#3 English Task (Reflection For Video English Math Tutor)

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.