Basic Mathematics Lesson : 4 Properties of Number

Basic Mathematics Lesson : 4

Properties of Number

With your host : Luis Antony Ast – The video Math Tutor

Hello, I’m Luis Antony Ast –The Video Math Tutor. Welcome to Basic Math Lesson Number 4 – Properties of number. Get start.

Introduction

All real numbers, variables and algebraic expression follow certain properties. They are little complicated. What I hope to do in this lesson is presents you in every kind way and provide you some certain example where necessary.

A special note.

When explain properties, I would use the variable ABC to present my number or my variable or my algebraic expression.

Properties of number.

1.      The Reflexive Property of Equality. A number is equal to itself.

Symbolically, this about thing. A equal to A itself. 2 is the same as 2 and 3 is same with 3.

This is very simple of this rule but it also important. In algebra, we have to check you word problem, we have A equals A, 2 equals 2.

2.      The Symmetric Property of Equality. If one value is equal to another, then that second value is the same as the first.

Symbolically, if 

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=B and B=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.  then  can be sunstituted for A in any expression.

5.      The Additive Property of Equality. We can add equal values to both sides of an equation without changing the validity of equation. To see this rule, we have 

we can add the same in both side of equation A+C=B+C

 and also C+A=C+B  . We can add the C infrontof or behind is not problem, it will same.

6.      The Concelation Law of Addition

If we have  , add both sides we subtract with  it will be cancel the equation. So,

                            

                             

7.      The Multiplicative Property of Equality. We can multiply equal values to both sides of an equation without changing the validity of the equation.

 , we can multiplied each sides with  . So,  it’s also same

8.      The Cancelation Law of Multiplication.

   we want to cancel the C, so we didvide the each sides with C.

9.      The Zero Factor Property. If two values taht being multuplied together equal zero, the one of the values, or both of them must be zero.

If   . So, A must be 0 or B must be 0 or both of them are 0. All numbers if times zero the result is 0.

Properties of Inequality

1.      The Law of Trichotomy. For any two values, only one of the following can be true about this values:

·         They are equal

·         The first has smaller value than the second

·         The first has a langer value than the second

Given any numbers A and B:

2.      The Transitive Property of Inequality. If one value is smaller than a second, and is less than a third, then we can conclude taht the first value is smaller than the third.

If

Properties of absolute value

·        

·        

·        

·        

·        

Property 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 real number + a real number = a real number.  

2.      The Closure Property of Multiplication. When you multiply real numbers to other real numbers, the product is real numbers.

Multiplication is a real “closed” operation. A real number • a real number = a real number. .

A Special Note.

Natural number is all number that positive,  . So, if we have . Negative 2 is not natural number.

Commutativity.

1.      The Commutative Property of Addition. It doesn’t matter the order in which number are added together.

 is the same thing as

Assosiativity

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 palced as we wish.

We can associate

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.

Identity

1.      The Identity Property of Addition. There exixt 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.

              , keep its identity.

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.

Of course no big surprise its also work multiplication. A • 1 = A. A kept its identity.

Like before if it’s switch around we still get the same result.

A special note.

Zero is the unique additive identity and 1 is the unique multiplicative adentity.

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 additive identity (the number of zero).

Symbolically, A + (-A) = 0, the (-A) we called the inverse because of A, so A + (-A) = 0. And if we switch around we get the same result (-A) + A = 0.

2.      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).

Symbolically we can say the A number times its multiplicated invers, which is 1/A the result is the multiplicated identity that is 1. And also if we switch around 1/A • A the result is same, 1.

By the way, there is one number that doesn’t have multiplicated numbers, can you guess? It’s 0. But why? Because if you divide anything number with zero, the result is undefined. So zero has no multiplicative numbers.

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 product.

So this is the first example, 5 ( 7 + 3 ) = ? If you want to simplify and solve we can write

5(7+3) = 5(10)

           = 50

Look at another situation

5 ( 7 ) + 5 ( 3 ) = 35 + 15

                        = 50

 So the result is same. But why it can be same? Because we use the distributive law of multiplication over addition. Symbolically we can write like this A ( B + C ) = AB + AC. A goes times the B and A goes times the C, so we get AB+AC. It called the left distributive properties. It’s same with ( A + B ) C = AC + AB.

2.      The Distributive Law of Multiplication Over Substraction. The distributive properties also work in substraction. A(B-C)=AB-AC

The General Distributive Property

Example : 2 ( 1 + 3 + 5 + 7 ) . From that example we can distribute 2 to the 1, 2 to the 3 and so on. So we get 2 ( 1 + 3 + 5 + 7 ) = 2 + 6 + 10 + 14

                                                                  = 32

Symboliccally, we get something like this

a(b1+b2+b3+...+bn)=ab1+ab2+ab3+...+abn (a is going distribute among all b’s terms).

3. The Negation Distributive Property. If you negate (or find the opposite) of sum, just “change the sign” of whatever is inside the parentheses.

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

=-A-B

Answer to the quiz questions

1.      Find the opposite

·         -5=5

·         2/3=-(2/3)

·         -1=1

·         0=0

2.      Find the multiplicated numbers

·         -5=-(1/5)

·         2/3=3/2

·         -1=1

·         0=none

3.      What is the additive identity? The answer of course zero (0).

4.      What is the multiplicative identity? The anwers 1

5.      Do all numbers have additive invers? The answer is yes, they all do.

6.      Do all numbers have invers multiplicative invers? No, zero is not

7.      Complete this equation

·         -4=4

·         8x7=7x8 (multiplication)

·         5(w-y)=5w-5y (distributive)

·         -3+(6+2)=(-3+6)+2 (addition)

·        

            a<b

8.      We have  (invers property of multiplication)

Since  and C are real numbers, so is  so that is the closer property of addition.

·          (addition)

·          (associative property of addition)

·         (identity properties in multiplication

·         If and , then  (transitive property of equality)

·          this kept the identity so it called identity property of addition

·          In this example, i’m negating something and distributing negative sign (negation distributive property)

9.      This little bit sneaky and difficult, so if you follow the property rule, its gonna be fine.

·          (assosiative propery of addition)

·          (assosiative property of multiplication)

·          (distributive law)

·          (the invers property of multiplication)

·          (commutative property of addition)

·          (identity property multiplication, the invers property of multiplication)

·          (this the invers propertyof addition)

·          (asssociate property of multiplication)

·           (properties of absolute value)

·          (distributive law of multiplication)

·          (the identity property of multiplication)

10.   

             (just switch it)

11. 

End of quiz.

Rachma Hanan Tiasto

11313244020

International Mathemathic Education ‘11