math in programming
TRANSCRIPT
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Marquez, Adonis
Gamilla, MelchorResource Speaker/s
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Objective At the end of the course, the participants will
be able to:Identify relevant mathematical formulas; and
Perform simple computations as required in
shop operations.
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Course OutlineI. Basic Elements
Arithmetic
Basic operations
Rounding off Numbers
II. Systems of Measurement System Internationale (SI or Metric)
English System (Imperial)
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Course OutlineV. Trigonometry
Types of Angles and Triangles
Trigonometric Functions
Pythagorean Theorem
VI. Taper Taper Ratio
Taper Calculation
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Basic Elements Arithmetic
Deals with the handling of numbers involving the
four basic operations such as:
Addition
Subtraction
Multiplication
Division
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Basic Elements Algebra
Is an extension of arithmetic and deals with handling
numbers in terms of equations and formulas.
Typical usage will involve:
Square Roots
Powers of a Number
Trigonometric Functions
Solving Formulas and Equations
Variable Data
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Basic ElementsSubtraction
is a natural result of an addition
called the inverse operation of addition.
Multiplication
represents the idea of repeated addition.
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Basic ElementsProperties of Multiplication
Division
represents the idea of repeated subtraction.
It is most commonly thought as the inverse of Multiplication.
Property Name Property Sample
Commutative Property A • B = B • A 5 • 8 = 8 • 5
Associative Property A • (B • C) = (A • B) • C 6 • (4 • 5) = (6 • 4) • 5
Identity Property A • 1 = 1 • A = A 8 • 1 = 1 • 8 = 8
Multiplication Property of
Zero
A • 0 = 0 • A = 0 3 • 0 = 0 • 3 = 0
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Basic ElementsOrder of Operations
In the field of mathematics, there is a precisely defined
order in which the calculations are performed. In times where there are combination of various algebraic
operations, the order of calculation will follow these rules:
Multiplications and divisions are always calculated first;
Additions and subtractions follow; and
Any roots, powers and operations within a parentheses are
always calculated before and multiplications and divisions.
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Basic ElementsSample
Perform the following operations:
a. (3 + 5) • 5
b. 4 • 5 + 2 – 1
c. 30 ÷ 2 – 1 + 3
d. 23 + 2 • (4 ÷ 2)
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Basic ElementsRules in Rounding Off Numbers
1. If the digit after that being retained is LESS than 5,
the retained digit is unchanged.Example:
Round off the following:
a. 27.73 to the nearest tenths
b. 254,400 to the nearest ten thousands
c. 944 to the nearest hundreds
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Basic ElementsRules in Rounding Off Numbers
3. If the digit after that being retained is EQUAL to 5,
what follows determines how to round the number:
a. If there are no nonzero digits after the 5, the
retained digit is made even (round it up if it is odd).
b. If there are nonzero digits after the 5, the retained
digit is increased by one.
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Basic ElementsRules in Rounding Off Numbers
Example:
Round off the following:
a. 27.752 to the nearest tenths
b. 255,400 to the nearest ten thousands
c. 15.350 to the nearest ones
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Basic ElementsExercise
Round each to the nearest tens.
1. 122.67 3. 36.2 5. 19.232. 521.8 4. 57.59
Round each to the nearest hundreds
1. 2382.1 3. 312.645 5. 34.32. 5371 4. 2016.2
Round each to the nearest thousands
1. 12,392.1 3. 9,920 5. 6,821
2. 84,625.455 4. 2,001.6
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Metric and English UnitsEnglish Units (Imperial Units)
Typical conversions seen in Imperial units:
LENGTH
1 yard 3 feet
1 foot 12 inches
1 mi 1760 yds
1 mi 5280 ft
AREA
144 in2 1 ft2
43,560 ft2 1 acre
640 acres 1 mi2
VOLUME
57.75 in3 1 qt
4 qt 1 gal
42 gal 1 barrel
32 qt 1 bushel
MASS
437.5 grains 1 oz
16 oz 1 lb
2000 lb 1 short ton
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Metric and English UnitsSystem Internationale (SI or Metric System)
Was developed in the 1790s by French scientists
It is comprise of seven base units and can be enlargeor reduce by a factor of 10
The factor of 10 can be represented by the use of
prefixes
Approximately 90% of the world is using this form of
measurement
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Metric and English UnitsSystem Internationale (SI or Metric System)
Base units of measurements are:
Kelvin (temperature)
Second (time)
Meter (length)
Gram (mass)
Candela (luminous intensity)
Mole (amt. of substance)
Ampere (electric current)
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Metric and English UnitsSystem Internationale (SI or Metric System)
Multiples
Name Kilo- Mega- Giga- Tera-
Prefix k M G T
Factor 100 103 106 109 1012
Fractions
Name Centi- Milli- Micro- Nano- Pico-
Prefix c m μ n p
Factor 100 10-2 10-3 10-6 10-9 10-12
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Metric and English UnitsMetric to English (Vice Versa)
LENGTH
1 inch 2.54 cm1 meter 3.28 feet
1 ft 30.48 cm
1 mi 1.61 km
AREA
1 in2 6.45 cm2
1 mi2 2.59 km2
1 m2 10.76 ft2
VOLUME
1 gal 3.7854 li1 m3 1000 li
1 in3 16.39 cm3
1 m3 35.31 ft3
MASS
1 lb 453 g
1 kg 2.2 lb
1 ton 907 kg
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Metric and English UnitsUNIT CONVERSION
A unit conversion expresses the same property as a
different unit of measurement.
A conversion factor is a number used to change one set
of units to another, by multiplying or dividing.
When a conversion is necessary, the appropriate
conversion factor to an equal value must be used.
For example, to convert inches to feet, the appropriate
conversion value is 12 inches equal 1 foot.
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Metric and English UnitsUNIT CONVERSION
A unit cancellation table is developed by using known units,
conversion factors, and the fact that a unit of measure ÷ the
same unit of measure cancels out that unit.
The table is set up so all the units cancel except for the unit
desired.
To cancel a unit, the same unit must be in the numerator and in
the denominator.
When you multiply across the table, the top number will be
divided by the bottom number, and the result will be the answerin the desired units.
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Metric and English UnitsUNIT CONVERSION
Sample: Convert 3 yards to meters
3 yards 3 feet
1 yard
1 meter
3.28 feet 2.74= meters
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Metric and English UnitsExercise
Convert the following:
1. 600 g to lb
2. 1.75 gal to mL
3. 2 ton to kg
4. 40 cm to yards5. 0.75 km to in
6. 10 kg to oz
7. 1.3 barrel to in3
8. 1550 m2 to acre
9. 1300 g to lb10. 150 m2 to in3
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GeometryThree entities in engineering drawing:
Points
Lines
Circles and Arcs
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GeometryPlane
A Plane extends in two dimensions.
It is usually represented by a shape that lookslike a tabletop or wall.
Collinear
are points that lie on the same line.
Coplanar
are points that lie on the same plane.
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GeometrySample:
1. Name three points that are collinear.
2. Name four points that are coplanar.
3. Name three points that are not collinear.
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Geometry Arcs
Are curved elements that have at least a
center or radius.Circles
Is a mathematical curve, where every point on
the curve has the same distance from a fixedpoint.
This fixed point is called a center point.
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GeometryTerms used in Circles:
Arc – is any part of the circle between two points on the
circumference of the circle.
Circumference – is the length of the circle.
Chord – is a straight line joining any two points on the
circumference of a circle.
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GeometryParts of a Circle:
Circumference
Arc
Center Point
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GeometryTerms used in Circles:
Sector
Segment
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GeometryPI ConstantIs a Greek letter used in mathematics to represent the
ratio of the circle circumference to the diameter.
Its symbol is represented by
Its numerical value is 3.1416
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GeometryQuadrants
IVIII
II I
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GeometryConcentric Circles Are circles that have the same center point.
Just because a circle is inside another circle doesn'tmean they are concentric, they must have the same
point as their center.
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GeometryPolygon
Is a common geometric element defined by a
number of straight line segments that are joined at each end point.
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GeometryPolygon
Common terms in polygons are the words
inscribed and circumscribed.
CircumscribedInscribed
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GeometryCommon Polygon
Number of Sides Common Name
3 Triangle
4 Square
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
12 Dodecagon
n N-gon
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Coordinate SystemTypes of Coordinate System
Rectangular Coordinates
Relative Coordinates
Polar Coordinates
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Coordinate SystemRectangular Coordinates
Also known as the Cartesian plane which was
developed by Rene Descartes
The vertical line represents the Y-axis while the
horizontal line represents the X-axis.
The point where the two lines intersect is called
the origin.
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Coordinate SystemParts of Cartesian Plane
X-AXIS
Y-AXIS
ORIGIN
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Coordinate SystemRectangular Coordinates
There are four distinct areas of the Cartesian plane.
We name them quadrants because there are four of
them
Each one has its own number, which we write as a
roman numeral. That is, quadrants I (1), II (2), III (3)and IV (4).
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Coordinate SystemParts of Cartesian Plane
X-AXIS
Y-AXIS
ORIGIN
Quadrant I
Quadrant IVQuadrant III
Quadrant II
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Coordinate SystemRectangular Coordinates
The X coordinate is the horizontal distance from the y-axis.
if the horizontal distance is on the left side of the y-axisthe sign convention should be negative and if it is onthe right side of the y-axis it should be positive.
The Y coordinate is the vertical distance from the x-
axis.
if the vertical distance is on the lower side of the x-axisthe sign convention should be negative and if it is onthe upper side of the x-axis it should be positive.
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Coordinate SystemParts of Cartesian Plane
X-AXIS
Y-AXIS
ORIGIN
Quadrant I
Quadrant IVQuadrant III
Quadrant II
( + , - )( - , - )
( + , + )( - , + )
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Coordinate SystemPolar Coordinates
Is a two dimensional coordinate system which
is determined through the use of radii (R) andangles (θ).
This particular coordinate system is useful the
CNC drilling cycles.
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Coordinate SystemPolar Coordinates
the radius can be obtained through
Pythagorean theorem:
The angle origin is located at the positive x axis, if
the angle opens CCW it is positive if it opens CW it
is negative.
R = x + y
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Coordinate SystemPolar Coordinates
Sample
Get the following polar
coordinates:
a. R = 5 , θ = 53.13o
b. R = 5 , θ = 235
o
c. R = 5 , θ = -53.13o
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Coordinate SystemRelating Polar to Rectangular Coordinates:
Polar coordinates can be converted into rectangular
coordinates and vice versa, this is done by means ofdetermine the relationship of the sides with the angle.
θ
x
Ry
The relationships are:
Y = R sin θ
X = R cos θ
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TrigonometryTriangle/sIs the most common geometrical entity in CNC
programming. Are also integrated in polygons
Always has three (3) sides but does not needto have equal length nor angles.
Types of triangle are divided into two parts
Angle
Length of Sides
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TrigonometryTypes of Triangles (Angles)
a. Right Triangle One of the angles is equal to 90o
A < 90o
B < 90o
C = 90o
A
B
C
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TrigonometryTypes of Triangles (Angles)
b. Acute Triangle The angles are greater than 0o and smaller than
90o.
A < 90o
B < 90o
C < 90o
B
C A
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TrigonometryTypes of Triangles (Angles)
c. Obtuse Triangle One of the angles is greater than 90o but smaller
than 180o
A < 90o
B > 90o
C < 90o
B
A C
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TrigonometryTypes of Triangles (Sides)
a. Equilateral Triangle All sides of the triangle are equal also all angles
are equal to 60o
a = b = c A = B = C = 60o
B
A C
a
b
c
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TrigonometryTypes of Triangles (Sides)
b. Isosceles Triangle Two sides of the triangle are equal also their
corresponding angle are also equal
Side a = c Angle A = C
B
A C
a
b
c
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TrigonometryTypes of Triangles (Sides)
c.Scalene Triangle All of the sides have different length and all the
angles are different as well
Side a ≠ b ≠ c
Angle A + B + C = 180o
B
A C
a
b
c
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TrigonometrySolving Similar Triangles
W
L
UH
=
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TrigonometryTrigonometric Functions
Shows the relationships of the legs of the
triangle to their corresponding angles.
Main trigonometric functions (TF) are:
Sin-Opposite-Hypotenuse (SOH)
Cos-Adjacent-Hypotenuse (CAH)
Tan-Opposite-Adjacent (TOA)
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TrigonometryTrigonometric Functions
Parts of the Triangle to consider in TF
C
B
A
S i d e
( a )
Side (b)
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TrigonometryPythagorean Theorem Is a well known formula to get the relationships of the
sides of a right triangles
C
B
A
S i d e ( a )
Side (b)
General Formula : c = a + b
c = a
+ b
a = c − b
b = c − a
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TrigonometrySine Law and Cosine Law
If the triangle is no longer a right triangle the
Pythagorean theorem is no longer applicable for itwould present error in the computation.
be used to compute the remaining sides of a triangle
when two angles and a side are known a techniqueknown as triangulation
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TrigonometryCosine Law
C
B
Ab
c a
a = b + c − 2bc cos A
b = a + c − 2ac cos B
c = a + b − 2ab cos C
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Taper CalculationsTapers
d
L
D
1 : X
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Taper CalculationsTaper Ratio Is defined as the ratio of the difference between the
large and small diameter over the given length of thecone.
For example we have a taper ratio of 1:5, this means
that in every 5 mm length there will be a 1 mm
increase in the diameter.
1
X
=D − d
L
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Taper CalculationsTaper CalculationsTo calculate the small diameter d, with D, L and X:
To calculate the large diameter D, with d, L and X:
d = D −L
X
D = d +L
X
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Taper CalculationsTaper CalculationsTo calculate the length L, when D, d and X are known:
To calculate the ratio X, with d, D and L:
L = D − d × X
X =L
D − d
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