Basic Mathematics
Lesson : 4 (Properties of Numbers)
With host: Luis Anthony
Ast (The Video Math Tutor)
Welcome to Basic
Mathematics Lesson number 4, properties of numbers.
A special Note
For explain properties, I would use variable ABC,
to present my numbers or my variables or other my variable expression.
Now I’ll choose say number.
A.
Properties of numbers
1.
The Reflexive Property of
Equality
A number is equal to itself.
So, symbolically is about meaning what that A is
equal to A. Its self. And how ways is like thing 2 same 2, 3 same 3. This is
simple. So it’s good to know this.
2.
The symmetric Property of
equality
If one value is equal to another, then that
second value is the same as the first.
Symbolically, we can just say if A equal to B, then
this B equals to A, that like symmetrical.
The word problem its look like this, 3 equals X,
that’s fine. But to this usually for finally answer you wanted x to be the other side, so x is equal to 3.
3.
The Transitive Property of Equality
If one value is equal to a second, and the
second happens to be the same as a third, then we can conclude the first value
must also equal the third.
Symbolically if A equal to B and B is equal to C,
so that A is equal to C.
4.
The Substitution Property
If one value is equal to another, then the
second value can be used in place of the first in any algebraic expression
dealing with the first value.
If A is equals B, then B can be substituted for A
and A can be substituted for B.
5.
The Additive Property of
Equality
We can add equal values to both sides of an
equation without changing the validity of the equation.
To see this rule, now we can use A equals to B. We
can add the same thing to both sides of an equation without changing. So I can
actually A plus C equals B plus C. We also change this to C plus A equals C
plus B.
6.
The Cancellation Law of
Addition
We can subtract the same thing in the equation. if
we look like this A+C=B+C, so we can subtract each side with C. and then this
is gonna be A=B.
7.
The Multiplicative Property
of Equality
We can multiply equal values to both sides of an
equation without changing the validity of the equation.
If A equals to B, I can multiply both sides
equation with C. And then A times C equals B times C we also can change this to
be C times A equals C times B.
8.
The Cancellation Law of
Multiplication
If we want to cancel the equation like this A times
C equals B times C, we can divide both sides with C. so that can be A equals B.
9.
The Zero Factor Property
If two values that are being multiplied together
equal zero, then one of the values, or both of them, must equal zero
If A times B equals zero, what we can calculate
this situation? A is equal to zero or B is equal to zero, or both of them is
must be zero, to make this situation true.
B.
Properties of
Inequalities
1.
The law of Trichotomy
For any two values, only ONE of the following
can be true about these values:
They are equal.
The first has a smaller value than the second.
The first has a larger value than the second.
Given any numbers A and B. so that ca be A=B or
A<B or A>B.
2.
The transitive Property of
Inequality
If one value is smaller than a second, and the
second is less than a third, then we can conclude the first value is smaller
than the third.
If A is less
than B and B is less than C, so we can transit so that A less than to C.
C. Properties of Absolute Value
For number one |A| is greater than and equals to
zero. Number two, the absolute value of negative A is equals to the absolute
value of A, like this |-A|=|A|.
Number three |AB|=|A||B|. You can multiply the
numbers in sides the absolute value or you can check of the number absolute
value in the right side and multiply together. Number four |A/B|=|A/B|. The
denominator is not equal to zero. So you can divide the absolute value in the
left side or you can check the numbers of the absolute value in the left side
and you can divide it.
D.
Properties of Numbers (Closure)
1.
The Closure Property of
Addition
When you add real numbers to other real numbers,
the sum is also real. Addition is a “closed” operation.
A plus B is equals a real number. If A is real
number and B is real number, then the answer will gonna be a real number.
2.
The Closure property of
Multiplication
When you multiply real numbers to other real
numbers, the product is a real number. Multiplication is a “closed” operation.
Symbolically with multiplication, A times B is
equal to a real number. So A is real, B is real, and the result is real.
A special Note
Like this data, the real numbers or closed with
aspect addition or multiplication, if the problem is real numbers so the result
also real numbers.
For example 3 is natural number and 5 is natural
number too. Then three minus five is negative two. and negative two is not part
of the sets of the natural numbers, so is not closed. If want to be “closed”
the all sides must a natural numbers.
E.
Commutativity
1.
The Commutative Property of
Addition
It does not matter the order in which are added
together.
For example A plus B is the same thing of B plus A.
2.
The Commutative Property of
Multiplication
It does not matter the order in which numbers
are multiplied together.
Of this property, if A times B is exactly same with
B times A.
F.
Associativity
1.
The Associative Property of
Addition
When we wish to add three (or more) numbers, it
does not matter how we group them together for adding purposes. The parentheses
can be placed as we wish.
(A+B)+C , I can associate for group this together,
two the term (A+B) and about the third term C. or I can show this with the
different way, like this A+(B+C).
2.
The Associative Property of
Multiplication
When we wish to multiply three (or more)
numbers, it does not matter how we group them together for multiplication
purposes. The parentheses can be placed as we wish.
(A.B).C, I can associating the first two term (A.B)
and then the third term C. or I can associate the group with different way,
like this A.(B.C).
G.
Identity
1.
The Identity Property of
Addition
There exists a special number, called the
“additive identity.” When added to any other number, then that other number
will still “keep its identity” and remain the same.
Symbolically A plus zero equals to A. And if I put
zero first plus A is equal to A.
2.
The Identity Property of
Multiplication
There exists a special number, called the “multiplicative
identity.” When multiplied to any other number, then that other number will
still “keep its identity” and remain the same.
A times one is equals to A, if I exchange to 1
times A is also equals to A.
A special Note
Zero is the unique additive identity. And one is
the unique multiplicative identity. Don’t confuse.
H.
Inverse
1.
The Inverse Property of
Addition
For every real number, there exists another real
number that is called its opposite, such that, when added together, you get the
additive identity (the number zero).
Symbolically if A plus the inverse of A is equals
to zero. And if negative A plus A then the answer is zero.
2.
The Inverse Property of
Multiplication
For every real number, except zero, there is another
real number that is called multiplicative inverse, or reciprocal, such that,
when multiplied together, you get the multiplicative identity (the number one).
Symbolically we can say the number (A) times its
multiplicative inverse (1/A), so that the result is one. And then the
multiplicative inverse (1/A) times a number (A) is also equals to one.
By the way
There is one number it doesn’t have the
multiplicative number, it is zero. For example, if we divide one by zero, the
result is undefined. So, zero has no multiplicative number.
I.
Distributivity
1.
The Distributive Law of
Multiplication Over Addition
Multiplying a number by a sum of numbers is the
same as multiplying each number in the sum individually, then adding up our
products.
First example, look at this thing 5(7+3). Simplify
this, so I can choose 7 adding 3 is equal to 10. Then, 5 times 10 is equal to
50. I will check 5 times 7 is 35 plus 5 times 3 is 15. And then you added, so
we get 50. The result is same. What happen in both parts? Look at this five
times seven and seven with the three, I am distributing. The number is in the
bracket. The answer is same. The answer is same.
Symbol like we can see A(B+C)=AB+AC. A times the B
plus A times the C is equal to AB+AC. So A goes times the B then A goes times
the C.
We have (A+B)C=AC+BC. I can distribute it like
this, C times A is AC and C times B is BC. So it is easy.
2.
The Distributive Law of
Multiplication Over Subtraction
The distributed property occurs in addition and
subtraction. You can symbolic like this, A(B-C)=AB-AC. A times B is equal to AB
and A times C is equal to AC, so you can subtract it.
3.
The general distributive
property
If we have 2(1+3+5+7), so I can distributed two to
the 1, 3, 5, and 7. We will get this 2 times 1 is 2 plus 2 times 3 is 6 plus 2
times 5 is 10 plus and 2 times 7 is 14. And we get the answer is 32.
Suppose we have a(b1+b2+b3+…+bn). I can distributed
‘a’ with b1, b2, b3, and so on until bn.
4.
The negation distributive
property.
If you negate (or find the opposite) of a sum,
just “change the signs” of whatever is inside the parentheses.
For last property, we have –(A+B) is equals to
(-A)+(-B) and then we can simplify this so that -A-B.
Answer to Quiz Questions
1.
The question number one,
you wanna find the additive inverses. Let’s going to that the inverses -5 will
be 5, 2/3 would be -2/3, -1 is 1 and the additive inverse 0 is just say 0.
2.
The questions number two we
want to find multiplicative inverses. Let’s going to that the multiplicative
numbers of -5 is -1/5, 2/3 would be 3/2, -1 it’s actually itself -1, and the
multiplicative numbers of 0 it doesn’t have (none).
3.
The question number 3 ask,
what is additive identity. Yes the answer is of course 0.
4.
The equation number 4, what
is multiplicative identity. It is 1, of course.
5.
The question number 5, do
all numbers have an additive inverse, the answer is yes.
6.
Number 6 ask, do all
numbers have an multiplicative numbers, and the answer is no because 0 does
not.
7.
The question number 7, I
will completely each the equation. I will fill in the box, so the first line
that –u plus u will gonna be 0. For eight times seven, we can use
multiplication property. Its round, so the answer will gonna be seven times
eight. 5(w-y), I can distributed it, and then I get 5w-5y. -3+(6+2), I will
using associative property, the answer will gonna be look like this (-3+6)+2.
8.
The next one, I want to answer
about property of equality. So, Z is equal to Z. If a is not less than b, a is
not equal to be therefore a is greater than b.
9.
Identify first line here
there is m times 1/m equal to one. So this is numbers times inverse property
multiplication.
10.
The next one, since square
root of three and e are real numbers, so this is square root of three plus e. And
the answer is the real number. It is a closure property of addition.
11.
2 plus x square equals x
square plus two, it is the commutative property of addition.
12.
(Z+7)+2=Z+(7+2), This is a
associative of property addition.
13.
Y times 1 is y, this is
identity property of multiplication.
14.
If x=y and y=5 then we
conclude x=5 , it is the property of equality.
15.
The next one, square root
of two plus zero is equal to square root of two. It will the same identity
property of addition.
16.
For the last one,
-(x+2)=-x-2 look the sign, so this is the negation of distribution property.
17.
Square root of three times
(2 plus x) equal to square root of three times (x plus two). this is a
commutative property of addition.
18.
(ab)c=(ba)c. this is a
commutative property of multiplication.
19.
[z+(x-1)]y=2y+(x-1)y. this
is a distributive property.
20.
(1/x^2+4)(x^2+4). It is the
inverse property of multiplication.
21.
(x+y)+z=z+(x+y). it is
commutative property of addition.
22.
(1)(1)=1. This is inverse
property of multiplication.
23.
5+w+(-w)=5. This is inverse
property of addition.
24.
(2a)(bc)=2(ab)c. this is
associative property of multiplication.
25.
|-2/3|=|-2|/|-3|=2/3. This
is a property of absolute value.
26.
(x+1)(y+2)=(x+1)(y)+(x+1)(2).
We can distributed (x+1) to y and also (x+1) to 2. So this is the distributive
law of multiplication.
27.
1(y-2)=y-2. We can
calculate with 1 times y minus 1 times 2. So this is distributive law of
multiplication over subtraction.
28.
x+5=5+x, so this is a
commutative property.
29.
p times q so that q times
p.
30.
2y+8=8+2y. so this is the
same thing as y times 2 plus eight. And then 8+y(2).
31.
2-ab=2-ba.
32.
3+(w+z)=(3+w)+z, so this is
the associative property.
33.
3(wz)=(3w)z.
34.
-2(x+3) ia equal to
-2x+(-2)(3), next equals to -2x-6.
35.
–(2y-9), so we can multiply
the sign (-) to 2y and to nine. So that -2y-(-9), and then we get -2y plus nine.
End of Quiz. And this is the end of the lesson. If
you have the question, you can ask me in Luis-Ast@VideoMathTutor.com.
