by Richard A Karlin
PREFACE
This is NOT a math course. It is too compact and terse and has no practice problems. You get good at math by doing a lot of math. This is a review of the math that a discrete component analog circuit designer must know. If there are holes in your knowledge, look for courses. The Schaum outline series is very good, and it provides hundreds of practice problems. There are many good textbooks available.
Pure versus Applied
Circuit design requires the use of applied mathematics. You will need applied mathematics for computing component values, finding tolerances, solving circuit meshes, etc. The areas of math required include some basic geometry, algebra, some basic trigonometry, and some basic calculus. What is excluded is pure mathematics itself, such as constructing proofs. Proofs such as showing that equal sideangleside means two triangles are congruent, or proving that the square root of 2 is irrational (meaning the digits go on forever). These are areas of pure mathematics and we only need applied math.
The metric system, known internationally as SI (from the French Systeme International) is universal in all things scientific. A French team devised it under King Louis XVI. It was completed and accepted under the new French revolutionary government. Wikipedia has excellent articles on Metric / SI.
Our Base Ten Number System
Our number system is base ten. This means that in a multidigit number, each position from righttoleft is worth ten times the previous position. So we write seventysix as 76, sevenhundredsixtyfive as 765, and seventhousandsixhundredfiftythree as 7653. Seventysix and fiftythreeonehundredths would be 76,53.
I am using the comma as a decimal point. This is typical European and SI usage and distinguishes decimals from sentencestop periods. Also, the comma is less likely to get lost than a period. The period is often only a single pixel and disappears rather easily. Now to make a decimal number such as 76,53 ten times larger, we need only move the decimal point once to the right: 765,3. Remember, I am using the comma as the decimal point. We now have sevenhundredsixtyfive and threetenths, and we can make a shift like this with no math, no calculator or computer. In a 10base math system, to multiply or divide by ten (or 100, or 1000, or any number that is 10 to an integer exponent), you just move the decimal point. This makes a standards system that is ten based very desirable. If the kings foot had been a little shorter, or his hand a little bigger, we might have had ten inches to the foot. This would make conversions from inches to feet (or feet to inches) a nobrainer.
The MishMash System
Unfortunately, we inherited a mishmash system. There is some perversity in the British that made them use every base possible. The chart of base usage below is not complete, but it gives the idea.
BASE USES
2—pints to the quart OR tablespoons to the ounce OR cubits to a yard
3—feet to the yard OR teaspoons to the tablespoon OR hands to the foot OR inches to a palm OR miles to the league
4—quarts to the gallon OR inches to the hand OR palms to a foot OR spans to a yard
5—fifths to the gallon (I am not sure who created the fifth, it might be French orUSA)
6—teaspoons to the ounce OR feet to a fathom
7—(I suspect they used it; I just have not found it yet.)
8—furlongs to the mile OR fingers to a yard
9—inches to the span
10—chains to the furlong
11—(I suspect they used it; I just have not found it yet.)
12—pence to the shilling OR inches to the foot
13—(I suspect they used it; I just have not found it yet.)
14—pounds to the stone
15—fathoms to the shackle
16—ounces to the pound OR nails to a yard
18—inches to a cubit
20—shillings to the pound
22—yards to a chain
32—ounces to the quart
36—inches to the yard
37—inches to the clothyard
1760—yards to the mile
5280—feet to the mile
The mile is actually not British. It was onethousand double paces (a left, then a right) for a Roman soldier. The Roman men were a little shorter than the modern male, so their pace was a little shorter. Mile comes from the French mil for thousand. The famous mile stones along Roman roads were placed every thousand double paces.
If a base does not appear above, do not assume that it was not used; rather my minimal research simply did not uncover it.
MishMash Distance (Length) Units
Just to measure distance (length) they used INCH, FOOT, PALM, HAND, NAIL, FINGER, SPAN, CUBIT, YARD, CLOTHYARD, ELL, BOLT, LEAGUE, FATHOM, SHACKLE, FURLONG, CHAIN, LINK, ROD, POLE, PERCH, MILE, … .
Metric Distance (Length) Units
The metric system uses exactly one distance (length) measure: the METER.
MishMash Conversions
There were many more mishmash measures in use, but those above should make the point. Conversions within the above mishmash are going to need paper and pencil or calculator. Linear conversions are simplest. If a lot is 50 feet long, it is 50/3 = 16.67 yards long. Area conversions can require multiplying or dividing twice. A square yard converts to 3 x 3 = 9 square feet. A square mile contains 5280 x 5280 = 27 878 400 square feet. I am using spaces instead of commas, every three numerals, both directions from the decimal. This is recommended for SI usage, and it works very well. Converting linear dimensions for a volume conversion requires cubing the conversion factor. For example, a cubic yard (a volume one yard wide by one yard deep by one yard high) contains 3 x 3 x 3 = 27 cubic feet (a cubic foot is a volume one foot wide by one foot deep by one foot high). Converting 100 cubic feet to cubic yards requires dividing by 3 x 3 x 3 = 27 to get 3,704 (rounded). (Once again, that comma is a decimal.)
The Metric System (SI)
Metric System Goals
The goals of the metric system (known internationally as SI) were to reduce the number of measuring units, and to extend those units with a system of 10 based (powers of ten) prefixes. It was a further goal to tie the units to the earth and relate them to each other. One more goal was to standardize, hopefully all of earth. They mostly succeeded. Where they failed has become unimportant in modern usage.
SI Base Units
The metric system has seven SI base units:
the meter, m for distance,
the kilogram, kg for mass, and also for weight
the second, s for time,
the ampere, A for electric current,
the kelvin, K for temperature,
the mole, mol for amount of a substance, and
the candela, cd for intensity of light.
The letters following each base name are NOT abbreviations. They are symbols, each for its unit. They remain exactly the same for all languages and all quantities. Thus, the symbol for amperes is A for one ampere or for 1000 amperes, and it remains the same in English, French, Cyrillic, Japanese, Chinese, Korean, Arabic, etc, etc. It is capitalized, because it comes from the name of a scientist. The symbol for the meter is m and because it is not from a scientist’s name it is lowercase (always lower case). The word may change in different languages, meter—metre, etc., but the symbol remains m.
SI Derived Units
The metric system currently has 22 SI derived units:
the radian, rad for angles,
the steradian, sr for solid angles,
the newton, N for force,
the pascal, Pa for pressure,
the joule, J for energy,
the watt, W for power,
the degree Celsius, °C for everyday temperatures,
the coulomb, C for electrical charge,
the volt, V for electrical potential,
the farad, F for electrical capacitance,
the ohm, Ω for electrical resistance,
the siemens, S (formerly the mho—ohm spelled backward) for electrical conductance (the reciprocal of the ohm); the final s of siemens is part of the scientist’s name, not a plural,
the weber, Wb for magnetic flux,
the tesla, T for magnetic flux density,
the henry, H for electrical inductance,
the lumen, lm for light flux,
the lux, lx for illuminance,
the hertz, Hz for the frequency of regular events,
the becquerel, Bq for rate of radioactivity, and other random events,
the gray, Gy and the sievert, Sv for radiation dose, and
the katal, kat for biological catalytic activity.
The following units are permitted:
For measuring angles: degree (°), arcminute (‘), and arcsecond (“),
For everyday time: minute (min), hour (h), day (d), and year (yr),
For volume and large mass: liter (l) and tonne (t),
For logarithmic units: bel (B), decibel (dB), and neper (Np),
For scientific use: atomic mass unit (u) and electronvolt (eV).
The total number of units may seem large, but this covers all of daily life, commerce, production, civil engineering, and science.
Each and every one of these units can be prefixed with a prefix from the list below (which is not complete because the smallest and largest prefixes are of interest only to particle physicists and astronomers) to create a vast range of sizes. There are currently 20 prefixes on the complete list. Thirteen are listed below.
SI PREFIX TABLE
NAME SYMBOL SCALE DECIMAL 1 BECOMES:
tera T Trillion times larger Magic Number 12 1 000 000 000 000
giga G Billion times larger Magic Number 9 1 000 000 000
mega M Million times larger Magic Number 6 1 000 000
kilo K Thousand times larger Magic Number 3 1 000
hecto H Hundred times larger Magic Number 2 100
deka Da Ten times larger Magic Number 1 10
no prefix Unit itself Magic Number 0 1
NAME SYMBOL SCALE DECIMAL 1 BECOMES:
deci d Ten times smaller Magic Number 1 0,100
centi c Hundred times smaller Magic Number 2 0,010
milli m Thousand times smaller Magic Number 3 0,001
micro µ or u Million times smaller Magic Number 6 0,000 001
nano n Billion times smaller Magic Number 9 0,000 000 001
pico p Trillion times smaller Magic Number 12 0,000 000 000 001
femto f Quadrillion times smaller Magic number 15 0,000 000 000 000 001
The Prefix Table Discussed
Let us have some important discussion about the prefix table. The table uses the comma (,) as the decimal (.) point. This is very SI. I have done a couple nonSI things to it which I strongly recommend. I have capitalized the K for kilo, the H for Hecto, and the D of the Da for deka. I recommend capitals for prefixes greater than 1 and lower case for prefixes less than 1. All less than 1 prefixes are already officially lower case. To achieve my goal, kilo, hector, and deka need to shift to upper case. (I think the SI committee will eventually come to this, but they tend to be a little bit slow.) This will help your brain and your readers’ brains. Brains need all the help you can give them. Hecto, deka and deci are hardly ever used in engineering or science. Scientists tend to use 10tothepowerof an integer exponents. Engineers prefer 10 to an exponent which is a multiple of 3: Kilo, Mega, milli, micro, etc. The Da for Deka turns a one letter prefix system into a mixed one and two letter system. This is very stupid. The answer is, don’t use deka. (Alternately, D for Deka and d for deci. It seems to work for Mega and milli.) There are 26 letters in the Latin alphabet, yet somehow, the designers, managed to use m three times. The SI committee has fixed this by using M for mega, m for milli, and the greek letter µ (mu) for micro. You can get a greek mu (µ) with alt code 230 (for Windows), but this is a pain. The engineering and scientific communities have fixed this by using u (which is close enough if you don’t suffer from OCD; of course, if you are truly nitpicky, the mu [µ] has a tail and the [u] does not, but if you have human friends, consider that they are tailless). The SI committee would do well to formalize this as an acceptable alternate. The medical community has fixed this by using mc, a two letter prefix, for micro. This is a regrettable error, and I hope they fix it.
Meter, Millimeter, Centimeter
An exception to the rule of 10toanexponentwhichisamultipleof3 is centi. For many everyday and lab length measurements, the meter is too large (a meter stick is about 40 inches—10% larger than a yard stick. See a yardstick in your closed eyes. Now next to it, a meter stick just 3.6 inches taller. For most purposes, you can consider the meter to be 10% larger than the yard. You can consider the yard to be 9% smaller than the meter. Now you own the meter—it is yours forever.) The millimeter, one thousandth of a meter, is the thickness of about 58 sheets of 20# bond paper. The millimeter (mm) is too small a unit for most everyday and lab measurements. The centimeter is the thickness of a thin finger (thickness, not width which is larger!). This is a really dandy unit for around house or lab. There are 30.5 (rounded) centimeters to the foot. The centimeter (cm) is to metric what the inch is to the footpoundetc system. There are 2.54 centimeters to the inch. The centimeter will never go away.
Use the Prefix to Track the Decimal
The whole purpose of the prefix is to let us keep track of the decimal point. Let us say something is 0,000 001 meter thick. We can write this as we just have. What about 0,000 000 000 001 meter thick. Counting zeros now, are we? Well, there is always 10 to some power. For the first one, 10^{6} meter and for the second one 10^{12}meter. It works, but it takes a lot of key and/or mouse stroking. How about, we use 1 micrometer thick for the first one, and 1 picometer thick for the second one. A small capacitor used in electronics can be expressed as 0,000 000 000 010 farad or 10 picofarad. Which of the foregoing two works better for you? What the prefix is doing is keeping track of the decimal for you. It also gives you a feel that the number alone misses. I confess that 0,000 000 000 010 farad does not do it for me. 10 picofarad resonates. The image of the part instantly springs up in my head.
SI is Cool
Note again the use of spaces between 1000 groups and comma for decimal point, and no plurals, no matter how many there are. It is cool to be very SI. (I will slip many times because I learned electronics long before most of the SI rules came about. Retraining the brain is a drag, so learn SI from the start.)
The Prefix Magic Number
This takes us to the Prefix Magic Number. If you use a prefix to change the size of the unit, like meter changing to millimeter, since the unit got much smaller, the same measurement will have to yield a larger number. So we have something which is 1 meter long, and we want to go to millimeters. But it takes 1000 millimeters to cover the length of 1 meter. So we need to move the decimal point 3 places taking 1,0 to 1000,0. Size of unit goes down, number of units goes up. Size of unit goes up, number of units goes down. You can do a bunch of thinking about it, or you can just learn that little mantra and apply it. I recommend the mantra. It leaves the brain free for other, more important thinking.
Memorize the magic numbers along with the name and symbol. The magic number tells you how far the decimal moves. Meter has a magic number of 0 and milli is 3. The decimal must move three places, and since milli is smaller, number must get bigger, so move the decimal three places to the right and 1 becomes 1000. Remember our capacitor? It was 0.000 000 000 010 farad. If we restate it in picofarad, pico has a magic number of 12, so it becomes 10 picofarad. Suppose we have a capacitor that is 100 millifarad, and we wish to restate it as farad. Bigger unit, smaller number. Milli is 3 and farad (no prefix) is 0, so we move three decimals smaller (left) and get 0,1 farad. Note that there are no plurals in measurements. 10 picofarad, NOT 10 picofarads. So now we have 0,001 millimeter and want to get a whole number (1 or larger). We need to move the decimal 3 places, and we are coming from milli which is 3. If we go to micro which is 6, we get a difference of 3 and we are going to a smaller unit, so we will get a larger number. 0,001 millimeter becomes 1,0 micrometer, a three position shift.
Relating SI Distance to the Earth
The meter was originally supposed to be onetenmillionth of the distance from the North Pole to the Equator along the meridian that passes throughParis. Limits of the then technology made a truly accurate survey of this distance impossible, but they got pretty close. Around the equator, the earth is 40 075 kilometer, so onequarter would be 10 019 kilometer. Around the poles, it is a little less because spin makes the earth a little bit apple shaped. But their number was very respectable. The standard was a bar of platinumiridium engraved with two lines, 1 meter apart. The meter standard today is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. We have advanced.
Related Units: Volume, Distance, Mass
Volume is really not a separate unit, because it is a container of a certain size as measured with the meter. For convenience, a container 1,0 decimeter on each side, which works out to be 10 by 10 by 10 centimeters, thus 1000 cubic centimeters, (symbol cc,) is called a liter (symbol L). The liter is just a little larger than the quart: 5,7% larger. For purchasing and storing purposes, a liter is a quart, a quart is a liter. You now own the liter—it is yours. SEE!! In no time at all, you have come to really know the liter and the meter. Who said metric was hard??
The milliliter (ml) and the cubic centimeter (cc) should be identical. In fact, there is a standards error of 3 parts in 100,000. I wouldn’t sweat it. Pretty much, only standards people themselves work to 3 parts in 100,000. (Well, plus a few physicists and a few astronomers, but not us circuit types.) Part of the tying together of units was tying the volume to the linear, and part was that a cc or a ml of water weighs 1 gram. This is handy.
The kilogram was defined to be the mass of one liter (1000 cc) of water at (initially 0°C—0 degrees Celsius), revised for practical reasons, to water at 4°C (4 degrees Celsius, which is where water has its greatest specific density). Today, the kilogram is equal to the mass of the international prototype of the kilogram. It is a physical standard, presumable platinumiridium. The kilogram is bigger than a pound: 2,2 times bigger. So close your eyes, and on a two pan scale see 1 kilogram on the left pan and 2,2 pounds on the right pan. It balances. You now own the kilogram. Double plus 10% gets you pounds. For many uses, just 2to1 works.
The Second
Initially, the second was defined as a fraction of the mean solar day. Fortunately for us, the earth rotates pretty regularly, but it was not quite regular enough, so a new definition was based on atomic behavior. Now, the second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.
An SI Caution
Do remember that micro in medical use is mc, at least for now.
SI Road Distances
For longer distances, SI uses the kilometer (symbol Km). It stays with the kilometer for all road distances. 1 Km equals 0,621 37 mile, or rounding 0,62 mile. Going the other way, 1 mile converts to 1,61 Km. If you use 0,6 and 1,6 for conversions, the error is only a few percent.
UNDERSTANDING MATHEMATICS
Problems in the Math World
Some of what we call math is really just shorthand notation. Let us start with the real numbers. This includes whole numbers, rational numbers (whole number fractions such as 5/6 9/10 112/254 etc. Anything that can be written as a ratio of integers), decimals (including irrationals – numbers which never end, such as Pi, the square root of 2……), both positives and negatives of all types, and zero. You may be surprised to learn that the introduction of negative numbers and of zero were both extremely controversial, with the best minds then around protesting both. How, they shouted and screamed, could you have negative apples? How could you have zero anything? How could these brilliant men get so confused? In part they were worrying about formal theorems and proofs. In the applied world, if I am giving a luncheon for 20 persons, and I want each to get an apple, and there are 17 apples in the pantry, I am short three apples. I understand minus 3 apples (3 apples!). If I go to My clean socks drawer, and it is empty, I understand zero pairs of clean socks! Yet all these famous brilliant minds were terminally confused!! A flip answer is that it is easy to educate someone beyond their intelligence, so try to bring your IQ up as you become more educated!! In the applied world, if it gives us useful answers, it rocks. Applied math begins with counting:    . One can optionally diagonally cross every fifth one. Next is learning to represent the count with symbols. 5 5 5. Next we go to a decimal representation: 15. A larger one: 1 234,567 8. I am still using a comma as the decimal point and a space to separate the numbers into three digit groups. The first math operation is addition: 12,34 + 34,63 = 46,97.
The second math operation (arguably) is sign change. 46,97 becomes 46,97 and 46,97 becomes 46,97. The third math operation (arguably) is subtraction, but one can eliminate subtraction by using a sign change for the second number and then addition. (One can argue that there is a zeroth math operation: counting. See!! Zero is good for some things.)
Math or Shorthand
Next is multiplication, but is it a new operation, or just a shorthand. Let us do the product 47,97 x 23,12:
479,7
+479,7
+ 47,97
+ 47,97
+ 47,97
+ 4,797
+ 0,4797
+ 0,4797
——————————
= 1 109,066 4
So is it a new operation, multiplication, or is A X B shorthand for add A to itself B times??
De We Care Which It Is??
If we are doing applied math, do we really care which it is?? Punch it into a scientific calculator and out comes the answer. Of course, knowing how to set the problem up, how to punch it into the calculator and how to interpret and verify the answer are all important! As we move forward, we will see a lot more of what is (arguably) shorthand, and we do not want to be frightened or confused by something which for us is only shorthand. Remember that no one is asking us (or looking to us) to verify mathematics. We are just a user of mathematics.
The Factorial—More Shorthand
Another famous shorthand is the factorial, written X!, for X factorial. The factorial (symbol !) is shorthand for the product of all integers from 1 to the factorial number. So 3! = 1 x 2 x 3 = 6. 4! = 1 x 2 x 3 x 4 = 24. Why start at 1? Well, if you start at 0 all factorials = 0. Not useful. What use is the factorial?? Probability theory. You have five cards: star, diamond, spade, club, heart. How many ways can you arrange the five in an inline order. (Only order changes). You have five choices for the first position, four for the second, three for the third, two for the fourth, and only one for the fifth and last. Total is 5! (5 x 4 x 3 x 2 x 1 = 1 x 2 x 3 x 4 x 5 = 5!). 5! = 120, so there are 120 ways to arrange your hand. Unfortunately, all 120 hands rank exactly the same. The number of different (different by cards held, not by card order!) seven card hands you can be dealt from a 52 card standard deck is 52! / (45! X 7!) = 133 784 560 (rounded). Some of these hands will have equal rank.
Different Fields—Different Displays
In engineering practice the answer above, 133 million, would get written as 133 x 10^{6} . This requires several mouse or keyboard visits to format and is a pain. An alternative is 133 x 10 exp 6. 10 exp 6 signifies 10 to the power 6. Another option is 10^6. The caret is uppercase 6 on the standard QWERTY keyboard. It appears just to the right of the base (follows the base) and is followed by the exponent. Thus onemillion can be written 10^6. Any of these will work, but metric does this with a prefix. Our unit here is a sevencardhand. (A derived unit; clearly not an SI base unit! Perhaps we can invent the symbol 7ch for sevencardhand.) Our answer in SI terms is then 133 M7ch, or 133 mega sevencardhand. NOTE: Capital M for the mega symbol (to diff it from milli or micro) and no plural s.
Exponent (Power) Shorthand
The power superscript (example:^{ 6 }) or its equivalents exp 6, or ^6 is another shorthand. Just as 2 X 3 means add 2 three times, 2^{3 }means multiply 2 three times ( 2 x 2 x 2 ). In this case, we get 8. The number to be multiplied (2 in the last example, 10 in the paragraph above) is called the base. The number of times to multiply the base and other important details are controlled by the superscript, called the exponent or power. So 10^{3} has a base of 10 and an exponent of 3. In this case, the base and exponent are both real positive integers, but this is not a restriction as we shall see. We can also write this as 10 exp 3 or as 10^3. We can pronounce this as base 10 raised to power 3, base 10 raised to exponent 3, 10 to the third power, 10 to the third, or 10 cubed. 10^2 is often referred to as 10squared, because the linear measure ten has been used to define a 10 by 10 square area, and 10^3 is called 10cubed because the linear measure 10 has become a 10 by 10 by 10 cubic volume. There are other variations. English, she is a very flexible language; yes?? The important point here is that 10^{3} is shorthand for 10 x 10 x 10. This should in no way puzzle us or cause us fear.
Stuff about Bases
Further, BASE^{1}, which can also be written as BASE exp 1, or BASE^1 equals BASE, for any BASE. A mathematician would refer to this as an IDENTITY. BASE^{0 }also written as BASE exp 0, or BASE^0 is equal to 1, for any BASE. Why?? The short answer is, it is useful. It works. We will come back to the long answer.
BASE 
BASE NAME 
To exp 0 
To exp 1 
To exp 2 
To exp 10

2 
BINARY 
1 
2 
4 
1024 
2.718 
NATURAL 
1 
2.718 
7,389 
22,026 000 
10 
DECIMAL 
1 
10 
100 
10 000 000 000 
A few things about this exponent shorthand. First, there are only three bases in common use: Base 2 (Binary) for all things computer or “IT” (Information Technology), Base 10 (Decimal) for all things everyday, and Base ε (The natural base signified by Greek letter epsilon—alt 238 on IBM—but usually changed to a lower case e for typing convenience) for all things natural and scientific. Base e is an irrational number. The value of e to 10 places is 2,718 281 828. The comma is being used as a decimal point. In use 2,718 usually suffices. Some calculators save display space by showing something like 4,56 x 10^{6} as 4,56E6. You are supposed to understand and fill in the x10 part in place of the E and understand the 6 as a superscript. It may be confusing at first, but it does save display space. Scientific calculators usually have an EE key for Exponent Entry. Be aware that such calculators automatically assume that the base is 10 and they are only equipped to receive integers (whole numbers, plus or minus, usually including zero). No fractions. No decimals. Most of these calculators require that a negative be indicated with a special () key or a changesign key, NOT with the subtract function key. If you want to enter a non10 base or use a fraction or decimal exponent, locate the X^{Y} key and use it. It may appear as ^ indicating you should enter BASE^EXPONENT, for example 4^2.5= on a TI30X IIS produces 32. More on this below.
Hopefully, the author will always tell you what the base is. If not, it may be clear from context. Make your bases clear to your readers.
Adding or Subtracting with Numbers Expressed As Powers
The short answer is, you cannot. Let us say that we are told the company annual budget will be 56,090 x 10^{6 }dollars (56 million, 90 thousand dollars). IT informs you that it is adding an approved server purchase for $20 000. How do you add 56 x 10^{6 }and 20 000?? You do not. You must convert the power shorthand to a fully written number: 56 090 000 dollars. Now you can add the 20 000 dollars. You will get: 56 110 000 dollars. (56 million, 110 thousand dollars). This can be written as 56,11 x 10^{6} dollars. An alternative is to convert the 20 000 into 0,020 x 10^{6}. Now you can add 56,090 to 0,020 to get 56,110 x 10^{6}. We can only add or subtract if the power suffixes are identical. The same rule applies to metric prefixes. If something weighs 1,23 mega gram and the packaging weighs 13 kilo gram, to get a total for shipping, we must convert both to either kilo gram or mega gram.
Multiplication in Power Format
Multiplication is a whole different story. Say we want to double our 56,11 x 10^{6} budget. We just do 56,11 x 2 = 112,22 x 10^{6} dollars, and we are done. Now suppose we have one million of something and we want to repeat it 1000 times. One million is 10^{6 }and 1000 is 10^{3}. Their product is 10 ^{(6+3)} = 10^{9}. Why?? Well, we have ten multiplied 6 times and then another 3 times for a grand total of 9. Sure enough, a million times a thousand is a billion which is 10^{9}.
Multiply by Adding—Logarithms
We have discovered a way of multiplying by adding. In its full form it is called logarithms. Before the creation of all these lovely pocket sized, battery operated, calculators, sales people had books of logarithms. To multiply A x B you would find the logarithm of A, add it to the logarithms of B, and find out what number {log (A)+log(B)} represented. That number was the product of A x B. There are two types of logarithms: natural logs use Base e (2,718 etc.) and common logs use Base 10. The logarithm of a number is the power you have to raise the base to in order to get that number. So, using common (Base 10) logs, the log of 10 is 1 because 10^{1} = 10, and the log of 100 is 2 because 10^{2} = 100. For commerce the Base was 10.
How about Base to the 2,5 power (Base^{2,5}) or 1,5 power??
Reminder: This: , is a decimal point!!
Let us start with BASE^{1/2}, also written BASE exp ½, or BASE^1/2. This is the square root of BASE. If the exponent is a fraction, the numerator is a power and the denominator a root. NOTE: FRACTION!! NOT DECIMAL!! So BASE^1/2 is the square root of BASE. 10^{5/2} is the square root of 10 to the fifth power. 10^{1,5} is the square root of 10 to the third power, 10^{3/2 }= (10^{3})^{1/2 }= 31,6 (the square root of 1000). NOTE: THE DECIMAL WAS CONVERTED TO A FRACTION!! To understand 10^1,5, you may well need to convert the 1,5 to 3/2. Now you have the square root of (10 cubed), which is the square root of 1000 = 31,6. Unless you are log scale familiar, do not try to work from the decimal. Cherish this. It is probably the only place in all the universe where fractions have any purpose or value. {If you are log scale familiar 10^1.5 lies between 10 and 100 and is half a log decade up from 10. The midpoint of a log decade is 3.1 and 10^1.5 is indeed 31.}
Your scientific calculator should understand decimal exponents such as 2,5 1,5 or whatever, perfectly well. If not, you need a new calculator. It is only us people who will have trouble interpreting. (Caution: This assumes you enter numbers into the calculator using Y^{X} or ^ and not using EE or EEX. EE and EEX and any other of their ilk automatically assume BASE = 10 and understand only integers!!!! If you enter something like 2 EE 1.5 you will probably get something like ERR, ERROR, or SYNTAX ERROR. These Scientifics are smart, but not that smart.)
Exponent Addition and Multiplication
We made the point earlier that to add A to B where A and B are of the form A = (C x BASE^EXPONENT), and B = (D x BASE^EXPONENT) both A and B had to have identical (BASE^EXPONENT). The sum is then (C + D) (BASE^EXPONENT).
But now, suppose we have (BASE^EXP1) x (BASE^EXP2) We are looking for a product, same base but different exponents. (BASE^EXP) is BASE x BASE x BASE x …EXP times. Thus, 8 EXP 4 is 8 x 8 x 8 x 8 = 4096. 8^3 is 8 X 8 x 8 = 512. The product of these two multiplication chains is 8 x 8 x 8 x 8 x 8 x 8 x 8 or more compactly stated, 8^7, just the sum of the exponents (4+3). If there are coeffecients C and D before the BASES, then we get (C x D) (BASE^(EXP1+EXP2)).
For example (4,5 x 22^3) x (2,2 x 22^5) = (4,5 x 2,2) x (22^(3+5)) = 9.9(22^8) = 5,43 x 10^11, a sizeable number.
Two rules you need to remember:
When faced with nested parentheses as just above, work from the innermost outward. Thus for the above sum EXP1 and EXP2, and then raise BASE to their sum. If faced with a BASE raised to a BASE which itself is raised to some power, if there are parentheses follow the innermost to outer rule, if there are no parens work from the top (highest or rightmost superscript) down. Example: 2^(2^8) = 2^256 = 1.158 x 10^77. If you do it wrong, you get 4^8 = 65 536. This is a gigantic difference. Faced with 2^2^8 with no ‘parens’, following the rightmost/highest rule, we get 2^256 = 1.158 x 10^77 which is correct for no parens, so if you want some other operation order, use parens!!!
Exponents: The full Monty—Negative Exponents
If you thought there had to be more to exponents, you are SOOO right. So far, we have only looked at positive bases with positive exponents, both in real numbers (meaning no imaginaries). Suppose the base is positive, but the exponent is negative. For example, 10^2. This is 1/10^2 = 1/100 = 0,01. The negative sign gives us the reciprocal of the raised base. 10^Exponent 3 = 0,001. 10^Exponent 0,5 = 0.316. Stated another way: BASE 10 to the +2 (10^+2) = (10 x 10) = 100. BASE 10 to the 2 (10^2) = (1/10 x 1/10) = 0,01. So positive exponents control the number of base multiplies and negative exponents control the number of base divides. Suppose we have 5^3 x 5^2. Per the foregoing, this should be 5 x 5 x 5 ÷ 5 ÷ 5. (The division symbol, named the obelus, ÷, alt code 0247 on windows.) The answer is 5^1, or just 5. If we add exponents, we get 5^(32) = 5^1 = 5.
So our shorthand, for all real numbers and for positive bases is correct, consistent, elegant, useful, and in short, it works. As applied mathematicians, that is all we can ask.
Negative Bases
So what happens if the base is negative?? Here things can get very confusing. If the exponent is a whole number, even integers (2, 4, 6, etc.) should produce a positive because our standard math is that a negative times a negative is positive. Odd integers (1, 3, 5, etc.) should produce a negative. But, we are at the tender mercies of the calculator. The TI 30X IIS produces all negatives. However, it does come up with a correct numeric, and, after the answer is delivered, the user is free to flop the sign as needed. The TI even comes up with a correct numeric for negative base and positive noninteger exponents, such as 1.5 or 2.5 (but negative base and negative noninteger exponent produces ‘syntax error’). Again, the sign is a task for the user. There is also a legitimacy issue here, because interpreting 10^1,5 as the square root of 10 cubed should produce +or i(31,623); an imaginary number and both a + and a – root.
The Greeks were Builders
The early Greeks were builders, and those who would build must measure. One of the first of their learnings was the theorem of Pythagoras. This theorem applied to triangles which had one 90 degree angle. Such an angle is called a ‘right angle’, and a triangle having one is a ‘right triangle’. Since every triangle has to have 180 degrees total angle, a right triangle has 90 degrees combined in the two nonright angles. Thus, once we specify an angle, all three angles are fixed. Thus, a right triangle with a 30 degree angle has angles of 30, 60, and 90. The shape is now totally fixed. Specify the length of any one side and the triangle is totally defined.
(Trig may have come to the west via India and Arabia—translating from Sanskrit to Arabic to Ancient Greek, but Pythagoras is credited with C^2 = A^2 + B^2 and I find it hard to believe one could develop trig and not develop C^2 = A^2 + B^2.)
The Very Famous Pythagorean Theorem
Pythagoras learned that the sides of a right triangle a, b, and c, were related by
csquared = asquared plus bsquared.
c^{2 }= a^{2 }+ b^{2}, or in caret notation, c^2 = a^2 + b^2
So for any right triangle, if you have any two sides, you can compute the third. c is the hypotenuse, the side opposite the 90 degree angle.
Given: a right triangle with A = 3 meters and B = 4 meters. How large is the hypotenuse c?? a^2=9, b^2=16, a^2 + b^2 = 9 + 16 = 25 = c^2
Therefore, c = square root of 25 = 5 meters. This can be considered the start of trigonometry.
The Triangle
Basic trigonometry deals with right triangles, just as the Pythagorean Theorem does. Our early Greek builders realized that if you took a right triangle and specified one of the other angles, let us call it angle A, and set it, for example to 30 degrees (1/3^{rd} of a right angle, in this case), you would have specified all three angles since they must add to 180 degrees. So if one of them is 90 degrees, and one of them is 30 degrees, the third must be 180 90 30 = 60 degrees. So the only difference between this 306090 triangle and all other 306090’s is size. They are all similar but differ in size. The angles of a triangle are usually labeled A B and C. For a right triangle, C is used for the right angle. A and B will add to 90 degrees. The sides are usually labeled a b and c, each according to the angle it is opposite. Thus the hypotenuse is opposite C, the 90 degree, and is labeled c.
Angle Measurement
There are three measurements in use for angles. Most scientific calculators can be set for any one of them.
They are:
Degrees, Minutes, Seconds (But both my TI and HP Scientifics accept decimal degrees). There are 360 degrees to a circle, 180 in a triangle, and 90 in a right angle. The degree symbol ° is Alt 0176. Each degree comprises 60 minutes. The minute symbol is either the prime or the single quote: ‘. Both degrees and minutes can be decimal: 30,123 degrees (30,123°) (still using the SI decimal point: ,). For minutes, 12,34’ . Finally each minute comprises 60 seconds: 60” (symbol, double prime or double quote.) We owe this sexagesimal (base 60) system to the Babylonians. Alas, they are no more, so we cannot return it. Our time system is also Babylonian sexagesimal: hours, minutes, and seconds.
Grads, also called gradians, or gons, are each 1/400^{th} of a circle, thus there are 100 to a right angle. This is handy because all trig functions cycle every 90 degrees (once through a right angle). The symbol for grad is now gon, but in the past, it has been g, gr, and grd. The grad / gon is always decimal (no sexagesimal, you will have to get your 60’s fix from your watch).
Radians are the angle measurement units preferred by mathematicians. There are 2 π (alt 227) radians (symbol rad) to a circle. If the radius of a circle rotates through a 1 radian angle, its tip moves over a length R (one radius) on the circumference. A half circle (180 degrees) is Pi (π) radians. A right angle (90 degrees) is Pi/2 radians (π/2). As one gets into more complex math, radians are required.
Degrees and grads (gon) present a problem. They are very close to each other. A wrongly set calculator will deliver up wrong answers which may appear to be correct. This can happen with radians, but it is less likely. Make sure your calculator is set to the angle measure you are using. Degrees are entrenched in navigation and maps. Radians are entrenched in math. The grad is still around, and it is the official SI angle unit, but it is on life support.
Beginning of Trig
Image a long wood or stone beam flat on the ground. This is the triangle b side. At one end imagine another long beam tipping up at 30 degrees (π/6 radians) this tip up angle is the A angle, and the beam is the c side (hypotenuse). The open side, opposite angle A is side a. As you walk down the beams and run a plumbbob down from c to b, thus making the plumbbob string side a, you realize that the ratio of side a, the plumbbob length, to side c, the tippedup hypotenuse, is constant. It is 0.5 (= a/c) and will be for any 30 degree angle anywhere you drop a plumbbob. The ratio will be different for any angle A. We call this ratio sine (A), or for short sin (A), or on calculator buttons just sin. It is zero for angle A = 0 degrees, (a parallel with c). It is 1,0 for a perpendicular to c (angle A = 90 degrees, approachable but not quite allowed). Sin (45) = 0,707. Sin (60) = 0,866. Once you have the Pythagorean Theorem and you are building and measuring triangles, the trig relationships sort of kick you in the face. It is hard to miss, especially with 30 degrees yielding a twotoone between c and a. Pythagoras dates to about 500 BC. If trig came west fromIndiaviaArabiais about 450 AD, it took the west over 900 years to spot something poking them in the eyes. I find this hard to believe. I suspect independent development.
Six Named Trig Functions
There are six named trig functions, but in fact there is really only one: sin (A) says it all. The cosine is side b over side c (side along the angle divided by the hypotenuse). But cos (A) = sin (90A). So cos is not an independent function. Tangent is side a over side b, but this is sin (A) / cos (A); sin (A) / cos (A) = a/c ÷ b/c = a/b = tan (A). {{A familiar relationship is sin^2 (a) + cos^2 (a) = 1. This comes from a^2/c^2 + b^2/c^2 = 1 which becomes a^2 + b^2 = c^2 by multiplying everything by c^2, and low and behold, the Pythagorean Theorem is with us again.}} The remaining three functions are all reciprocals of the three we have just learned. Cotangent (cot) is 1/tan = b/a. Secant (sec) is 1/cos. Cosecant (csc) is 1/sin. My HP49 and TI30X do not have buttons for cot, sec, csc. It is easy to find the reciprocal and invert it with the X^{1} function, and there are better uses for buttons.
“Inverse” Functions
Pay attention here. If you can go from angle to side a ÷ side c ratio (sine), you can take the sine and go back to the angle it came from. Thus sin (30) is 0.5, A side a to side c ratio of 0.5 (in other words, a sine of 0.5) says we have an a angle of 30 degrees. Scientific calculators provide sin, cos and tan, and they provide the inverse of sin, cos, and tan. My HP49 calls these asin, acos, and atan. These are arcsin, arccos, and arctan, though I prefer antisin, anticos, and antitan. Put a ratio in, like put 0.5 in, hit asin, and you get 30 degrees. My TI30, in what has to be a stupidity which makes three metric prefixes beginning with m, or medicine’s mc for micro seem almost sane, calls these inverse keys sin^{1}, cos^{1}, and tan^{1}. This illogic is found on many scientific calculators. You just have to persuade your brain that these 1 superscripts have nothing to do with exponents. Hopefully either HP’s axxx or something like a left caret <xxx will replace these misguided notations.
There are inverse operation pairs all over math. Subtraction is the inverse of addition. If 3+4=7, 74=3. Of course, equally addition is the inverse of subtraction. They are a pair. Division is the inverse of multiplication and multiplication the inverse of division. For example, 5 x 6 = 30 and 30 ÷ 6 = 5. Division and multiplication are a pair. Taking a root of a number is the inverse of raising it to a power. 2,3 raised to the 3^{rd} power is 12.167. The third root of 12,167 is 2,3. Another pair. Log and antilog are a pair.
These last two pairs run into a little trouble with signs. Raising to even powers loses the sign. For example, both +2 and 2 yield +4 when squared. Therefore, when we take the square root of +4, we have to state it as + or – 2, because we do not know what it once was. This problem exists for all even powers. There is disagreement on what the log of a negative number is and whether it even exists. In many cases, you can just make it the same as the log of the equivalent positive number. Thus, the log of +100 is +2 (10’s base). The log of 100 can be set to +2.
In trig we have six more inverse pairs. We have sine and anti sine (arcsin), cosine and anti cosine (arcos), tangent and anti tangent (arctan), cotangent and anti cotangent (arccot), secant and anti secant (arcsec), and last, cosecant and anti cosecant (arccsc). Use arc or anti, whichever is clearer to you. For brevity, you can even go to a simple A. For example, asin for arcsine. HP prefers the A which is compact and very clear.
If you need to show something like 1/sin(30), use parens: (sin(30))^{1 }The outer parens makes it clear that the sin function of 30 degrees is what is being inverted.
Scientific Calculators understand parens. For example, to raise 10 to a fraction with the TI, I can enter 10^(5÷2)= and I get 316,227…. (REMINDER: That comma is a decimal point.) The division symbol on the TI display will be a fraction bar, not an obelus, but you get the fraction bar with the divide function symbol, which is an obelus. Thus (4÷8)+(1÷8)= yields 0.625 (which is 5/8 in fraction form). If you want fraction format for an answer, you need something like the HP49.