English Assignment - Reflection from Videos (part 1)

Reflection for English – Math Videos

Last meeting, Mr. Marsigit gave us some videos and told us to make a reflection from the videos. He wanted us to make one reflection that he picked and three others that we picked ourselves. Here're some videos.

A.      Basic Mathematics Lesson 4: Properties of Numbers (VideoMathTutor.com)

This video is hosted by Luis Anthony Ast. The material is shown bellow.

1.       Introduction

All numbers, variables, and algebraic expressions have properties. So in this video, we will discuss about properties of numbers, variables, and algebraic expressions.

        Special Note: We use variable A, B, and C to represent numbers, variables, and algebraic expressions. But to simplify we just use the word number or value.

2.       Properties of Numbers

a.       The Reflexive Property of Equality

A number is equal to itself.

A=A, 3=3, 5=5

The reflexive property means that a number is equal to that number itself. So if we have A, we know that A is equal to A itself. Or if we have a number, for example, 3 so number 3 is equal to 3 itself. It's simple but you may pay more attention when this form is in algebraic expression. In algebra we're not done yet on a word problem before the expression is equal to itself.

b.      The Symmetric Property of Equality

If one value is equal to another, then the second value is the same as the first.

A=B, B=A. 3=x, x=3

Symbolically we write A is equal to B (A=B) then B is equal to A (B=A). In algebra, especially in a word problem, for final answer you may get result like 5=x or 3=y. It's okay to write like that. But traditionally we write the variable first then the number. So we switch the answer become x=5 or y=3.

c.       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.

A=B and B=C, then A=C

The value A is equal to B, and B is equal to C. Because B is equal to C then A is equal to C. So the value C is moving from the second equation to the first.

d.      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.

We have A is equal to B (A=B) then B can be substitute to any expression that dealing with A. So if we have A=B and we have expression A+2B so we can simplify the expression by substitute the A to the form. So we get A+2B=B+2B=3B.

e.      The Additive Property of Equality

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

A=B                add by C

A+C=B+C

If we have A is equal to B we can add both sides with some value but we're not changing the validity of the equation. So we have A=B then we add both side by C. It becomes A+C=B+C. We're not changing the basic concept that A=B. We can also write the equation C+A=C+B. You can add the number on the left or right side. It does depend on you. Same thing.

f.        The Cancellation Law of Addition

We can subtract the equation from The Additive Property of Equality. So from A+C=B+C we subtract each side by C. The result will be A=B.

A+C=B+C

-C         -C

A=B

g.       The Multiplicative Property of Equality

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

A=B                multiply by C

A.C=B.C

If we have A=B we can multiply both sides by a number without changing its validity. For example, we're going to multiply both sides by C so A=C becomes A.C=B.C. But just like the addition, we can write the form like this: C.A=C.B.

h.      The Cancellation Law of Multiplication

If we have A.C=B.C and we want to cancel C on the form, so we can use the cancellation law. We just need to divide each side by C. From A.C=B.C becomes A=B.

i.         The Zero-Factor Property

If two values that are being multiplied together equal to zero, then the one of the values or both of them must equal zero.

A.B=0

If we have the form A.B=0 then we can assume that A is equal to zero, or B is equal to zero, or both of them are equal to zero. It's like the condition that if any number times zero is equal zero. So if A is zero then A times B equal zero. If B is zero then A times B is also zero. If A and B are zero then A times B is zero too.

3.       Properties of Inequality

a.       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

A=B                                               A<B                        A>B

If we have numbers A and B, then there are three possibilities that can be happen. First, A can be equal to B, or A can be smaller than B, or A can be larger than B.

b.      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.

The concept is same as the equality. If we have A<B and B<C, then A<C.

4.       Properties of Absolute Value

a.       For absolute value there's no negative number. So the value must be zero or positive.

|A|≥ 0

b.      The absolute value of opposite number is the same as the absolute number of the number.

|-A|=|A|

c.       We can multiply the numbers inside the absolute value first or we take the absolute value to each number first then multiply it.

|A.B|=|A|.|B|

d.      We can divide the number inside the absolute value or we take the absolute value first then divide it.

|A/B|=|A|/|B|

But remember that B cannot be zero because we know that any number divide by zero is undefined.

5.       Properties of Numbers

a.       Closure

                                                               i.      The Closure Property of Addition

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

If we have A is a real number and B is a real number then A+B is a real number too.

                                                             ii.      The Closure Property of Multiplication

When you multiply real numbers to other real numbers, the product is a real number. Multiplication is "closed" operation.

If we have A is a real number and B a real number then A.B is equal to a real number too.

Special Note

Addition and multiplication are "closed" operation. It means if you put a real number for first number and a real number for second number, then the result is a real number too. This can be not work when we use it in subtraction. For example, we use the natural number. We have 3 as the first number and 7 for the second number, both are natural numbers. If we subtract them like this: 3-7; the result is -4. Like we know -4 is not part of natural number. Then this operation can't be called as closed operation.

b.      Commutativity

                                                               i.      The Commutativity Property of Addition

It does not matter the order in which numbers are added together. Like when we have A+B, it's the same with B+A.

A+B=B+A

                                                             ii.      The Commutative Property of Multiplication

It doesn’t matter the order in which numbers are multiplied together. Like when we have A times B, it's the same as B times A.

A.B=B.A

c.       Associativity

                                                               i.      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=(A+B)+C                   A+B+C=A+(B+C)

For example, we have A+B+C. We can group the first two-number and add then up first. Then add the result by the last number. Or we can add the last two-number, and then add the result by the first number.

                                                             ii.      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 replaced as we wish.

A.B.C=(A.B).C                        A.B.C=A.(B.C)

For example we have A.B.C, we can group the first term like this: (A.B).C. Se we multiply the first two-number and the result times the last number. Or we can write the term as A.(B.C). So we multiply the last two-number first. The result times the first number.

The commutative and associative aren't for subtraction and division.

d.      Identity

                                                               i.      The Identity Property of Addition

There exist a special number, called the "additive identity" when added to any other number will still "keep its identity" and remain the same.

A+0=A                      0+A=A

So if we have A+0 then it is equal to A itself. It's the same as 0+A, because it is equal to A itself. So we get that zero is the identity of addition.

                                                             ii.      The Identity Property of Multiplication

There exist a special number, called the "multiplicative identity" when multiplied to any other number, then the other number will still "keep its identity" and remain the same.

A.1=A                        1.A=A

A number time one is that number itself, and if we switch the position becomes one times a number so it stills the same number. So we get that one is the identity of multiplication.

Special Note

                Zero is the identity of addition. One is the identity of multiplication.

e.      Inverse

                                                               i.      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).

A+(-A)=0                 (-A)+A=0

If we have a number plus the opposite of the number, it's equal to zero. It's the same as if we add the opposite of the number by the number, it's zero.

                                                             ii.      The Inverse Property of Multiplication

For every real 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).

A.(1/A)=1                                (1/A).A=1

A number times its multiplicative inverse the result is one. And if we multiply the multiplicative inverse by the number, it's equal to one.

Zero is the number that doesn't have multiplicative inverse. If we put zero as A, it becomes 1/0. We know that any number divided by zero is undefined. So 1/0 is undefined. Zero has no multiplicative inverse.

f.        Distributivity

                                                               i.      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.

A.(B+C)=A.B+A.C

So before we discuss the form, we just get to the example first. If we have form 5.(2+8) we can solve it with two ways. The first we add up the number inside the bracket, and then multiply it by the other number. So from the form, we add up 2 and 8 first, its equal 10. Then we have 10 times 5, equals 50.

The second way is we multiply the numbers first like this: 5(2)+5(8). From 5(2) we get 10, from 5(8) we get 40. After that we add up the numbers, so it becomes 10+40 equals 50. The result is the same as the first way.

So we can say that if we have numbers A, B, and C, then we have form A.(B+C)=A.B+A.C then it's like we spread the number A into the bracket. Just like the proof we know that the result at right side and left side is same.

It's the same thing when we have form like this: (B+C).A=B.A+C.A.

                                                             ii.      The Distribution Law of Multiplication over Subtraction

The concept is same as the distribution law of multiplication over addition. So if we have A(B-C)=A.B-A.C. Then it's the same with (B-C).A=B.A-C.A.

                                                            iii.      The General Distributive Property

If we have a number A and a term, let it called B₁, B₂, until B_n. Because we don't know how many the term are, we just say it 'n'. We have the form like this: A(B₁+B₂+…+B_n) so it's become A.B₁+A.B₂+…+A.B_n.

                                                           iv.      The Negation Distributive Property

If you negate (or find the opposite) of a sum, just "change the signs" or whatever is inside the parentheses.

-(A+B)=(-A)+(-B)=-A-B

So if we have number A and B added up inside the bracket, and we negate it, so all we need is only change the sign of the number inside the bracket. If –(A+B) so it becomes (-A)+(-B). Then we can simplify as –A-B.

All we need to do is only change the sign. If it is positive, so we change it into negative. If it is negative, so we change it into positive.