5. กลุ่มที่ 5 วชาญ & อัญชุ ลี
ิ
Let a, b, and c be real numbers, variables, or algebraic
Properties of Real Numbers
expressions.
Property Example
1. Closed Property of Addition
2 + 3∈ ℝ
a+b∈ ℝ
2. Closed Property of Multiplication
2•3∈ ℝ
a•b∈ ℝ
3. Commutative Property of Addition
2+3=3+2
a+b=b+a
4. Commutative Property of Multiplication
2•(3)=3•(2)
5. Associative Property of Addition
a•b=b•a
2+(3+4)=(2+3)+4
6. Associative Property of Multiplication
a+(b+c )=(a+b)+c
2•(3•4)=(2•3)•4
7. Additive Identity Property
a•(b•c)=(a•b)•c
3+0=3
8. Multiplicative Identity Property
a+0=a
3•1=3
9. Additive Inverse Property
a•1=a
3 + (-3) = 0
a + ( -a ) = 0
10. Multiplicative Inverse Property
Note: a cannot = 0
11. Distributive Property
2•(3+4)=2•3+2•4
12. Zero Property
a•(b+c)=a•b+a•c
5•0=0
a•0=0
6. กลุ่มที่ 6 สรนันท์ & เจนจิรา
Properties of Set
1. Identity Laws
A ∪φ = A
A ∩U = A
2. Domination Laws
A ∪U = U
A ∩φ = φ
3. Idempotent Laws
A∪ A = A
A∩ A = A
4. Commutative Laws
A∪ B = B ∪ A
A∩ B = B ∩ A
5. Associative Laws
A ∪ ( B ∪ C ) = ( A ∪ B) ∪ C
A ∩ ( B ∩ C ) = ( A ∩ B) ∩ C
6. Distributive Laws
A ∪ ( B ∩ C ) = ( A ∪ B) ∩ ( A ∪ C )
A ∩ ( B ∪ C ) = ( A ∩ B) ∪ ( A ∩ C )
7. De Morgan's laws
( A ∪ B )' = A'∩ B'
8. If A ⊆ B and C ⊆ D ,then A ∪ C ⊆ B ∪ D ,and A ∩ C ⊆ B ∩ D
( A ∩ B)' = A'∪ B'
9. If A ⊆ B ,then A ∪ B = B and A ∩ B = A
10. A ∪ ( B − A) = A ∪ B
11. A ∩ ( B − A) = φ
12. A − ( B ∪ C ) = ( A − B) ∩ ( A − C )
A − ( B ∩ C ) = ( A − B) ∪ ( A − C )
13. B = A' if and only if A ∪ B = U and A ∩ B = φ
14. ( A')' = A
7. กลุ่มที่ 7 นัฐพล & นฤมล
Absolute Value
only how far a number is from zero:
Absolute Value means
"6" is 6 away from zero,
and "-6" is also 6 away from zero.
So the absolute value of 6 is 6,
and the absolute value of -6 is also 6
More Examples:
• The absolute value of -9 is 9
• The absolute value of 3 is 3
• The absolute value of 0 is 0
• The absolute value of -156 is 156
So in practice "absolute value" means to remove any negative sign in front
No Negatives!
of a number, and to think of all numbers as positive (or zero).
To show that you want the absolute value of something, you put "|" marks
Absolute Value Symbol
either side (they are called "bars" and are found on the right side of your
keyboard), like these examples:
|-5| = 5 |7| = 7
Sometimes absolute value is also written as "abs()", so abs(-1) = 1 is the
same as |-1| = 1
8. กลุ่มที่ 8 วศิษฏ์ & วชุลดา
ิ ิ
Inequality tells you about the relative size of two values.
Introduction to Inequalities
Mathematics is not always about "equals"! Sometimes you only know that
something is bigger or smaller
> greater than x+3>2
Symbol Words Example
< less than 7x < 28
≥ greater than or equal to 5≥x-1
≤ less than or equal to 2y + 1 ≤ 7
Our aim is to have x (or whatever the variable is) on its own on the left of
Solving
the inequality sign:
Something like: x<5
or: y ≥ 11
We call that "solved".
Solving inequalities is very like solving equations ... you do most of the
How to Solve
same things ...
... but you must also pay attention to the direction of the inequality.
Direction: Which way the arrow "points"
These are things you can do without affecting the direction of the
Safe Things To Do
inequality:
• Add (or subtract) a number from both sides
• Multiply (or divide) both sides by a positive number
9. กลุ่มที่ 9 สาธิต & ศิริลกษณ์
ั
Add (or subtract) a number from both sides
x+3-3<7-3
Solve: x If we subtract 3 from both sides, we get:
x<4
And that is our solution: x < 4
In other words, x can be any value less than 4.
Multiply (or divide) both sides by a positive number
If we divide both sides by 3 we get:
Solve: 3y < 15
3y/3 < 15/3
y<5
And that is our solution: y < 5
10. กลุ่มที่ 10 เดชา & ปรียาภรณ์
Geometry
Geometry is all about shapes and their properties. If you like playing
with objects, or like drawing, then geometry is for you!
Geometry can be divided into:
11. กลุ่มที่ 11 กิตติพงษ์ & สุภัทรา
Triangles
There are three special names given to triangles that tell how many sides (or angles) are
Equilateral, Isosceles and Scalene
equal.
There can be 3, 2 or no equal sides/angles:
Triangles can also have names that tell you what type of angle is inside:
What Type of Angle?
Sometimes a triangle will have two names, for example:
Combining the Names
15. กลุ่มที่ 15 วสรรค์ & อนุภา
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Finding a Central Value
When you have two or more numbers it is nice to find a value for the "center".
2 Numbers
With just 2 numbers the answer is easy: go half-way in-between.
3 or More Numbers
You can use the same idea when you have 3 or more numbers:
16. กลุ่มที่ 16 สรศักด์ิ & อารัรัตน์
The Mean
So far we have been calculating the Mean (or the Average):
But sometimes the Mean can let you down:
The Mean was accurate, but in this case it was not useful.
17. กลุ่มที่ 17 อดิศักดิ์ & จิรนันท์
The Median
But you could also use the Median : simply list all numbers in order and
choose the middle one:
18. กลุ่มที่ 18 ชัยพฤกษ์ & ผ่องนภา
The Mode
The Mode is the value that occurs most often:
19. กลุ่มที่ 19 ปัณณวชญ์ & ฐิติมา
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The Range (Statistics)
The Range is the difference between the lowest and highest values.
The Range Can Be Misleading
The range can sometimes be misleading when there are extremely
high or low values.
20. กลุ่มที่ 20 กนกวรรณ & นัทมล
Standard Deviation and Variance
Deviation just means how far from the normal
The Standard Deviation is a measure of how spread out numbers are.
Standard Deviation
Its symbol is σ (the greek letter sigma)
The formula is easy: it is the square root of the Variance.
The Variance is defined as:
Variance
Here are the two formulas, explained at Standard Deviation Formulas
Formulas
if you want to know more:
