กลุ่มที่ 5 วชาญ & อัญชุ ลี ิ Let a, b, and c be real numbers, variables, or algebraic Properties of Real Numbers expressions. Property Example1. Closed Property of Addition 2 + 3∈ ℝ a+b∈ ℝ2. Closed Property of Multiplication 2•3∈ ℝ a•b∈ ℝ3. Commutative Property of Addition 2+3=3+2 a+b=b+a4. Commutative Property of Multiplication 2•(3)=3•(2)5. Associative Property of Addition a•b=b•a 2+(3+4)=(2+3)+46. Associative Property of Multiplication a+(b+c )=(a+b)+c 2•(3•4)=(2•3)•47. Additive Identity Property a•(b•c)=(a•b)•c 3+0=38. Multiplicative Identity Property a+0=a 3•1=39. Additive Inverse Property a•1=a 3 + (-3) = 0 a + ( -a ) = 010. Multiplicative Inverse Property Note: a cannot = 011. Distributive Property 2•(3+4)=2•3+2•412. Zero Property a•(b+c)=a•b+a•c 5•0=0 a•0=0
กลุ่มที่ 6 สรนันท์ & เจนจิรา Properties of Set1. Identity Laws A ∪φ = A A ∩U = A2. Domination Laws A ∪U = U A ∩φ = φ3. Idempotent Laws A∪ A = A A∩ A = A4. Commutative Laws A∪ B = B ∪ A A∩ B = B ∩ A5. Associative Laws A ∪ ( B ∪ C ) = ( A ∪ B) ∪ C A ∩ ( B ∩ C ) = ( A ∩ B) ∩ C6. Distributive Laws A ∪ ( B ∩ C ) = ( A ∪ B) ∩ ( A ∪ C ) A ∩ ( B ∪ C ) = ( A ∩ B) ∪ ( A ∩ C )7. De Morgans laws ( A ∪ B ) = A∩ B8. If A ⊆ B and C ⊆ D ,then A ∪ C ⊆ B ∪ D ,and A ∩ C ⊆ B ∩ D ( A ∩ B) = A∪ B9. If A ⊆ B ,then A ∪ B = B and A ∩ B = A10. A ∪ ( B − A) = A ∪ B11. A ∩ ( B − A) = φ12. 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 = φ14. ( A) = A
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 6More 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 156So in practice "absolute value" means to remove any negative sign in frontNo 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 "|" marksAbsolute Value Symboleither side (they are called "bars" and are found on the right side of yourkeyboard), like these examples: |-5| = 5 |7| = 7Sometimes absolute value is also written as "abs()", so abs(-1) = 1 is thesame as |-1| = 1
กลุ่มที่ 8 วศิษฏ์ & วชุลดา ิ ิ Inequality tells you about the relative size of two values. Introduction to InequalitiesMathematics is not always about "equals"! Sometimes you only know thatsomething 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 ≤ 7Our aim is to have x (or whatever the variable is) on its own on the left ofSolvingthe inequality sign: Something like: x<5 or: y ≥ 11We call that "solved".Solving inequalities is very like solving equations ... you do most of theHow to Solvesame 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 theSafe Things To Doinequality: • Add (or subtract) a number from both sides • Multiply (or divide) both sides by a positive number
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
กลุ่มที่ 10 เดชา & ปรียาภรณ์ Geometry Geometry is all about shapes and their properties. If you like playingwith objects, or like drawing, then geometry is for you! Geometry can be divided into:
กลุ่มที่ 11 กิตติพงษ์ & สุภัทรา TrianglesThere are three special names given to triangles that tell how many sides (or angles) areEquilateral, Isosceles and Scaleneequal.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
กลุ่มที่ 15 วสรรค์ & อนุภา ิ Finding a Central Value When you have two or more numbers it is nice to find a value for the "center".2 NumbersWith just 2 numbers the answer is easy: go half-way in-between.3 or More NumbersYou can use the same idea when you have 3 or more numbers:
กลุ่มที่ 16 สรศักด์ิ & อารัรัตน์ The MeanSo 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.
กลุ่มที่ 17 อดิศักดิ์ & จิรนันท์ The MedianBut you could also use the Median : simply list all numbers in order andchoose the middle one:
กลุ่มที่ 18 ชัยพฤกษ์ & ผ่องนภา The ModeThe Mode is the value that occurs most often:
กลุ่มที่ 19 ปัณณวชญ์ & ฐิติมา ิ 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 extremelyhigh or low values.
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 FormulasFormulasif you want to know more: