math in programming

8/10/2019 Math in Programming

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Marquez, Adonis

Gamilla, MelchorResource Speaker/s

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.

Course OutlineI. Basic Elements

Arithmetic

Basic operations

Rounding off Numbers

II. Systems of Measurement System Internationale (SI or Metric)

English System (Imperial)

Course OutlineV. Trigonometry

Types of Angles and Triangles

Trigonometric Functions

Pythagorean Theorem

VI. Taper Taper Ratio

Taper Calculation

Basic Elements Arithmetic

Deals with the handling of numbers involving the

four basic operations such as:

Addition

Subtraction

Multiplication

Division

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

Basic ElementsSubtraction

is a natural result of an addition

called the inverse operation of addition.

Multiplication

represents the idea of repeated addition.

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

A • 0 = 0 • A = 0 3 • 0 = 0 • 3 = 0

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.

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)

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

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.

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

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

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

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

437.5 grains 1 oz

16 oz 1 lb

2000 lb 1 short ton

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

Base units of measurements are:

Kelvin (temperature)

Second (time)

Meter (length)

Gram (mass)

Candela (luminous intensity)

Mole (amt. of substance)

Ampere (electric current)

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

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

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

1 lb 453 g

1 kg 2.2 lb

1 ton 907 kg

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.

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.

Sample: Convert 3 yards to meters

3 yards 3 feet

1 yard

1 meter

3.28 feet 2.74= meters

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

GeometryThree entities in engineering drawing:

Points

Circles and Arcs

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.

GeometrySample:

1. Name three points that are collinear.

2. Name four points that are coplanar.

3. Name three points that are not collinear.

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.

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.

GeometryParts of a Circle:

Circumference

Center Point

GeometryTerms used in Circles:

Sector

Segment

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

GeometryQuadrants

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.

GeometryPolygon

Is a common geometric element defined by a

number of straight line segments that are joined at each end point.

GeometryPolygon

Common terms in polygons are the words

inscribed and circumscribed.

CircumscribedInscribed

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

Coordinate SystemTypes of Coordinate System

Rectangular Coordinates

Relative Coordinates

Polar Coordinates

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.

Coordinate SystemParts of Cartesian Plane

X-AXIS

Y-AXIS

ORIGIN

There are four distinct areas of the Cartesian plane.

We name them quadrants because there are four of

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).

X-AXIS

Y-AXIS

ORIGIN

Quadrant I

Quadrant IVQuadrant III

Quadrant II

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-

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.

X-AXIS

Y-AXIS

ORIGIN

Quadrant I

Quadrant IVQuadrant III

Quadrant II

( + , - )( - , - )

( + , + )( - , + )

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.

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

Sample

Get the following polar

coordinates:

a. R = 5 , θ = 53.13o

b. R = 5 , θ = 235

c. R = 5 , θ = -53.13o

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.

The relationships are:

Y = R sin θ

X = R cos θ

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

Length of Sides

TrigonometryTypes of Triangles (Angles)

a. Right Triangle One of the angles is equal to 90o

A < 90o

B < 90o

C = 90o

b. Acute Triangle The angles are greater than 0o and smaller than

A < 90o

B < 90o

C < 90o

c. Obtuse Triangle One of the angles is greater than 90o but smaller

than 180o

A < 90o

B > 90o

C < 90o

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. Isosceles Triangle Two sides of the triangle are equal also their

corresponding angle are also equal

Side a = c Angle A = C

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

TrigonometrySolving Similar Triangles

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)

TrigonometryTrigonometric Functions

Parts of the Triangle to consider in TF

S i d e

Side (b)

TrigonometryPythagorean Theorem Is a well known formula to get the relationships of the

sides of a right triangles

S i d e ( a )

Side (b)

General Formula : c = a + b

a = c − b

b = c − a

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

TrigonometryCosine Law

a = b + c − 2bc cos A

b = a + c − 2ac cos B

c = a + b − 2ab cos C

Taper CalculationsTapers

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.

=D − d

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

D = d +L

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

D − d

math in programming

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