" REFLECTION 3 "
by : Elsa Winda Prastiana
( 09313244015 / International Mathematics Education )
At
April 9th 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
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
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)
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
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
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
