Reflection 3

" REFLECTION 3 " 

by : Elsa Winda Prastiana 

( 09313244015 /  International Mathematics Education )

At April 9^th 2012, Mr.Marsigit showed video about property of numbers. That video explain about all property of numbers.

1.      The reflextive property of Equality.

A number is equal to it self

Symbolicly : A = A

2.      The symmetric property of equality.

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

Symbolicly : if A = B then B = A

                                    3 = X then X = 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.

Symbolicly : if A = B and B = C then A = C

4.      The substitution property of equality.

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.

Simbolicly : If A = B then B can be substituted A for any expression

                  If B = A then A can be substituted B for any expression

5.      The additive property of equality.

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

Symbolicly : if A = B then A + C = B + C

                                                        C + A = C + B

6.      The cancelation law of addition.

Symbolicly : C + A = C + B
If we cancel C in the both side B, the final equation A = B

7.      The multiplicative property of equality.

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

Symbolicly : A = B (times C in both side)

                               AC = BC

                                CA = CB

8.      The cancelation law of multiplicative.

Symbolicly : AC = BC (both side divide by C)

                                 A = 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.

Symbolicly : if AB = 0 , there are 3 possibility :
                                     A = 0

                                                             B = 0

                                                            A and B = 0

10.  The law of trichotomy.

For any two values, only one of the following can be true about these values :

Ø  They are equal

Ø  The first has a smaller value than the second

Ø  The first has a larger value than the second

Given any numbers A and B :

                        A = B

                        A < B

                        A > B

11.  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.

Simbolicly : if A < B and B < C we can conclude A < C

12.  Properties of absolute value.

Symbolicly :  
1. |A| ≥ 0
2. |-A| = |A|
3. |AB| = |A| |B|
4. |A/B| = |A|/|B|  ; (B ≠ 0)
 

                                                            

Properties of numbers :

A.    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

If A and B is real number, then A+B = 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

If A and B is real number, then A B = a real number

B.     Commutativity.

1.      The commutative property of addition.

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

Symbolicly : A + B = B + A

2.      The commutative property of multiplication.

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

Symbolicly : A B = B A

C.     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.

Symbolicly : (A+B)+C = 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 then parentheses can be placed as we wish.

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

By the way, commutativity & associativity can’t apply substraction & division.

D.    Identity.

1.      The identity property of addition.

There exists a special number called the “additive identity” when added to any other number. Then that order number will still “keep its identity” and remain the same.

Symbolicly : A+0 = A

                    0+A = 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.

Symbolicly : A . 1 = A

                    1 . A = A

So, 0 is the unique additive identity.

       1 is the unique multiplicative identity .

E.     Inverse.

1.      The inverse property of addition.

For every rela number, ther exists another real number that is called its opposite. Such that, when added together, you get the additive identity (the number zero).

Symbolicly : A + (-A) = 0

                   (-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)

Symbolicly : A . 1/A = 1
                  1/A . A = 1

Zero has not multiplicative identity.

F.      Distributive.

1.      The distributive law of multiplication over addition.

Multiplying a number by a sum of numbers is the same as multipliying each number in the sum individually, then adding up our products.

Example : 5(7+3) = 5(10)

                              = 50 …….(1)

                  5(7)+5(3) = 35+15

                                   = 50 ……(2)

                  1&2 is same

The formula : A(B+C) = AB+AC

                        (A+B)C = AC+BC

2.      The distributive law of multiplication over subtraction.

The formula : A(B - C) = AB - AC

3.      The general distributive property.

Example : 2(1+3+5+7) = 2.1+2.3+2.5+2.7

                                      = 2+6+10+14

                                      = 32

The formula :a(b_1+b_2+b_3+b_4+…+b_n)
             ab_1+ab_2+ab_3+ab_4+…+ab_n

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.

The formula : -(A+B) = (-A)+(-B)

                                    = -A-B