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Shiawassee Amateur Radio Association [SARA]

Established: January, 1958 an ARRL Affiliated Club since 1961

"Whiskey 8 Quack Quack Quack"

Meets at: James P. Capitan Center, Lower Level; 149 E. Corunna Ave.; Corunna, MI 48817 Monthly: 2nd Tuesday @ 7:00 PM

Club station located in the James P. Capitan Center - Lower Level.
IARU: 2 Grid Square EN72wx   Latitude: 42.9819 N   Longitude: -84.1164 W   Alitude: 760 ft.

Contact us at:   SARA / W8QQQ <Email>

You're invited to a club meeting!  7:00 PM the 2nd Tuesday of each month in Corunna, MI.


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SARA's Science & Mathematics Reference Page

Shiawassee Amateur Radio Association presents some Science & Mathematics Reference Information which we hope is useful to viewers. Mathematics is one's first tool to higher understanding of real world conditions. It starts with counting and progresses through very high levels of processing, but the learning steps can be small and the ability to predict with high accuracy the outcomes of things is very powerful.

Using computers for efficient mathematics leads quickly to "Spreadsheet" Programs. Using Windows, the first is Microsoft Excel, but that requirees a purchase. It is part of the Microsoft Office programs, so many have and use it. But if you want to go the free route, then there are some freeware spreadsheet programs we suggest you review. All are fully functional, free programs, so what do you have to lose?

This list is NOT all you can find, just a small sample. There are many 'free' spreadsheet programs for you to choose from. All operate on similar concepts, so pick one and dive on in. Note these are all 'Windows Based' products, Apple and Linux have ones available also --> just go a searching!

We have an Electronics Reference Page for basic Frequency & Wave Length; Capacitance & Inductance with Reactance; and Resonance basic information. When you get ready, buy a Raspberry Pi [RPi] and use the Mathematica program (free on a RPi). Is a simple to learn program that can do amzazingly complex mathematics! You should check it out. Follow the link.

We attempt to provide a quick connection to various math equatons and constants for your use on this page. Actually, it is one of my 'memory jogger' locatons to hold the concepts in a single location for me to quickly reference. I use computers in the 'ham shack', on the electronics workbench and in the computer shop when working various projects, they have become 'needs/necesary units in each of these areas. I have them connected to the home server just to make things easy. A couple are RPi units running in a GUI, but what a way to find needed information and update the inventory list(s).

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A Comment on Mathematics

Modern educated man uses Mathematics and calculations to describe his world and how everything interacts (thus forward from very early times, say 1000 BC). He has defined the methods used for counting and quantification of his complete world. The proper usage and defined calculations are important! It is used internationally to quickly and precisely convey information between people. It is used to explain how things work together, from very simple to extremely complex interactions. When properly used it explains how a current state evolved from pervious states and how/when the current state will evolve into some future state. It does this with known and defined accuracies and precision. I find it hard to believe that today's schools do not inform students of how important mathematics is to everyone's life. In my opinion, they do not explain and make the usage understood to all students, they let students avoid the topic because "it is hard and I do not understand" - so lets just skip it. The basics are NOT complex they just need careful explanations and proper importance placed on learning them. Simple counting can seem 'beyond the understanding of a person' and so it is 'okay'. Many people today cannot calculate change without the use of a computer/calculator... why is that 'okay'. {Okay, off the soap box .}

Be advised mastering basics is not difficult. You need to understand exactly what basic item you are missing in your understanding and fix it, then things will become very clear. If you can watch a 'step behind' and project logic unto the 'next step' you can master any mathematical challenge. Do not fear approaching some learning issue because of a small mathematical concept you need to understand. Mathematics is simply logical and you need the proper background to move logically to the 'next step'. Some level of mathematics understanding is required for a ham license, but many 8~10 year old students do it without strain, you can to. Simple multiplication and division suffice at the operational level (okay some squaring and squre roots can be brought into play).Just try to understand where your knowledge is lacking and set out to learn that "missing" step.

Decimal Equivalents (& Drill Sizes)

Not necessarily a math only issue but conversion of fractions of an inch to decimals (same value conversions as drill sizes). Seems like I am always looking for these and it takes me a long time (minutes) to find the information, so I put it here for my use! Works for both our purposes. Therefore I added in 'Number' & 'Letter' sized drill bit table. I know I can quickly find the information it is handy any time - ENJOY. Then the math work stuff really begins.

Fractions & ANSI Size Drill Bit Selections

    Fraction       Decimal       Fraction       Decimal       Fraction       Decimal       Fraction       Decimal  
    (in.)     (in.)     (in.)     (in.)
1/64 0.0156 17/64 0.2656 33/64 0.5156 49/64 0.7656
1/32 0.0313 9/32 0.2812 17/32 0.5312 25/32 0.7812
3/64 0.0469 19/64 0.2968 35/64 0.5469 51/64 0.7969
1/16 0.0625 5/16 0.3125 9/16 0.5625 13/16 0.8125
5/64 0.0781 21/64 0.3281 37/64 0.5781 53/64 0.8281
3/32 0.0937 11/32 0.3437 19/32 0.5937 27/32 0.8437
7/64 0.1093 23/64 0.3594 39/64 0.6094 55/64 0.8594
1/8 0.0125 3/8 0.3750 5/8 0.6250 7/8 0.8750
9/64 0.1406 25/64 0.3906 41/64 0.6406 57/64 0.8906
5/32 0.1562 13/32 0.4062 21/32 0.6562 29/32 0.9062
11/64 0.1719 27/64 0.4219 43/64 0.6719 59/64 0.9219
3/16 0.1875 7/16 0.4375 11/16 0.6875 15/16 0.9375
13/64 0.2031 29/64 0.4531 45/64 0.7031 61/64 0.9531
7/32 0.2187 15/32 0.4687 23/32 0.7187 31/32 0.9687
15/64 0.2344 31/64 0.4844 47/64 0.7344 63/64 0.9844
1/4 0.2500 1/2 0.5000 3/4 0.7500 1 1.0000

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Numbered and Letter Size ANSI Drill Bit Chart

     Decimal         Decimal         Decimal         Decimal   
   Drill #     (in.)    Drill #     (in.)    Drill #     (in.)    Drill Letter    (in.)
80 0.0135 53 0.0595 26 0.1470 A 0.2340
79 0.0145 52 0.0635 25 0.1495 B 0.2380
78 0.0160 51 0.0670 24 0.1520 C 0.2420
77 0.0180 50 0.0700 23 0.1540 D 0.2460
76 0.0200 49 0.0730 22 0.1570 E 0.2500
75 0.0210 48 0.0760 21 0.1590 F 0.2570
74 0.0210 47 0.0785 20 0.1610 G 0.2610
73 0.0240 46 0.0810 19 0.1660 H 0.2660
72 0.0250 45 0.0820 18 0.1695 I 0.2720
71 0.0260 44 0.0860 17 0.1730 J 0.2770
70 0.0280 43 0.0890 16 0.1770 K 0.2811
69 0.0292 42 0.0935 15 0.1800 L 0.2900
68 0.0310 41 0.0960 14 0.1820 M 0.2950
67 0.0320 40 0.0980 13 0.1850 N 0.3020
66 0.0330 39 0.0995 12 0.1890 O 0.3160
65 0.0350 38 0.1015 11 0.1910 P 0.3230
64 0.0360 37 0.1040 10 0.1935 Q 0.3320
63 0.0370 36 0.1065 9 0.1960 R 0.3390
62 0.0380 35 0.1100 8 0.1990 S 0.3480
61 0.0390 34 0.1110 7 0.2010 T 0.3580
60 0.0400 33 0.1130 6 0.2040 U 0.3680
59 0.0410 32 0.1160 5 0.2055 V 0.3770
58 0.0420 31 0.1200 4 0.2090 W 0.3860
57 0.0430 30 0.1285 3 0.2130 X 0.3970
56 0.0465 29 0.1360 2 0.2210 Y 0.4040
55 0.0520 28 0.1405 1 0.2280 Z 0.4130
54 0.0550 27 0.1440        

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Some Important Mathematical Constant Values

"Pi" is the ratio of Circumference to diameter for a circle.

Name   SymbolNumeric Value
Pi Pi symbol  3.14159265...
Pi^2 Square of Pi  9.86960440...
Pi^.5 Square Root of Pi  1.77245385...
1/Pi Reciprocal of Pi  0.31830989...
Pi/180 Pi divide by 180  0.01745329...
Pi/360 Pi / 360  0.00872665...
e e  2.7182818...
Degrees in 1 Radian Degrees in 1 Radian  57.29577951...
Radians in 1 degree Pi / 180  0.01745329...
Speed of Light c  299,792,458 m/sec or
  Speed of Lightc  983,571,056.4 ft/sec or
  Speed of Lightc  188,857.7297 mi/sec
Newtonian Gravitational Constant G  6.67384 x 10^-11 m^3/(kg x sec^2)
Plank constant h  6.62606957 x 10^-34 Jsec
Coulomb's constant  Coulomb's Constant   8.98755178 x 10^9 Nm^2/C^2
Wavelength lamda or Reciprocal of Pi   c is 'Speed of Light' and f is frequency in Hertz (cycles per second)
Characteristic Impedance of vacuum Reciprocal of Pi  376.73031346... Ohms

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Web Math Programs

Convert Units

'Convert for Windows' is a very useful free program to convert from one value of measurement into another type of unit. It covers: Force; Light; Mass; Power; Pressure; Speed (velocity); temperature; Time; Volume; Volume - Dry; Accelleration; Angle; Area; Density; Distance and Flow. Also, a 'custom' area allows you to enter your own conversions (if required by you... example money conversions). Convert for Windows Freeware Program

SMath Studio

'SMath Studio' is a free math studio for entering math equations and doing ploting, design, and other mathematical investigations in topics of your choosing. SMath Studio Tiny, powerful, free mathematical program with WYSIWYG editor and complete units of measurements support. It provides numerous computing features and rich user interface. The application also contains integrated mathematical reference book. A 'Built-in' Extensions Manager tool allows you to get access to hundreds official and third-party resources. Students and Engineers get a great tool for free!

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Formulas for Area

Rectangle ~ Area (Straight Sides)

The most basic area formula is the formula for the area of a rectangle. Given a rectangle with length (l) and width (w), the formula for the rectangle's area is:

A = l w  (rectangle)

That is, the area of the rectangle is the length multiplied by the width. As a special case, as l = w which is a square, the area of a square with side length s is given by the formula:

A = s 2 (square)

Triangle ~ Area

It follows that a triangle's area is shown to be one half the area of a rectangle (or parallelogram), namely: {b is base and h is height}

A = 1 (b * H) / 2  (triangle)

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Dissection of parallelograms into triangles

Image of a Parallelogram with height (h) and width (b)

Many simple formulas for area follow from the method of dissection. This involves cutting a shape into pieces, whose areas must be summed to attain the area of the original shape.

For an example, any parallelogram can be subdivided into a trapezoid and a right triangle, as shown in the figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure becomes a basic rectangle. It follows that the area of any parallelogram has the same formula (thus same area) as a rectangle with same width and height.(area is just the product of the length and width).

A = l w (parallelogram)

However, the same parallelogram can be cut along one of the diagonals into two congruent triangles, as shown in the figure to the right. Each triangle has height, h, and width, b. You can use the triangle area formula to find the area of one triangle, and then multiply by 2 (two triangles) to get the parallelpgram's area.

Image Parallelogram cut into two equal triangles with height (h) and width (b)

It is easy to see that the area of each triangle is half the area of the parallelogram:

A = 1 (b * H) / 2  (triangle)

Similar arguments and methods can be used to find area formulas for any trapezoid, as well as many more complicated polygons. If one tries the method of dissection and looking at results, usually the correct formula can be found. The polygon's area will be the sum of all the areas from each of the small divisions summed up. As you do these disections, triangles and the ability to work with them becomes important. This leads to the field of "Trigonometry" (see below).

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Circle ~ Area & Parts (Curved Sides)

A = Pi r^2  (Circle with Radius, r) or A = Pi d^2 / 4 (Circle with diameter, d)

Image of a Circle cut into sections with height (r) and total width 2 Pi;

The formula for the area of a circle can be based on a similar method as used in parallelograms above. Given a circle of radius r (a disk), it is possible to partition the circle into sectors, as shown in the figure to the left. Each sector is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is r, and the width is half the circumference of the circle, or (Pi)r. Thus, the total area of the circle is r x (Pi r), or Pi r^2, as 2r = d the formula can be rewritten as Pi d^2 / 4

The side [s] of an inscribed square becomes: Square Root of 2 dia


0.7070 dia


Length of an arc (small curved piece of the circle) becomes: [L ]:

(deg) Pi (dia) / 360

   or   (deg) (dia) / 0.0087267

Area of a sector [A s]:

(deg) Pi (dia) / 360


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List of Common Formulas for Area

ShapeFormula for AreaVariables
Regular Equilateral TriangleSQRT 3 (s^2)/ 4 s is the length of one side of the triangle.
Triangle 1SQRT s(s-a)(s-b)(s-c) s is half the perimeter, a, b and c are the length of each side.
Triangle 2(ab(sin(C0)/2 a and b are any two sides, and C is the angle between them.
Triangle 3ab/2 a and h are the base and altitude (measured perpendicular to the base).
Isosceles triangle=(b(SQRT(a^2-(b^2/4))/2 or (b/4)SQRT(4a^2-b^2) a is the length of one of the two equal sides and b is the length of the different side.
Rhombus/Kiteab/2 a and b are the lengths of the two diagonals of the rhombus or kite.
Parallelogrambh b is the length of the base and h is the perpendicular height.
Trapezoid(a+b)h/2 a and b are the parallel sides and h the distance (height) between the parallels.
Regular hexagon3(SQRT(3))s^2)/2 s is the length of one side of the hexagon.
Regular octagon2(1+SQRT(2))s^2 s is the length of one side of the octagon.
Regular polygon 1(n l^2/4)(cot(Pi/n)) l is the side length and n is the number of sides.
Regular polygon 2(p^2/(4 n))((cot(Pi/n)) p is the perimeter and n is the number of sides.
Regular polygon 3(n R^2/2)(sin(2Pi/n)  OR  nr^2(tan(Pi/n)) R is the radius of a circumscribed circle, r is the radius of an inscribed circle, and n is the number of sides.
Regular polygon 4ap/2  OR  n s a)/2 n is the number of sides, b is the side length, a is the apothem, or the radius of an inscribed circle in the polygon, and p is the perimeter of the polygon.
CirclePi r^2  OR  Pi d^2/4 r is the radius and d the diameter.
Circular sectortheta r^2/2  OR  L r /2 r and theta are the radius and angle (in radians), respectively and L is the length of the perimeter.
EllipsePi a b a and b are the semi-major and semi-minor axes.
Total surface area of a cylinder2 Pi r(r + h) r and h are the radius and height.
Lateral surface area of a cylinder2 Pi r h r and h are the radius and height.
Total surface area of a sphere4 Pi r^2  OR  Pi d^2 r and d are the radius and diameter.
Total surface area of a pyramidB +((P L)/2) B is the base area, p is the base perimeter and L is the slant height.
Square to circular area conversion4 A / Pi A is the area of the square in square units.
Circular to square area conversionPi C / 4 C is the area of the circle in circular units.

The above calculations show how to find the areas of many common shapes. Hope they help you.

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Sphere Formulas (Area & Volume)

Where: d is diameter; r is radius; and 2r = d

Surface area [As] of a sphere:

As = Pi d^2

Volume [V] of a sphere:

V = (4 Pi r^3) / 3

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Trigonometry Formulas for Triangles

Trigonometry is the mathematical branch dealing with relationships of sides and angles in triangles. It also deals with a "unit vector" rotating around a point of origin, which has an angle of rotation, a unit vector and the 'x axis' and 'y axis' values for the vector's end pont (a triangle with sides, 'x', 'y'; hypotenuse ~ 'c' and the rotational angle {theta}).

In triangle terms:

Sine = Opposite side / Hypotenuse         Cosine = Adjacent Side / Hypoternuse         Tangent = Oposite Side / Adjacent Side

In 'unit circle' terms:

Sine = Opposite side / Hypotenuse         Cosine = Adjacent Side / Hypoternuse         Tangent = Oposite Side / Adjacent Side

Also defined is the reciprocal of each function:

Cosecant = 1/ Sine  OR  Hypotenuse / Opposite Side

Secant = 1/ Cosine  OR  Hypotenuse / Adjacent Side

Cotangent = 1/ Tangent  OR  Adjacent Side / Opposite Side

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Given a triangle with angles A, B, and C  that are respectively opposite the sides a, b, and c.

Image acute triangle with sides a, b, & c and opposite angles A, B, & C.

NOTE: Some of the following equations can have two 'correct' solutions, so some judgement is required !  Then the following results can be determined:


"Law of Sines"

[sin (A) / A ] = [sin (B) / B ] = [sin (C) / C ] = D {a constant}


"Law of Cosines"

Most useful if you have any two sides and the included angle.

c^2 = a^2 + b^2 - a a b cos (C)

and rearrange to two other forms:

a^2 = b^2 +c^2 - 2 b c cos (A)


b^2 = a^2 + c^2 - 2 a c cos (B)


"Law of Tangents"

b^2 = (a-b/(a+b) = (tan (alpha-beta) /2) / (tan (alpha + beta)/2)

The law of tangents is most useful if you have two sides and an included angle or two angles and the included side.

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Trigonometry functions expressed in terms of the other basic trigonometry functions:

 In terms of:      Sin (theta)    Cos (theta)    Tan (theta)
  Sin (theta)    Sin (theta)  =/- SQRT(1- cos^2 (theta))    +/- tan (theta) / SQRT (1 + tan^2 (theta))  
  Cos (theta)  +/- SQRT(1 - sni^2 theta)      Cos (theta)  +/- (1 / (SQRT( 1 + tan^2 (theta)))  
  Tan (theta)  +/- (sin (theta)) / (SQRT (1 - Sin^2 (theta)))    +/- (SQRT ( 1 - Cos^2 (theta)))/ Cos (theta))      Tan (theta)


Inverse Functions

Inverse functions are also called "arc functions".

arcsin (theta)


arccos (theta)


arctan (theta)


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Various Online Calculators

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Wolfram Mathematica / Language

The Wolfram Language is a computer language and gives you a way to communicate with computers, in particular so you can tell them what to do. It is considered a "higher level" language to express to the computer what you want and then displays results back to you. As an aside, you may want to look up details on Mathematica {math solving software} and Wolfram Alpha {solution software} that are 'earlier' conceptual efforts on this process. Be advised that you should start with version 11 or higher if just beginning. Oh, lets mention that it runs on almost all operating systems - Windows, MAC, Linux, etc. (included free on the RPi). On SARA's pages you will find some links to resources to get you going fast. Most probably you will need to 'signup' for a Wolfram ID along the way. It is pretty painless.

A powerful math tool is using "Wolfram Mathematica" for doing calculations and display. Many start out with the book "Hands-On Start to Wolfram Mathematica", 484 pages, by Cliff Hastings, Kevin Mischo, & Michael Morrison. The New Version 11 includes content for 3D printing and all the Version 11 graphics capabilities, including word clouds; customizing/labeling; and new content for working with data, audio, linguistic data and dates; and includes an all-new index. Publisher: Wolfram Media, Inc. (2016); ISBN: 9781579550127 (Paperback) [about $36.00] and includes website access. You can checkout Chapter 7 for free at: Wolfram Mathematica, Hands On, Chapter 7.

There is a companion video that goes through some concepts used in the book. uTube video on Wolfram Mathematics concepts.

There is free access to "An Elementary Introduction to the Wolfram Language" which works more like a reference guide than a series of tutorials, with each couple of pages outlining a concept or feature of Wolfram Language. The whole programming language is too vast to be covered in a single book, so it doesn't feature everything. Instead, it's more like edited highlights of concepts you really need to know. [Click the 'Read Online' Ribbon on the book cover.]

Then there is 'Wolfram U' which has training lectures and interactive courses that cover the language. Your first stop should be the full interactive course for the book "An Elementary Introduction to the Wolfram Language" that was listed in the previous paragraph by Stephen Wolfram.

Then try looking through some of these sites:

There are many other links for you to scan and use. Try searching the internet. Wolfram is very powerful language and it is 'free' on every RPi distribution over the last few years. Enjoy!

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Misc. Information

Electronics Reference Page from here you can get directly to the electronics reference page.

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How we did it -- The equations shown on this page are presented by using "Latex" and the equation writing capabilities from the Code Cogs Web Site. Images are then stored "locally" to allow proper HTML5 validation.

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