
The inverse will be three because one third times three is equal to one.
And all the real numbers except for zero, do have the inverse element for multiplication.
So these are the computational rules and, well, as I told you later, when we analyze the rules for
matrix multiplication and matrix addition, we will compare those roles well to this list.
And we will look if all the rules are preserved or maybe there are some differences, there will be
differences.
Statement by, by.