# SARA Mathematic Reference

**SARA Officers ~ last update: ****05**** Nov 2023**

**SARA's Science & Mathematics Reference**

**SARA's Science & Mathematics Reference**

Shiawassee Amateur Radio Association presents some Science & Mathematics Reference Information which we hope is useful to viewers. Mathematics is one of the most important tools used to gain a higher understanding of real-world physics. From simple counting, then geometry, trigonometry, and on into complex differential equation solving. Everyone learns counting and progresses through adding, subtraction, multiplying and division (usually by the 5th grade in school. The learning of advanced steps can be small and the ability to predict a future state with high accuracy is very powerful and difficult to master. A good understanding of the mathematical language is really required in today's world. 'Required' is not an understatement. Computers, cameras, communication systems, construction, mechanics, cooking, and all other human activities are made easier with a little math thrown-in. Start using small steps and understand and use the language of mathematics, as it is the secret of quick, concise communication of advanced concepts in all areas of life. The better you know the language, the easier it becomes to discuss and communicate with others in a concise manner.

Using computers for efficient mathematic solutions, leads quickly to "Spreadsheet" Programs and other more specific computer programs. Using a "Windows system" (an example), the first program is usually is 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, gong for the free route, look for some freeware spreadsheet programs. We suggest you review starting here. 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 Type' products, Apple and Linux have programs available also --> just go internet searching for spreadsheet and look for the system of your choice!

SARA has an Electronics Reference Pages for basic Frequency & Wave-Length; Capacitance & Inductance with Reactance; and Resonance basic information that you may wish to 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 should check it out. 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).

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)**.

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 holds many various concepts in a single location for me to quickly reference. I use computers in the 'ham shack', on the electronics workbench, in the computer shop/lab and at a couple of 'desk' locations at my house. When working on various projects, they have become a 'much needed' unit in each of these areas. I have the computers connected through a home server just to make things easy ~ everything is available everywhere in my home and internet connected. A couple of inexpensive RPi units running in a GUI serve as the terminals on the workbenches (about $100 each), but what a way to find needed information and update the inventory list(s) from any position. I can order the stuff I need for project at any location and keep it organized when materials come in. Then follow the project with computer notes and pictures. Keeping updated notes on the various projects helps make it a little more organized and connects things over time. Some projects are spanning multiple years, then I decide to make a change. There is literally tons of information stored on my local network. In today's world the computer has passed from a 'wanted' to a 'needed' item in most households. They have become interwoven into my ham radio efforts and add to the enjoyment of this vast hobby. Mathematics is interwoven into the complete mental process and is essential in understanding the complexity of it all.

**Some General Comment on Mathematics**

**Some General Comment on Mathematics**

Modern educated man uses Mathematics (math) and calculations to describe his world and how * everything* interacts. Thus from very early times, say 1000 BC or much earlier ~ say making pyramids or something similar like early detailed calendars. Mankind has defined the methods used for counting and quantification and an understanding of his complete world (and defining his universe). The proper usage and defined calculation methods are important for all mankind! Mathematics 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/condition evolved from previous states and how/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 them 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 a teacher's lack 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 needs to be taught. Careful explanations and proper importance placed on learning mathematical skills is 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 STEM scores show a trend of 'lacking knowledge' in students training moving them to lower levels - this is just not acceptable. Society needs improvement in these student skills as our future problems become more complex!

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. 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 get into more complex concepts (trigonometric and geometric concepts), but they really can be mastered with very little effort and some practice. 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 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 mathematics 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 Calculat**or. 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.

Ham Radio Math ~ What is needed for a Ham license? We start with converting units and determining basic interactions {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 Numbers' (which are not complex to use) and some simple trigonometry, for doing alternating current applications. 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, throw in a few simple algebra tricks and we will complete the required mathematics for all of ham radio and you will swimming with all the big math sharks!. 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.

**Decimal Equivalents (& Drill Sizes)**

**Decimal Equivalents (& Drill Sizes)**

Not necessarily a math only issue but an 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) to find the information, so I put it here for my personal use! ~ less than 30 seconds. The chart works for both purposes, so it was added here. The 'Number' & 'Letter' sized drill bit table fills a dual purpose. I know I can now quickly find the information it is handy any time - * ENJOY*. We will then start the math work stuff. Oh, did I say that computers seem to be 'everywhere' in my home now... in the kitchen appliances, television, ham shack, workshop, kitchen, office, etc. ~ I remember when having a single computer was a very big luxury in the my home. Now, 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 everywhere / everything topics in real time and it is just a standard part of life.

**Fractions & ANSI Size Drill Bit Selections**

**Fractions & ANSI Size Drill Bit Selections**

**Numbered and Letter Size ANSI Drill Bit Chart**

**Numbered and Letter Size ANSI Drill Bit Chart**

** We are 'Working in here -- 05 Nov 2022!**

**Numbers / Numbering**

We teach toddlers their numbers at an early age (integers used for basic counting. Then late we start teaching other "types of numbers" and basics of number systems. Seldom do we chart out the modern number system types, so lets do a summary view here so we have a common understanding of what we use for 'number systems' and classification of a number.

Complex {Counting} _____

Real Numbers _Imaginary__

Rational Numbers Irrational Numbers

Fractions _Integer Algebraic irrational Transcendental

Finite decimal Dyadic (finite binary) Repeating decimal Natural Numbers Negative Integers

Zero: 0 One: 1 Prime Numbers Composite Numbers

Try a 'Wikipedia' search for 'number systems' for a more complete view and understanding. Each type of number has a specific definition (meaning) and has a relationship on how numbers are grouped int various 'sets of numbers' for classification. That is 18 different 'type of number' that is need for including all types. This concept is usually not presented to student in a complete view, it is shared as each piece is brought forward in teaching mathematics.

It is very 'unfortunate' that the names of 'complex' and 'imaginary' types are so named. They are NOT complex or imaginary numbers when looking at the uses of these types of numbers. Imaginary numbers represent the square roots of negative numbers, mainly consider the 'square root of minus one'. Or reverse to say "Give the number whose squared value is negative one". We simplify the number by choosing an algebraic number of "* i*" (or in electronics we use "

*" to be different from "*

**j***" which is used as a short term for current amplitude). A famous mathematician, Euler, proved that:*

**i**e^{-}^{i}^{pi} + 1 = 0 {Euler's Identity ~ search 'Euler' on Wikipedia}

and to Euler's Formula:

This brings out the concept of trigonometry to 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.