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Basic Curriculum
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Basic Maths
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Maths for Computer Graphics
Trigonometry (lines, triangles, circles)
3D Maths
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http://nccastaff.bournemouth.ac.uk/jmacey/MathsForCG/index.html
Vectors, Matrices, Affine Transforms
How Animation Systems work
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The Computer Image, The Animation Pipeline (how XSI, Maya, Houdini, Shake etc really work)
Some Basic Computer Principles, How the computer Actually works
Rendering
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How a renderer actually produces and image, Lighting Models, Shading Models.
And a whole lot more besides.
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Tuesday, 13 October 2009
Numbers
Using Python as a Calculator
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To demonstrate some of the principles of Maths for CG we will be
using the python console as a calculator
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This will allow you to get a feel for how computers process
numbers and is advantageous when beginning to create scripts.
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A number is an abstract entity used originally to describe quantity.
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i.e. 80 Students etc
The most familiar numbers are the natural numbers {0, 1, 2, ...} or {1, 2, 3, ...},
used for counting, and denoted by N or N
This set is infinite but countable by definition.
To be unambiguous about whether zero is included or not, sometimes an
index "0" is added in the former case, and a superscript "*" is added in the
latter case:
No = {0, 1, 2, ...}N ∗ = {1, 2, ....}
Tuesday, 13 October 2009
Tuesday, 13 October 2009
Integers
Integers in Computers
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The integers consist of the positive natural numbers (1, 2, 3, -), their
negatives (-1, -2, -3, ...) and the number zero.
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The set of all integers is usually denoted in mathematics by Z or , Z
which stands for Zahlen (German for "numbers").
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They are also known as the whole numbers, although that term is
also used to refer only to the positive integers (with or without
zero).
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Most programming languages have an integer data type
They are usually used for counting and index values
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An integer is usually in the range -32,768 to 32,767
.
//C++ / C
int a=1;
// Maya Mel
int $a=1
// Python
a=1
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Rational Numbers
Number Lines
!"#$#%
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'+
'*
')
'(
&
(
)
*
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+
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The number line is a diagram that helps visualise numbers and their
relationships to each other.
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The numbers corresponding to the points on the line are called the
co-ordinates of the points.
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The distance between to consecutive integers is called a unit and is
the same for any two consecutive integers.
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The point with co-ordinate 0 is called the origin.
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In mathematics, a rational number (or informally fraction) is a ratio
or quotient of two integers
a
It is usually written as the vulgar fraction b or a/b, where b is not
zero.
Each rational number can be written in infinitely many forms, for
example
3
2
1
= =
6
4
2
Tuesday, 13 October 2009
Real Numbers
"('&
"&
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•
"%
"$
"#
#'$
!
!$
#
Real Numbers
"
$
%
&
For every rational number there is a point on the number line
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For example the number 2 corresponds to a point halfway between 0 and 1 on
the number line.
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and − 4 corresponds to a point one and one quarter units to the left of 0.
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The set of points that corresponds to all points on a number line are called the
set of Real Numbers.
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Since there is a correspondence between the numbers and points on the
number line the points are often referred to as numbers
Tuesday, 13 October 2009
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Real numbers may be rational or irrational; algebraic or
transcendental; and positive, negative, or zero.
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Real numbers measure continuous quantities.
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Measurements in the physical sciences are almost always conceived as
approximations to real numbers.
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Real Numbers
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Tuesday, 13 October 2009
They may in theory be expressed by decimal fractions that have an
infinite sequence of digits to the right of the decimal point;
floating point data types
Computers can only approximate most real numbers with rational
numbers; these approximations are known as floating point
numbers or fixed-point numbers.
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Computer algebra systems are able to treat some real numbers
exactly by storing an algebraic description (such as "sqrt(2)") rather
than their decimal approximation.
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Mathematicians use the symbol R or alternatively R to represent
these numbers
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•
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Real numbers are represented using floating point data types.
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In most languages we have two sizes for our real data types.
Most computer systems have several ways of representing them
The different representations are dependent upon the number of
bit the operating system use (32 vs 64)
Symbols
real data types
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• Mathematicians use all sorts of symbols to substitute for natural language
expressions.
• Here are some examples
// A float is usually in the range +/- 3.4 e +/- 30 (˜7 Digits)
// C++ / C
float a=2.5;
<
>
≤
≥
≈
≡
%
=
// Mel
float $a=2.5;
// Python
a=2.5
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# less than
a<b
# greater than
a>b
# less than or equal to
a<=b
# greater than or equal to
a>=b
# Not equal
a!=b
# equality
a==b
Tuesday, 13 October 2009
Boolean Values
• Boolean values are used to hold truth values.
• Typically in a computer these values are stored
as
• 1 == true
• 0 == false
• Some modern languages actually use the values
true and false to make the code more readable.
• We use boolean values to store the results of
Comparing Floating point values
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#!/usr/bin/python
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False
False
True
True
False
True
a=1
b=1
print
print
print
print
print
print
a<b
a>b
a<=b
a>=b
a!=b
a==b
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floating point data values are not accurate and comparing them may
lead to strange problems (especially in Python)
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Consider the following python example
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comparisons
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less than
greater than
less than or equal to
greater than or equal to
approximately equal
equivalent to
not equal to
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#!/usr/bin/python
# declare a to be equal to 0.2+0.1 (should be 0.3)
a=(0.2+0.1)
# declare b as 0.3 directly
b=0.3
print "testing equality"
if( a == b) :
print a,b," are the same"
else :
print a,b," are different"
Irrational Numbers
Better Comparison
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Usually with floating point values we compare to see if we are close
To do this we use an error margin (usually called epsilon)
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#!/usr/bin/python
def FCompare(a,b) :
epsilon = 0.0000000000001
return ( ((a)-epsilon)<(b) and ((a)+epsilon)>(b) )
# declare a to be equal to 0.2+0.1 (should be 0.3)
a=(0.2+0.1)
# declare b as 0.3 directly
b=0.3
print "testing equality"
if( FCompare(a,b)) :
print a,b," are the same"
else :
print a,b," are different"
Tuesday, 13 October 2009
All rational numbers are real numbers, but there are points on the
number line that do not correspond to rational numbers.
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Those real numbers that are not rational are called irrational
numbers
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An irrational numbers cannot be written as a ratio of integers.
√
It can be show that numbers such as 2 and π are irrational
Irrational numbers are not as easy to represent as rational numbers
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Irrational Numbers II
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Usually they are represented using symbols such
√ √
as 2, 3 and
π
When we perform computations with irrational numbers we use
rational approximations for them
• For example :
√
• 2 ≈ 1.414 and π ≈ 3.141
√
• Note that not all square roots are Irrational for example 9 = 3
Absolute Value
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The absolute value of a number is the numbers distance from 0
(origin) on the number line.
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For example the numbers 5 and -5 are both five units away from 0
on the number line.
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So the absolute value of both these number is 5
We write |a| for the absolute value so
|5| = 5 and |−5| = 5
because 3 × 3 = 9
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•
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The notation |a| represents distance, and distance is never negative
so |a| is greater than or equal to zero for any real number
abs() in python
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from math import *
a=5
b=-5
fa=5.02234
fb=-5.02234
print
print
print
print
"abs(a) = ",abs(a)
"abs(b) = ",abs(b)
"abs(fa) = ",abs(fa)
"abs(fb) = ",abs(fb)
[jmacey@neuromancer:Lecture1]$./absolute.py
abs(a) = 5
abs(b) = 5
abs(fa) = 5.02234
abs(fb) = 5.02234
Tuesday, 13 October 2009
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Two numbers that are located on opposite sides of zero and have
the same absolute value are called opposites
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The number 5 and -5 are opposite to each other.
The symbol "-" is used to indicate opposite as well as negative.
When the negative sign is used before a number it should be read
as "negative"
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•
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Opposites
When it is used in front of parentheses or a variable it should be read as
"opposite"
-(5) = -5 means the "opposite" of 5 is "negative" 5
In general -a means "the opposite of a"
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If a is positive, -a is negative.
If a is negative, -a is positive.
opposite.py
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#!/usr/bin/python
from math import *
a=5
b=0.234
c=-234
d=-0.4345
print
print
print
print
Using this we can give a symbolic definition of absolute value
|a| =
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Opposites
#!/usr/bin/python



a
−a
if a is positive or zero
if a is negative
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"-a
"-b
"-c
"-d
=
=
=
=
",-a
",-b
",-c
",-d
negate.py
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#!/usr/bin/python
from math import *
a=5
print
print
print
print
print
print
print
print
print
print
"-a = ",-a
"--a = ",--a
"---a = ",---a
"----a = ",----a
"-----a = ",-----a
"------a = ",------a
"-------a = ",-------a
"--------a = ",--------a
"---------a = ",---------a
"----------a = ",----------a
•
Arithmetic Expressions
The result of writing numbers in a meaningful combination with the
arithmetic operations is usually called and arithmetic expression or
expression
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For example
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The parentheses (or brackets) are used as grouping symbols and
indicate which operation to perform first
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Because of the parentheses these expressions have different values
(3 + 2) × 5 and 3 + (2 × 5)
arithmetic.py
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#!/usr/bin/python
from math import *
a=3
b=2
c=5
print "Using Numbers"
print "(3+2) * 5 =",(3+2)*5
print "3+(2*5) = ",3+(2*5)
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[jmacey@neuromancer:Lecture1]$./arithmetic.py
Using Numbers
(3+2) * 5 = 25
3+(2*5) = 13
Using variables
(a+b)*c = 25
a+(b*c) = 13
print "Using variables "
print "(a+b)*c = ",(a+b)*5
print "a+(b*c) = ",a+(b*c)
(3 + 2) × 5 = 5 × 5 = 25
3 + (2 × 5) = 3 + 10 = 13
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Tuesday, 13 October 2009
Exponential Expressions
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Exponential Expressions
An arithmetic expression with repeated multiplication can be
written by using exponents
For example
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2 × 2 × 2 = 23 and 5 × 5 = 52
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In general an expression of the form an is called an exponential
expression and is defined as follows
For any counting number n,
If the exponent is 1, it is usually omitted
e.g. 91 = 9
an = a × a × a × . . . × a
!
"#
$
n factors
we call a the base and n the exponent
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The expression an is read "a to the nth power"
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power.py
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#!/usr/bin/python
"Powers"
"3 raised to the power of 2 = 3*3 =",pow(3,2)
"5 raised to the power of 5 is ",pow(5,5)
"9 raised to the power of 1 is ",pow(9,1)
print "Floating Point powers"
print "2 raised to the power of 1.5 is ",pow(2,1.5)
[jmacey@neuromancer:Lecture1]$./power.py
Powers
3 raised to the power of 2 = 3*3 = 9.0
5 raised to the power of 5 is 3125.0
9 raised to the power of 1 is 9.0
Floating Point powers
2 raised to the power of 1.5 is 2.82842712475
Tuesday, 13 October 2009
When we evaluate expressions, operations within grouping symbols are always
performed first
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For example
•
To make expressions look simpler we sometimes omit some or all of the
parentheses
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To do this we must agree in which order the operation are always carried out
(3 + 2) × 5 = (5) × 5 = 25 and (2 × 3)2 = 62 = 36
We do multiplication before addition and exponential expressions before
multiplication.
2
3 + 2 × 5 = 3 + 10 = 13 and 2 × 3 = 2 × 9 = 18
Tuesday, 13 October 2009
Order of Operations
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If no grouping symbols are present, evaluate expressions in the
following order:
1. Evaluate each exponential expression (in order from left to right).
2. Perform multiplication and division (in order from left to right).
3. Perform addition and subtraction (in order from left to right).
•
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•
from math import *
print
print
print
print
Order of Operations
For operations within grouping symbols, use the above order within
the grouping symbols.
Tuesday, 13 October 2009
Algebraic Expressions
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Since variables (or letters) are used to represent numbers, we can use variables
in arithmetic expressions.
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The result of combining numbers and variables with the ordinary operations of
arithmetic is called an algebraic expression
•
For example
x + 2, πr2 , b2 − 4ac, and
a−b
c−d
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Expressions are often named by the last operation to be performed
in the expression.
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For example the expression x + 2 is a sum because the only
operation in the expression is addition
Translating Algebraic Expressions
Algebraic Expressions II
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The expression a − bc is referred to as a difference because
subtraction is the last operation to be performed
• The expression 3(x − 4) is a product
is a quotient
• The expression
• The expression (a + b) is a square because the addition is performed
3
x−4
•
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Algebra is useful because it can be used to solve problems.
•
Consider the following examples of verbal expressions and their corresponding
algebraic expressions
Since problems are often communicated verbally, we must be able to translate
verbal expressions into algebraic expressions and translate algebraic expression
into verbal expressions
2
before the square is found
Verbal Expression
Algebraic Expression
The sum of 5x and 3
The product of 5 and x + 3
5x + 3
5(x + 3)
The
The
The
The
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8+
8+x
3
x
3
or (8 + 3)/3 or (8 + x) ÷ 3
3 − x2
(3 − x)2
Tuesday, 13 October 2009
algebra.py
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sum of 8 and x3
quotient of 8 + x and 3
difference of 3 and x2
square of 3 − x
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#!/usr/bin/python
•
from math import *
"5x+3 = ",5*x+3
"5(x+3) = ",5*(x+3)
"8 + x/3 = ",8+(x/3)
" (8+x) / 3 = ",(8+x)/3
"3-xˆ2 = ",3-(x*x)
"(3-x)ˆ2 = ",(3-x)*(3-x)
The value of an algebraic expression depends on the values given to
the variables.
For example, the value of x − 2y where x = −2 and y = −3
is found by replacing x and y by − 2 and − 3 respectively
print "Enter a value for x"
x=float(raw_input(">"))
print
print
print
print
print
print
Evaluating Algebraic Expressions
x − 2y = −2 − 2(−3) = −2 − (−6) = 4
If x = 1 and y = 2, the value of x − 2y is found by replacing x by 1 and y by 2
x − 2y = 1 − 2(2) = 1 − 4 = −3
Tuesday, 13 October 2009
Expression.py
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Equations
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#!/usr/bin/python
from math import *
print "Enter a value for x"
x=float(raw_input(">"))
print "Enter a value for y"
y=float(raw_input(">"))
•
•
print "x-2y = ",x-2*y
An equation is a statement of equality of two expressions
For example
11 − 5 = 6 , x + 3 = 9 , 2x + 5 = 13 and x
are all equations.
x
−4=1
2
In an equation involving a variable any number that gives a true statement when
we replace the variable with a number is said to satisfy the equation and is
called a solution or root to the equation
For example in the equation 2x + 5 = 13 the solution for x is x = 4 as (2 × 4) + 5 = 13
Tuesday, 13 October 2009
Tuesday, 13 October 2009
The Commutative Properties
Properties of the Real Numbers
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There are a number of different properties associated with numbers
They are used in arithmetic.
In Algebra we must understand these to allow us to manipulate
equations
We get the same results whether we evaluate 3 + 5 or 5 + 3
This is an example of the commutative property of addition
The fact that 4 × 6 and 6 × 4 show that multiplication is
commutative.
In General :
For any real numbers a and b
a + b = b + a and ab = ba
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Commutative.py
#!/usr/bin/python
The Associative Properties
from math import *
print "Enter a value for x"
x=float(raw_input(">"))
print "Enter a value for y"
y=float(raw_input(">"))
print "Multiplication is Commutative "
print "x*y =",x*y
print "y*x =",y*x
print "y-x = ",y-x
print "x-y = ",x-y
Consider the computation 2+3+6 using the order of operations we
add 2 and 3 to get 5 and then add 5 and 6 to get 11
•
If we reverse the order we get the same result so
(2 + 3) + 6 = 2 + (3 + 6)
print "Addition is Commutative"
print "x+y = ",x+y
print "y+x = ",y+x
print "Division and Subtraction are not"
•
•We also have an associative property of multiplication.
(2 × 3) × 4 = 6 × 4 = 24 or 2 × (3 × 4) = 2 × 12 = 24
So For any Real Numbers a, b and c
print "y/x =",y/x
print "x/y =",x/y
(a + b) + c = a + (b + c) and (ab)c = a(bc)
Tuesday, 13 October 2009
Tuesday, 13 October 2009
Associative.py
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The Distributive Property
•
#!/usr/bin/python
from math import *
print "Enter a value for a"
a=float(raw_input(">"))
print "Enter a value for b"
b=float(raw_input(">"))
print "Enter a value for c"
c=float(raw_input(">"))
Example :
3(4 + 5) = 3 × 9 = 27 or 3 × 4 + 3 × 5 = 12 + 15 = 27
• We say the multiplication by 3 is distributed over the addition.
Distributive Property
So For any Real Numbers a, b and c
print "Addative Association "
print "(a+b)+c =",(a+b)+c
print "a+(b+c) =",a+(b+c)
a(b + c) = ab + ac and a(b − c) = ab − ac
print "Multaplicative Association"
print "(ab)c = ",(a*b)*c
print "a(bc) = ",a*(b*c)
Tuesday, 13 October 2009
Distributive.py
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The identity properties
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#!/usr/bin/python
from math import *
print "Enter a value for a"
a=float(raw_input(">"))
print "Enter a value for b"
b=float(raw_input(">"))
print "Enter a value for c"
c=float(raw_input(">"))
The numbers 0 and 1 have special properties.
Multiplication of a number by 1 does not change the number
Addition of 0 to a number does not change the number.
That is why 1 is called the multiplicative identity and 0 is called the additive
identity
•
•
print "Distributive property "
print "a(b+c) =",a*(b+c)
print "ab+ac =",(a*b)+(a*c)
Identity Properties
For any real number a
a × 1 = 1 × a = a and a + 0 = 0 + a = a
print "a(b-c) =",a*(b-c)
print "ab-ac =",(a*b)-(a*c)
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The Inverse Properties
•
This is written
1
1
such that a × = 1
a
a
Inverse Properties
For any real number a there is a number -a such that a + (−a) = 0
1
For any nonzero real number a there is a number such that
a
a×
Tuesday, 13 October 2009
Identity.py
Every non-zero real number a also has a multiplicative inverse or reciprocal
1
=1
a
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#!/usr/bin/python
from math import *
print "Enter a value for a"
a=float(raw_input(">"))
print
print
print
print
"Inverse property"
"a+-a =",a+-a
"Reciprocal of a (1/a) = ",1/a
"a * 1/a = ",a*(1/a)
•
Properties of Integers
The following table lists some of the basic properties of addition
and multiplication for any integers a, b and c.
closure
associativity
commutativity
identity element
inverse element
distributivity
Tuesday, 13 October 2009
addition
Multiplication
a + b is an integer
a × b is an integer
a + (b + c) = (a + b) + c
a+b=b+a
a+0=a
a × (b × c) = (a × b) × c
a×b=b×a
a×1=a
a + (−a) = 0
a × (b + c) = (a × b) + (a × c)
Tuesday, 13 October 2009
References
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Elementary and Intermediate Algebra, Mark Dugopolski McGraw Hill 1st Ed
2002
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Engineering Mathematics, K. A. Stroud, Macmillan 3rd Edition 1987
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•
•
•
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http://www.cplusplus.com/doc/tutorial/variables.html
Collins Dictionary of Mathematics E. Borowski & J. M. Borwein Harper Collins
1989
http://en.wikipedia.org/wiki/Number
http://en.wikipedia.org/wiki/Rational_number
http://en.wikipedia.org/wiki/Real_numbers
http://en.wikipedia.org/wiki/Floating-point_number