Assigment 3 for English ( Reflection from Video part 2)

Reflection from English Math Tutor

Hello my name is Luis Anthony as the video  math tutor.

Welcome to basic math lesson number 4 properties of number introduction, variable, expression.  What I have to do in this lesson is presenting size, variable necessary.

A special Note

I explain property. I would use variable ABC, to present my number and variable expression.

A.  Properties of Numbers

1.    The Reflexive Property of Equality

A number is equal to itself. So, symbolically is about A is equal to A. It's self and how ways? It  is like thing 2 sames 2, 3 sames 3, So this is simple. So its good know this .

2.    The Symmetric Property of Equality

If one value is equal to another, then the second value is the same as the first. Symbolically, I just say if A equal to B, then this B equals to A, ,  because it is symmetric, so the second value will be same with the first value.

The word problem its look like this, 3 equals x, because 3 equals to x same with x equals to 3, for finally answer x is equal to 3.

3.    The Transitive Property of Equality

If one value is equal to 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, A equals to B and B equals C. We can change, A to the one, A is equal to C.

4.    The Substitusion Property

If one value is equal to another, then the second value can be used in place of the first in any algebraic expresion dealing with the first value.

Example:

See A is equal to B, then can be substitute, for A it's express dealing with that. If A equals to B, then A can be substitute B, any expression. Both of them, substituting one to another. More detail

5.    The Additive Property of Equality

We can add equal values to both sides of an equation without changing the validity of the equation.

Example :

To see this word, I will use  A equals B, we can add the same thing equation. I can actually add. Let's say A plus C equals B plus C. And this expression same thing A equals B. This is also change C plus A is equal to C plus B, you can so find  side to other side, same thing.When we add the left side with constanta C, so the right side must be add C. We can change the exhauted of the constanta ( number ). A + C = B + C same with C + A = C + B.

6.    The Cancellation Law of Addition

Most important properties of number, equation, algebraic equation, its one something that we want to do.  Look like this A plus C is equal to B plus C we substract by minus C, when we substraction (–C) on the left side, we must substraction the right side to be actually A is equal to B.

7.    The Multiplicative Property of Equality

We can multiply equal values to both sides of equation wwithout changing the validity of the equation.

If A is equal to B, I can move it to both side, so A times C is equal to B times C. Simply multiplication  next each other variable, so C times A is equal to C times B, say that, to see this law I say A times C is equal B times C.

8.    The Cancellation Law of Multiplication

A . C =  B . C . We want cancel C, what can I do? Assuming we can divided both side by C (A . C / C = B . C / C), now we get A is equal to 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 AB equal to zero, what can we do in this section? hmmmm. A is equal to zero or B is equal to zero or both of them, must be zero, to make this situation true, Then when you multiply two, two variable together, A times B the answer must be zero. When, any numbers multiply with zero, the result must bee zero.

B.  Properties of Inequality

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 valuee than the second.

Example :

Any number A and B or everything can happen A can be equals to B, or A is less then B or A is greater then B.

2.    The Transitive Property of Inequality

If one value is smaller than a second, and the second is the less than a third, then we can conclude the first valueis smaller than the third.There if A less then B and B less then C, we can transit A, here A less then C.

C.  Properties of Absolute Value

1.    All absolute value are'nt negative ( zero or positive)  |A| >0

2.    |-A| = |A|, absolute value on the number of negative is equal to absolute value on the number of possitive.The absolute value of opposite is same as the absolute value of the number

3.    |AB| = |A||B|. When we multiply A and B with absolute values, we can write A and B on the one absolute values or we can take A and B on individual abssolute valuess.. it is same.We can multiply the number inside the absolute value or you can take the absolute value individual then multiply together

4.    |A/B| = |A|/|B|, B is not equal to 0.

When we A divided B, we can write A divided by B on one absolute valuees or wwe can take the absolute valuess on individual absolute valuees.    B≠0 you can divide number A, inside the absolute value or you can take the absolute value individual first and then divided it

D.  Properties of Numbers

1.    Closure

a.    The Closure Property of Addition

When you add real numbers to other real numbers, the sum is also real. Addition is a “closed” operation.

Example :

A + B  equal to a real number, and then A is real number B is real, the answer will be real number. So A addded by B, the result is a real number.

b.    The Closure Property of Multiplication

When you multiply the real number  to ather real numbers, the product is a real number. Multiplication is a “closed” operation.

Symbolically the multiply A times B is equal real number so A is real B is real, the result is real number.

A Special Note

I like to say that the real number are closed with the addition and multiplication, A equal to real, B equal to real the result is real number. Now, for example  substraction with natural number. Let's say we are given 3 is natural number, right? and five is also natural number. three minus five is equal to minus two, minus two is not part of set of component natural number so its not closed. To be closed natural, natural, and natural number.  So, the natural number is not close with substraction.

E.   Commutativity

1.    The Commutative Property of Addition

It does not matter the order in which numbers are added together.

Example :

A plus B is the same  thing as B plus A.

2.    The Commutative Property of Multiplication

It does not matter the order in which numbers are multiplied together.

All this property, we have A times B is exactly same as 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 purpose. The parantheses can be placed as we wish.

Example :

( A + B ) + C same with A + ( B + C ). All the associative property addition I can associate the group together I can associate different ways, we can move the paratheses on the any place. I can put (A+(B+C), so the result is A + ( B + C ).

2.    The Associative property o Multiplication

When we wish to multiply three ( or more ) numbers, it does not matter how wwe group them together for multiplication porpose. The paratheses can be placed as we wish.

Example :

( A . B ). C same with A. ( B . C ), because the assosiative disposition.

It's same thing with multiply, the first two term, and the result time the third or I can associate indifferent way, the third times second term and the result times by the first. In any case it same at all, by the way, associative property doesn’t cover substraction and division.

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.

Example : A + 0 = A

Symbolically I would say number plus zero,  this is the number A. First zero plus A ia equal to A

2.    The Identity Property of Multiplication

There exixts 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.

Example:

A.1=A

A number times one is  this number. It's identity and like before this is round we still get. The same result,. A Special Note

0 is unique identity addition, only one word .

1 is the unique identity the multiplication.

H.  Inverse

1.    The inverse property of addition

For every real number, there axis another real number that is called its opposite, such that when added together. You get the additive identity(The number zero).

Example:

A+(-A)=0

(-A)+A=0

Symbolically A is the number and we add the inverse is equal to zero,  and when we turning the result is same, the opposite number  plus the number and the answer is zero.

2.    The inverse Property of Multiplication

For every number, except zero. There is another real number that is called its multiplicative inverse, or reciprocal, such that, when multiplied together, you get the multiplicative identity ( the number one).

Example:

A.1/A=1

1/A.A=1

Symbolically we can say, the number is times multiplicative identity  one position round, the multiplicative inverse, times the number is one. By the way finally there is one number doesn’t  have multiplicative number is zero, why?  Because I divided by zero, this is undifined, so zero has no multiplicative number So, zero is not inverse property of multiplication.

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.

Example :

a.    First example, 5 (7+3). look at this problem, First, we must addition the number on the brackets. So I can choose 7 adding 3 is equal to 10. Then, we get 5 times 10 is equal to 50.

b.    5(7) + 5(3)

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, distributive. The number is in the bracket. The answer is same. The answer is same with the first example. 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 to B then A times the C. We have (A+B)C=AC+BC. I can distribute 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 next topic is 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.

A( B-C) = AB-AC

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. So, we addition the answer from the multiplication 2+6+10+14=32. Suppose we have a(b1+b2+b3+…+bn). I can distributed ‘a’ with b1, b2, b3, and so on until bn.

From the case we get the formula :

a(b1+b2+b3+…………..+bn)=ab1+ab2+ab3+…………+abn.

4.        The negative distributive property

If you negate ( or find the opposite ) of a sum, just “ change the signs” of whatever is inside the parantheses. For last property, we have –(A+B)=(-A)+(-B)=-A-B.

Quiz

Answer  to quiz question!

The question number one, you wanna find added inverses. Let’s going to that the inverses -5 will be 5, 2/3 would be -2/3, -1 is 1 and -0 is just say 0.

The question number two, we want to find multiplied inverses. Let’s going to that the multiplied number of -5 is -1/5, 2/3 would be 3/2, -1 is actually itself -1, and the multiplied number 0f 0 it doesn’t have (none).

The question number 3, what is additive identity. Yes it’s of course 0. The equation number 4, what is multiplied identity. It is 1 of course. The question number 5, do all number has the inverse, the answer is yes. Number 6 ask, do multiplied numbers, and the answer is no because 0 does not.

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, then I get 5w-5y. -3+(6+2), I will using associative property, the answer will gonna be look like this (-3+6)+2.

The next one, I want to answer this problem.

Z is equal to Z. a is not less than b, a is not equal to be therefore a is greater than b. Identify first line here there is m times 1/m equal to one. So this is inverse property.The next one, since square root of three and b are real numbers, so this is square root of three plus e. and the answer is the real number. Two plus x to the power of two, it is the commutative property. this is a associative of property addition. Y times 1 is identity property multiplication.

If x=y and y=5 then x=5 is the property of equality.

The next one square root of two plus zero is equal to square root of two. it will the same identity property of addition. For the last one, -(x+2)=-x-2 look the sign, so this is the negation of distribution property.

We want to multiplicate the inverse. Lets going to that. There is two, there is x-1. I spread y out. Its distribute property.The last lesson was going on, we have 1/x2+4, (x2+4) is inverse of 1/x2+4 so the result is 1. It’s the inverse property. This is the next question (x+y)+z = z+(x+y) . the bracket in here and the bracket ib here, its commutative property addition. Well, I first group, so x and y still together (1)(1)=1, for this one I get 2 different answer, because one is identity property of multiplication, one is always same with its inverse. This is inverse property of multiplication. The next one, I have 5 + w + (-w)=5 , so what happen here? Is the inverse property of addition, and trhe last one here, (2a)(bc)=2(ab)c, we cand round so here I associate 2a and the associate bc, and then I associate ab, so is the associate property of multiplication. The next question, |-2/3|= |-2|/|3|=2/3. We know, tis is trhe absolute value, and this is property of absolute value. Its clear!

If we have (x+1)(y+2)=(x+1)y+(x+1)2. The distributing x+1 to y and also to the 2, so its distribution law of multiplication and addition. The last one here I have 1 times something is equal to something. This is identity law of multiplication. This 1 times y and 1 times 2 . distributed  law of distribution, and then substract it.

Question number 10.  If we have x+5 = x+5 . It has relation with commutative property, so the result is 5+x. The next p times q so its q times p, Its actually several way to do it. 2y plus 8 so I round it and I get 8+2y. well you can change position 2 and y, so I get y(2)+x and then I rounded it and equal to 8+y(2).  So the answer is the real number. 2x square = x square + 2, so it’s the commutative property. This is a associative property of addition. Y times 1 , identity property of multiplication. If x=y and y=5, then x=5 it’s the property of equality. The next one the root of 2 + 0 = the root of 2. It will the same. Identity property of addition. For the last one , -(x+2)=-x-2. So this is the negation of distribution property. The root of 3 (2+x) = the root of 3 (x+2), (ab)c=(ba)c, [2+(x-1)]y=2y+(x-1)y, (1/xsquare + 4)(xsquare+4)=1,a.b then c,for the last one 2-ab, be careful I can not round it because distributed property is not occur tin substraction. I round the multiplication, so I get 2-ba.

Question number 11, 3+(w+z) I use that associative property. So the first one w+z is already associate and then I round it. So the result is equal to (3+w)+z . I have this one. The second , group 3 to w and z. For the last question, I use the distributed law to simplify the question, -2 is times with x and with 3, and its equal to -2 (x+3) – 2 x – 6, -2(y-9) so I get -2y – (-2)(-9)  and I will simplify it and I get  -2y + 9, End of quiz. This is end of the lesson.