SARA Science & Mathematics Reference ~ last update: 25 Apr 2025

SARA's Science & Mathematics Reference

Shiawassee Amateur Radio Association presents some Science & Mathematics Reference Information. Mathematics is one of the most important types of 'human endeavor' tools ever. It used to gain a higher understanding of real-world physics and how things interact in both the past and in the future. From simple counting, then geometry, trigonometry, and into complex differential equation solving, all the concepts build on previous knowledge and are actually somewhat easy to master. Really, it just takes lots of active practice - it is NOT a read and learn type of process! Most people learn counting and progress through adding, subtraction, multiplying and division (usually by the 5th grade level in school). The additional learning of various advanced mathematical steps can be added by using small logical steps and leads into the ability to predict a 'future state' with very high accuracy. The schools in the United States do a varying level of justice in increasing the student's understanding beyond this '5th grade level'. Some school's do have some students that continue beyond, but most schools and students do not "get it". They think the higher mathematical levels are not needed or offer little to the adult world. {We call this 'lack of proper' STEM learning ~ STEM = Science-Technical-Engineering-Mathematics.} It takes dedication to master and lots of individual practice to truly master.

Mathematical ability is very powerful and to many can seem to be difficult to master. A good understanding of the mathematical language used is really a requirement in today's world. An understanding of 'Mathematical Requirements' is an understated issue for today's society. Computers, Artificial Intelligence [AI], cameras, communication systems, construction, mechanics, cooking, and all other human activities are made much easier with the math thrown-in. Start learning and using and advancing in small logical steps is key to the process. Using the language of mathematics is the real secret of quick, concise communication of most technical concepts. Math reduces spoken language barriers. The better you know the mathematical language, the easier it becomes to discuss and communicate with others in a concise and precise manner. This is a requirement for today's ham radio hobby. Geometry, Algebra, Calculus, Set theory, Matrices, n-dimensional spaces, etc. all have 'tricks' to learns and application issues to learn and master.

Using computers for efficient mathematical solutions, leads quickly to "Spreadsheet" Programs and other more specific computer programs {Mathematica, Matlab, etc.}. Using a "Windows system" (an example), the first program encountered is usually Microsoft Excel, but that requires a purchase. It is part of the Microsoft Office program set, so many have and use it. But others are set back by the cost. There are 'other routes' to be investigated, going for the free route, look for some freeware spreadsheet programs. We suggest you review some starting here. All are fully functional, free programs, so what do you have to lose?

• Google sheets

• Open Office Home

• Accel Spreadsheet

• Gnumeric

This list is just a start of what you can find and NOT even close to a good list, it is just a very small sample. There are many 'free' spreadsheet programs for you to choose from. All operate based on similar concepts, so pick one and dive on in. Note these are all 'Windows Type' products, Apple and Linux have similar programs available also. Just go on the internet searching for spreadsheet programs and look for the operating system of your choice!

SARA has an Electronics Reference Pages for basic Frequency & Wave-Length; Capacitance & Inductance with Reactance; Resonance Effects and basic information that you may check out. If / When you get ready, buy a Raspberry Pi [RPi] micro-computer and use the Mathematica program (it comes free on a RPi). It is a simple to learn computer program that can do amazingly complex mathematics! You really should check it out. It solves algebraic expression directly. Follow the link. You can get Linux, Apple, and Windows Mathematica versions for a large computer, but costs are much larger than a RPi (think 'hundreds' of dollars for the program alone). This one program makes a RPi a requirement for a modern ham.

Math for Hams does not need to get into difficult and complex operations, but some math can go a long way towards solving common ham problems. Checkout a 'DX Engineering' article for basic mathematics for hams. Brushing Up On Basic Math / Calculator Tools (DX Engineering).

Here we attempt to provide a quick connection to various math equations and constants for your use on this page. Actually, it is one of the 'memory jogger' locations to hold many various concepts in a single location for quick reference (especially for me). The usage of computers in the 'ham shack', on the electronics workbench, in the computer shop/lab and at a couple of 'desk' locations at a residence are the norm of today. When working on various projects, they have become a 'much needed' {think requirement} tool in each of these living areas. Having the computers connected through a home server just to make things easy ~ everything is available everywhere in a home and all internet connected. A couple of inexpensive RPi units running in a GUI server environment as the active terminal on the workbenches (about $100 each), but what a way to find needed information and update the inventory list(s) from any position in the residence. The ordering, stocking and tracking the stuff needed for project at any location is a very great benefit. Keep the status organized and current when various projects are being worked on, is a good usage of your computer's power. Follow the project status with computer notes, pictures and datasheets keep the various projects advancing and provides a historical record after the project is completed. These records are priceless if you start having issues and/or the project needs repair. Some projects can and do span multiple years, I have some approaching 60 years old. There can be literally tons of information stored on a good local network. In today's ham world the computer has passed from a 'wanted' to a 'required' item in most households. They have become interwoven into most ham radio efforts and add to the enjoyment of this vast 'Ham Radio' hobby. Mathematics must be interwoven into the complete mental process of modern mankind and is essential in understanding the complexity of it all that can be explained. Yep, we still have many things mankind cannot explain.

Raspberry PI - MATHEMATICA - Arduino

If you buy a Raspberry Pi [RPi] microcomputer you gain a software program called 'Mathematica' from Wolfram Aplha. It is free in the RPi operating system! The Microsoft Windows version is about $600.00 and there are versions for Linux and Apple (also pricey). Also, the Arduino microcontoller has the ability to do direct input/output into Mathematica. This is usually more powerful then using spreadsheets, but you can add them into the mix and use them all, all together! They can add to your mathematical skills and understanding at the highest levels, enjoy.

Mathematica is a very strong program {language} for doing many advanced mathematics. You can directly enter algebraic expressions and the program will 'solve' them for you. It is great for simple and 'stands out' for doing advanced math. The RPi and Arduino platforms are very inexpensive making them great for you own usage. Tutorials are great for learning Mathematica and mathematics in general ~ more uses for you computers and the internet.

Check these out:
RPI:        https://www.wolfram.com/raspberry-pi/
Arduino:  https://reference.wolfram.com/language/ref/device/Arduino.html
                https://library.wolfram.com/infocenter/ID/9294/
                https://library.wolfram.com/infocenter/BySubject/Engineering/
look for additional information and do some internet searching for more interesting concepts.

 Some General Comment on Mathematics

The modern educated man uses Mathematics (math) and calculations to describe his world and how everything interacts. Thus from very early times, say 1000 BC (and much earlier) ~ say like making pyramids or something similar like early detailed calendars. Mankind has defined the methods used for counting and quantification for an understanding of his complete world (and defining his entire universe). The proper usage and defined calculation methods are important for all mankind! Mathematics are 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 can explain how a current state/condition evolved from previous states/conditions and how and when the current state will evolve into some future state. It does this with known and well defined accurate and precise levels. I find it hard to believe that today's schools do not inform students of how very important mathematics is to everyone's life. In my opinion, 'schools do not explain and make students understand the proper usage of math'. It is an area of really misunderstood issue(s). Teachers let students avoid the topic because of an attitude that "it is so hard and difficult to understand - so we will just skip it" (or perhaps the teacher's lack of ability to explain the correct details). The mathematical basics are NOT complex or difficult,  the schools need better instructors and need to focus more on what really is required to be taught correctly. Careful explanations and proper importance placed on learning mathematical skills and concepts is simply a real-life requirement. Simple counting is not 'beyond the understanding of a person' (children at age two have the skill developed). It is not 'okay' to ignore society's needs to higher math skills, just because the path may be difficult to teach. Many people today cannot calculate change without the use of a computer/calculator... why is that accepted as 'okay'? {All right let me step off the soap box, but I am not sorry.} We are doing a major error in accepting this behavior on a social level. Recent dropping of student's  STEM scores show a trend of 'lacking basic knowledge' in training and a moving of students to lower levels of competence - this is just not acceptable. Society needs improvement in these student skills as our future problem solving skills become more complex and important in our society!

Be advised mastering mathematical basics is not difficult. At any step, you need to understand exactly what basic item (thought) you are missing and fix it (learn), then things will become clear and simple to understand (building your knowledge in small steps). If you watch for the 'step behind the thought' and then project logic unto the 'next step' - you can master any mathematical challenge, it just takes paying attention as you go along. Yes, it is work, but you can learn it with just a little effort. Do not fear approaching some learning issue because of a small mathematical concept you need to understand. Mathematics is always simple and applied logically and you only need the proper background (previous math topics) to move logically to the 'next step'. Some low level of mathematics understanding is required for a ham license, but many 8~10 year old students do it without strain and have licenses, you can to. Simple multiplication and division suffice at the operational level (okay some squaring and square roots can be brought into play) for a ham license. Just try to understand where is your knowledge lacking and set out to learn that "missing" step. As you advance in electronics the math requirements grow and depending on your needs may lead into more complex concepts (trigonometric, geometric and 'complex' math  concepts), but they really can be mastered with very little effort and a little practice. These higher levels are part of the Extra Class license study.  PLEASE checkout the YouTube videos from 3Blue1Brown they have video series that make the higher levels of math seem simple and teach Math in a very nice step-by-step format. Great for a review on a topic ~ remember "search" tools on a computer are your great friends. These free videos proceed from simple beginnings on to some of the highly complex mathematics concepts and allow you to slow learning down and learn some very deep thoughts on mathematical theory. Give them a shot. There are many, many videos from some extremely good teachers.

The internet provides a wealth of capabilities for information storage and retrieval. It can be a very useful training tool for learning mathematics and logical concepts . It is a great source to extend beyond what we have collected here. Check the reference pages used on the RF Cafe's web site for Mathematics References. I also use 'Wikipedia' for details on specific topics - use a simple read and understanding path.

You can review and learn from sites like: Interactive Mathematics - Mathematics Lessons from simple concepts to Calculus. This is a great place for reviewing mathematical concepts when you get stuck.

Another free online math place is DESMOS Math Calculator. You can plot all kinds of math stuff and make art shapes. It can consume lots of time, but the beauty behind math can be played around with and it is amazing! You really have no limits on learning and using mathematics to it's fullest capabilities. You did catch that "free" word in there - right? It just requires time and patience. Learning more math will lead to your ability understand and converse with others on detailed topics physics topics (radio and electronics being our major areas).

Ham Radio Math ~ What is actual minimum needed for a Ham license? We start with converting units and determining basic interactions - very basic Algebra {Example: feet or meters to inches, frequencies to wavelengths,.} Add some simple linear Algebra using Ohm's Law relationships, current requirements Vs wire sizes, electrical energy to heat or radiated power, etc.}. We then need to learn to make measurements (record those numbers) and use them to understand a 'system wide' examination of our circumstance, apply some physics and gain a solution and understanding our particular situation. We add in some 'Complex Number Theory'  (which are not complex to use) and some simple trigonometry, for doing alternating current applications and waveforms. All of these are simple logic and number crunching. We may need some trigonometry for understanding R.F. concepts (it is just understanding triangles and extends into "space concepts"), we stop before getting into the deep end of the math pool, but these should not cause you any fear. Later, we throw in a few simple algebra tricks and we will complete the required minimum mathematics for all of ham radio and you will swimming with all the big math sharks - no problem! That is a 'full requirement' to get your Amateur Extra Class license. Just do not stress about it and learn to enjoy your journey! Let it be part of your really enjoyable hobby. If (when) you start adding in some computer details, Matrix Algebra may add additional concepts to master, but lets not go there just yet.

An understanding of logarithms and scientific notation will be required. These allow for "scaling" from very, very large to very, very small numerical values in an understandable 'scalable' way. This is one of those 'hidden keys' to understanding real world physics and the way things get measured. In today's world there are just a 'few' defined constants, then everything you measure can be expressed in units of those constants. With the 2019 redefinition, the SI system is constructed around just seven defining constants, allowing all other units to be constructed directly from these constants. 

So the seven defined (exact value) constants are now {post 20 May 2019}: the Planck constant (h), the elementary electric charge (e), the Boltzmann constant (kB), the Avogadro constant (NA), the second, metre, and candela. All of the SI base units have been defined in terms of physical constants. As a result, five constants: the speed of light in vacuum, c; the Planck constant, h; the elementary charge, e; the Avogadro constant, NA; and the Boltzmann constant, kB, now have known exact numerical values when expressed in SI units. The first three of these constants are fundamental constants, whereas NA and kB are of a technical nature only: they do not describe any property of the universe, but instead only give a proportionality factor for defining the units used with large numbers of atomic-scale entities. They no longer have values based on physical measurements. An example is The speed of light c is exactly equal to 299,792,458 metres per second (approximately 300,000 kilometres per second; 186,000 miles per second; 671 million miles per hour). This is the speed of EM radiation in a vacuum (light just being EM at different frequencies than radio communication). It was set as one of the known physical 'constants' in 2019! {not so long ago!}

A total side journey for 'Real Life'

The modern educated man never gives second thoughts to his knowledge and the uneducated can not see the benefit of working to gain an understanding of mathematics.  Do you know your hat size? Yes all my ball caps have adjustments, but hat sizing is a real world thing. Hat sizes use mathematics. If you measure the circumference of your head {Use a wrap around length of string and then measure the length} and then divide by the value of Pi {approx ~ 3.14...} you will know your hat size! Usually to the closest 1/8 of a hat size. Nice to know, but unless you get into hat manufacturing, probably not very useful - but math is 'everywhere' in the real world. Learn it and you will be greatly rewarded.

Decimal Equivalents (& Drill Sizes)

Not necessarily a math issue but a useful aid in the conversion of fractions of an inch to decimals (same value conversion process as drill sizes). Seems like I am always looking for these in the shop and it takes me a long time (three or four minutes sometimes) to find the charted information (or lost seconds doing a couple of calculations), so I put these here for my personal use! ~ less than 30 seconds from when I need them and a 'no brainer'. The data works for many of my purposes, so lets share it here. The 'Number' & 'Letter' sized drill bit table fills another such a dual purpose. I know I can now quickly find the information at any time - ENJOY. We will then start the real math work stuff below. Oh, did I say that computers seem to be 'everywhere' in my home now... in the kitchen appliances, television, ham shack, workshop, kitchen, office, phone, etc. ~ I remember when having a single computer was a very big luxury in my home. Now (2024-25), literally every teenager has a cell phone which includes a camera and a very powerful computer ~ shades of Dick Tracey (younger users can search for him on the internet). Computers are used for searching everything/everywhere topics in real time and it is just a standard part of a modern life. They added 'voice control' so no typing is required. Using your face or a finger print for security on your device is becoming the 'preferred method' ~ truly the great information age is now!

Fractions & Decimals

     Fraction sizes (rational numbers) equivalent chart.

Fraction       Decimal      Fraction       Decimal      Fraction      Decimal      Fraction       Decimal      

  1/64           0.0156          17/64         0.2656          33/64        0.5156          49/64          0.7556
  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       0.5496          51/64          0.7969
  1/16           0.0625           5/16          0.3125            9/16        0.5625          13/16          0.7969
  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           0.1094          23/64         0.3594           39/64       0.6094           55/64         0.8594

  1/8             0.1250            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             0.250               1/2          0.5000              3/4        0.7500             1              1.0000

Numbered and Letter Size ANSI Drill Bit Chart

Drill #      Decimal      Drill  #      Decimal      Drill #      Decimal       Drill          Decimal

               Fraction                       Fraction                     Fraction      Letter        Fraction

  80          0.0135          53          0.0595          26         0.147              A           0.234

  79          0.0145          52          0.0635          25         0.1495            B           0.238

  78          0.016            51          0.067            24         0.152              C           0.242

  77          0.018            50          0.070            23         0.154              D           0.246

  76          0.020            49          0.073            22         0.157              E           0.250

  75          0.021            48          0.076            21         0.159              F           0.257              

  74          0.0225          47          0.0785          20         0.161              G          0.261

  73          0.024            46          0.081            19         0.166              H          0.266

  72          0.025            45          0.082            18         0.1695            I            0.272

  71          0.0265          44          0.086            17         0.173              J           0.277

  70          0.028            43          0.089            16         0.177              K           0.281

  69          0.0292          42          0.0935          15         0.180              L           0.290

  68          0.031            41          0.096            14         0.182              M          0.295

  67          0.032            40          0.980            13         0.185              N          0.302

  66          0.033            39          0.0995          12         0.189              O          0.316

  65          0.035            38          0.1015          11         0.191               P          0.323

  64          0.036            37          0.104            10         0.1935            Q           0.332

  63          0.037            36          0.1065            9         0.196               R          0.339

  62          0.038            35          0.110              8          0.199              S          0.348

  61          0.039            34          0.111               7         0.201               T          0.358

  60          0.040            33          0.113              6          0.204               U          0.368

  59          0.041            32          0.116              5          0.2055             V          0.377

  58          0.042            31          0.120              4          0.209              W          0.386

  57          0.043            30          0.1285            3          0.213              X           0.397

  56          0.0465          29          0.136              2          0.221              Y           0.404

  55          0.052            28          0.1405            1          0.228              Z            0.413

  54          0.055            27          0.144                                                   

 Numbers / Numbering

We teach toddlers their decimal-integer numbers at an early age (integers are used for starting basic counting). Then later we teach other "types of numbers" and the basics {bases} of "number systems". The theory of "Sets", a grouping of numbers is concept that should be mastered. Seldom do we chart out the modern number system "types of numbers" to assist us in our learning, so to provide a summary chart, try a 'Wikipedia' search for 'number systems'.  That provides a very nice chart showing 18 types of 'number classification'. You should understand each of these 'number classification'. Each has a specific definition (meaning) and the chart shows the relationship on how numbers are grouped into various 'sets of numbers'. This concept is usually not presented to a student in a complete view, it is shared as each individual piece is brought forward in teaching mathematics. I think this actually slows down the student's progress. It becomes important to know math concepts: integer counting, odd-even; rational-irrational; real-imaginary-complex; transcendental, etc. Math characteristics of each type leads to specific methods to handle computations and a deeper understanding of various math concepts. Concepts of geometry, trigonometry, matrix operations, factorials and math "series", higher dimensional concepts (n-dimensional), vectors, calculus are all important. 

The use of 'zero' and negative numbers are great human milestones in math history. It is very 'unfortunate' that the names of 'complex' and 'imaginary' types are so named. They are NOT complex nor imaginary. When studying numbers, "Imaginary numbers" are the representation of the square roots of negative numbers. Mainly consider the 'square root of minus one' or possibly simpler by stating in reverse "Give the number whose squared value is negative one". We simplify the number notation by choosing an algebraic number of "i" or in electronics we use "j" (to be different from "i" which we use as a shortened term for current amplitude). This leads to each 'complex' number being divided into two types... a 'real' part and and an 'imaginary' part. These lead into vectors and operations on vectors. Leading to 'sets of equations' and matrix theory.

A famous mathematician, Euler, proved that: {Which is One of mankind's most beautiful equations.}

e-ipi  + 1 = 0      {known as Euler's Identity ~ search 'Euler' on Wikipedia}

and to Euler's Formula:

This brings in the concept of trigonometry into the complex plane and shows the 'real part' value is a cosine and the 'imaginary part' is a sine. These lead to 'pi' being brought out through this concept as a part of the overall 'complex space' and ties much of higher mathematics together. No need to go further here, just remember taking small learning steps will lead to a great method to discuss, calculate and truly understand our physical world. We actually live within a magical galaxy of time and space.

SI Constants

The International System of Units (SI), commonly known as the metric system, is the international standard for measurement. The SI rests on a foundation of only seven (7) defining constants:
The speed of light in vacuum, the Planck constant, the elementary charge (i.e. the charge on a proton), the cesium hyperfine splitting frequency, the Boltzmann constant, the Avogadro constant, and the luminous efficacy of a specified monochromatic source. Definitions of all seven (7) SI base units are expressed using an explicit-constant formulation and experimentally realized using a specific practical measurement techniques.

A redefining of the SI – In November 2018, the world's measurement experts voted and unanimously approved a revision of the SI Standards that established a measurement system entirely based on known physical constants of nature. The changes became effective on World Metrology Day, May 20, 2019. (NIST)  In everyday applications "ALL physical measurements now can be expressed 'exactly' and 'precisely' in values of these seven constant values and only the desired precision needs to be figured out by the various calculations" or stated differently "Everything inm our universe can be now calculated exactly, just use a large enough calculation to meet your precision requirements". Everything is 'precisely defined' to infinite precision and accuracy (unless we humans change it later... Haha!).

Values the Seven SI Defined Constants:
Name                                                   Symbol            Value                              Units
Speed of Light                                           c                  299,792,458                   meters / second 
Planck Constant                                        h                  6.626 070 15 × 10-34      Joule seconds
Elementary Charge                                   e                  1.602176634 × 10-19      coulombs
Hyperfine transition of Cesium-133          ∆νCs              9,192,631,770                hertz
Boltzman constant                                     k                 1.380649 × 1023              joules/kelvin
Avogadro constant                                     NA               6.02214076 × 1023          particles / mole
Luminous Efficacy Radiation 540x10^12   Kcd                683                                lumens / watt
  Monochromatic radiation of frequency hertz
  "Green Light"

The Seven SI base units:
Name                                               Typical Symbol         Name                    Symbol
Time                                                     t                              second                     s
Length                                                 l, x, r, etc.                meter                       m
Mass                                                    m                            kilogram                   kg
Electric Current                                    I, i                           ampere                     A
Thermodynamic Temperature              T                             kelvin                        K
Amount of Substance                           n                             mole                         mol
Luminous Intensity                                Iv                            candela                    cd

A nice chart showing how these seven base units are related to many derived units can be found at the NIST site {upper right column icon}. Also, the Wikipedia's site as lots of great information. {Wikipedia does not want direct links into their information ~ so use the following "search terms" for some great reading:

International_System_of_Units       Lots of details on unit definitions.

Metric_prefix                                   The names used and powers of ten to be used.

Scientific_notation                           Use 'Engineering Notation' ~ what most calculators & computers use.

Metric Prefixes & Scientific Notation

The SI definitions for prefixes cover a range 10-30  to 1030 , the powers of ten for 0, 1, 2, and 3 (both positive and negative go by integer counts and then switch to ranges of 3 units {3, 6, 9, 12, ...  30}. See the Wikipedia site. It is common to use "Engineering Notation", a form of 'Scientific Notation' for doing many, many practical calculations in the area of Ham Radio. See the sub-title for "Engineering Notation" on Wikipedia the page for "Scientific Notation".
A point usually not recognized early in scientific notation learning is the use of 'precision'. Trailing zeros of the base number translates to increased precision of the measurement. Do not add or drop those zeros! Make all your calculations as precise as possible. The 'SI unit definitions' eliminates any accuracy issues for calculations. Hang in there and learn this in small steps ~ it will assist you greatly in the future. Both understanding and communicating large details quickly can be done with mathematical notation. This is 'scaling' of numbers, sizes, distances, and many other values used in describing physics in a universal manner where very small and very large numbers get involved in your concepts.