Math-speak

[R.G.]

This one is for the guys who I regularly mystify with equations.

Math is a special jargon. The concepts are simple, but like any highly specialized technical field, they have invented special words for the unique processes, and the words are confusing because they seem to be normal words, but they have another special meaning. Part of the confusion is that math language also uses special symbols that are not available on a computer keyboard in many cases, so they get posted as non-standard, even for math. What follows is what I tend to use, having picked it up in a long period of math courses, computer science courses, and professional practice. The colon ":" separates the symbol from the definition.

= : this is the equal sign. It means that two numbers are the same value, as "Vbias = 4.25" means that if you measure Vbias, it will have a numeric value of 4.25 Volts. Pretty much everyone gets this one.

+ : Plus sign or addition. It means take the two numbers it separates and add them together. In algebra, it may not be possible for you to know the numbers yet, so it means "when you finally know what x and y are, add the two numbers to get the result". It is important to separate this meaning from the other meaning - "greater than zero, or positive, or positive polarity". In one context, it means an action (addition) and in the other it means a condition (greater than zero). There is no way other than by context to sort these out as the symbol is the same.

- : minus sign. It means to subtract the two numbers it separates, much like the plus sign means to add. Likewise, it is also used for "less than zero, or negative, or negative polarity". Context is important.

* : multiplication. It means "multiply the two things together". Example, 2*3 = 6; also 2*X will be twice the value of X when we eventually find out what number X represents. Very often, we get tired of writing 2*X or the equivalent. There is another version of this notation that uses a dot, like a period, but up in the middle between two numbers to indicate that we should multiply. And last, mathematicians get lazy too, so instead of writing all the *s and dots, they sometimes just leave them out, writing 2*X as "2X". If you happen on something like "2C" it means "multiply C by 2". This version, with no dots or asterisks is always used with symbols like X and Y, because "24" is ambiguous. Is that twenty-four or 2*4? It's always take as twenty-four, not 2*4. But 2X is "two times X".

/ : division. It means "divide the first number by the second one". This is a bit of a trick, because while the addition, subtraction, and multiplication symbols are not order sensitive ( that is, 2+3 is the same as 3+2 and 2*3 is the same as 3*2) division IS order sensitive. 2 / 3 is not the same value as 3 / 2.

Which brings up ordering.

The order of operations matters.

To properly evaluate what a mathematical expression means, you have to get the order of operations right. The correct order is:
1. Do all the multiplications
2. Do the divisions
3. Do additions and subtractions

For instance, if X = 2*3+4/5 we get different answers if we just go left to right than if we do multiplication and division first. In one case, (left to right) we get X = 2 and the other (multiply and divide first, then add) we get X = 6.8. If you see something without parentheses to force ordering, do all the multiplies and divdes first, then the adds and subtracts. In an algebra classe, the right answer for X would be 6.8.

Which brings up grouping.

( ) : the grouping operator. This means "carry out all of the math inside the parentheses first, then use that as a single number for what's outside it. So if X = 2*(3+4)/5, X = 2*(7)/5 = 14/5 = 2.8  and if Y = (2*3)+(4/5), Y = 6+0.8 = 6.8, which is what we got from the rule about doing multiply and divide first. 

Sometimes when we're calculating stuff we don't know all the numbers. For instance, I've already been mathematically impure by using "2*Y" and "2X" to explain things. My bad. If we have a situation where we know we're going to come out with some number, but we just don't know what the exact value is, we just use a symbol, like a single letter. For instance, how tall are you? I don't know exactly either. I'd have to go measure. But if I wanted to calculate how many times as tall as I am that a football field is long, I could do it the lazy way and write:
N (which stands for how many times as long as I am a football field is; we get to use symbols however we feel like it. 8-) )... er...
N = 100yards*3 ft/H where I'm using H as my present height. So I can write up the math symbol, go measure my height, and plug that number in for H and get the final result.

A symbol is just a number you don't know the exact value of yet. We use it in math because we can then do a lot of wrangling around with the symbols to get things down to a final form that's easier to calculate than where we started, and we can do all that without running out and measuring things first.

Now off into the fancy stuff: exponentiation and roots. The square of a number shows up all the time in electronics math. It's usually written in normal math by using a superscript, a smaller number written to the right and near the top of another number or symbol.
X² : means "X squared" which in turn means "multiply X by itself". So if you have X =4, X²=16. If the special superscript font comes out, you'll see a capital X with a littler "2" near it's top right corner. Not all computer fonts can do that, so there's another way it's written - the exponentiation operator.

^ : means "raised to the power of..." as in 2^3 means "two raised to the third power" or 2*2*2 = 8; and 3^2 means "three raised to the second power - which is the same as squared" or 3*3 = 9. So it's common to write "squared" without the superscripts as "X^y" where X is the number to be exponentiated and y is the power it's to be raised to.

What happens if the exponent is not greater than one? It turns out that raising a number to a fractional power is the same as taking it's root.

Right. What's a root?

If I have a number, like, say 4. What is the square root of 4? By now you've figured out that mathematicians use "square" for "second" a lot when they're talking about exponents. This is the same as saying "what is the number which when squared equals four?". Or
another way is X^2 = 4. X is called the "square root of 4". It's the same as four to the 1/2 power. We know this one by experience: 2*2 = 4, so 2 is the square root of 4. This can be expressed two ways:

2 = 4^0.5

or

2 = SQRT(4)

This second way is what was adopted back in the days when computers still had tails and there were NO printers that could make a square root symbol the right way like there are today. But it is unambiguous. It's pronounced "The square root of four". Or X or whatever is inside the parentheses. Again, parentheses are for grouping. Calculate whatever is inside the parentheses and then take the square root.

So - I hope this helps some of you with my blithering on in math-speak. To those of you who already know this, I apologize for boring you to tears.

[leonhendrix]

Jeremy: It's just simple math!

Mark :...Sss! It's maths

[heyniceguy]

and dont ever.... ever... ever.... divide by zero.

[rockgardenlove]

All this stuff I've got down...the part where I get lost is in all the little abbreviations.  Stuff like Vbe and so.  (Though I *think* I know that one.)

[Joe Kramer]

Hey RG!

Humbling to say, I'm one of the guys you must have had in mind here.      Just south of the minus and plus signs, it's Greek to me.  I'm copying this to my reference files.  From my seat at the back of the class, I thank you for sharing and teaching. . . .

Remedial Joe

[Mark F]

Thanks for posting that, R.G. I'm sure it helped alot of people & not just us who come to school on the "Short Bus"

[johngreene]

and dont ever.... ever... ever.... divide by zero.

But that's where all the mojo is!

[johngreene]

2 = SQRT(4)

And in electronics, complex impedance is R + jX where j = SQRT(-1)

guess that's why they call them 'imaginary' numbers!



--john

[petemoore]

  Has this been put in the additions to FAQ yet?
  One I'm trying to get past is "l vee l''
  ..as used in calculating the mute pin of LM3886's...the upright lines look longer than symbols ['l'...which is lower case L's] I used, what does it mean when two straight, upright lines are 'parenthesizing' letters...and
  Vee...I should be able to figure out what that is...

[Seljer]

2 = SQRT(4)

And in electronics, complex impedance is R + jX where j = SQRT(-1)

guess that's why they call them 'imaginary' numbers!



--john

http://xkcd.com/c179.html

[johngreene]

  Has this been put in the additions to FAQ yet?
  One I'm trying to get past is "l vee l''
  ..as used in calculating the mute pin of LM3886's...the upright lines look longer than symbols ['l'...which is lower case L's] I used, what does it mean when two straight, upright lines are 'parenthesizing' letters...and
  Vee...I should be able to figure out what that is...

That means 'absolute value' or just the magnitude. What they are saying is ignore the '-' of the Vee and just add the voltages together as though they were both positive. Another way of saying it would be Vcc -(-Vee) < 84V. With regard to the mute pin, they are saying ignore the minus sign of the Vee voltage when calculating the resistor value using the formula.

--john

[jrc4558]

I thought: i = SQRT(-1) 
No wonder internet search turned up nothing...

[Seljer]

I thought: i = SQRT(-1) 
No wonder internet search turned up nothing...

from Wikipedia

j : "The imaginary unit (\sqrt{-1}), in fields such as physics and electrical engineering where i is traditionally used to denote a changing current"

i was already taken, so they just moved one letter forward

[R.G.]

Symbols are a big part of algebra-speak.

Think of symbols as a box you put stuff in and mark with a big pen.

For instance, I have here a box with a big "K" on it. What's inside? I don't know - the box is taped closed. But the guy who gave it to me told me that it had a kitchen utensil in it. So it could be a fork, knife, spoon, spatula, wire whisk, potato masher, and so on. But it could not be a doorknob. We can use "K" as the name for whatever is inside and pass "K" around just like we would whatever is inside until we actually have to use it to cook.

I have another box here. It's marked "N". The guy who gave it to me told me it had one of the natural numbers in it. That means it could be 1, 2, 3, 4, ...

(... : means "and so on in this same progression to infinity"; so this series extends 5,6,7, ever increasing, but with NO FRACTIONS )

and so I can use "N" anywhere I'd use 1, or 2, or 3 or - well, you see.

I have another box marked "R". It contains an integer. An integer is the set of all non-fractional numbers, positive and negative, and including zero.

So a symbol is a "container" for some value. The kind of value you can put inside needs some explaining if you are thinking it is constrained in some way.

Algebra has traditionally used "x" as an unknown which may take on any value at all; positive, negative, fractional, irrational, you name it. Traditionally, if you used two completely unconstrained values, we'd use "y" for the second; likewise, "z" for the third. We have tended to use "N" for an integer or natural (counting) number. But there is no reason we can't say that "X" stands for any integer.

So a symbol is just a container, a convenient way to manipulate a number that's not completely specified or know somehow, until we can plug in values and compute what its value really is.

Where the symbol-letter is not one of the x-y-z or l-m-n or a-b-c sets, it's also traditional to make the letter you pick be somehow related to the thing being measured. So in electronics, "V" is usually used for a voltage, "R" for a resistance, "C" for a capacitance, "L" for an inductance (for historical reasons), and "I" for a current (again for historical reasons). This is somewhat culturally dependent. You see European texts using "U" for voltage, for instance. "E" is sometimes used for voltage, derived from "electromotive force", but it's more commonly used in the USA for electric field strength.

But you can use any letter as a symbol for any numeric quantity.

In figuring out voltages in a circuit, you can use V for voltage - but there are lots of them. How do you note the differences so you don't get confused with which voltage is which.

Enter subscripts. Mathematicians use subscripts as a kind of sticky note they tag onto a main symbol to help them keep it straight which is which. So "V" is voltage. "V_supply" is the supply voltage. "V_collector" is the collector voltage. "V_base" is the base voltage. Less ambiguous. Now we can name voltages at special places. Usually, mathematicians being lazy, this is contracted to "V_c", "V_b", etc. the voltages being assumed to be measured between our universal reference point, ground, and the point in question.

But what if we want to measure across two points, one of which is not ground, how do we specify that? Traditionally, we put in the major symbol (like "V") and then a first subscript (like "b") and a second subscript (like "e") and we get "V_be", which stands for "voltage from base to emitter" because we usually are lazy and call it just b for base and e for emitter. Likewise, we rever to V_ce (voltage from collector to emitter) and V_gs (voltage from gate to source).

In less-refined places where we cannot type subscripts, this gets to just being Vbe, Vce, Vgs, etc.

So the first letter is a pointer to what kind of thing it is we are representing by the symbol. The second letter is a pointer to "from here...". If there is no second letter, it's understood that it's to ground - literally 0V. If there is a second letter, it stands for "... to here".

But this is all just the traditional meaning. Math symbology is very much invented. It's a shorthand that most people have agreed on and use because it's faster and more precise than writing all this mess out in words.

[jrc4558]

For instance, I have here a box with a big "K" on it. What's inside? I don't know - the box is taped closed. But the guy who gave it to me told me that it had a kitchen utensil in it. So it could be a fork, knife, spoon, spatula, wire whisk, potato masher, and so on. But it could not be a doorknob. We can use "K" as the name for whatever is inside and pass "K" around just like we would whatever is inside until we actually have to use it to cook.

I have another box here. It's marked "N". The guy who gave it to me told me it had one of the natural numbers in it. That means it could be 1, 2, 3, 4, ...

I beleive it was Wittgenstein who argued that psychology commints a semantic fallacy. Once discovered, any psychological phenomenon is given a name, and in the next generations of scientists (the ones that learn from those who made the initial discovery), this name is what signifies the original phenomena. What Wittgenstein was concerned with was the fact that with time, we loose the original understanding of the phenomenon, and start using the name given to it to describe it.
In case of mathematical designators, I haven't ever formed a proper understanding of them back in school, so I can't say that I lost any knowledge, however. It is pretty hard sometimes to conceptualise that the flow of electrons is 'I'. I still have problems with load lines...

[rockgardenlove]

So on this page, what's Rd?
http://www.interq.or.jp/japan/se-inoue/e_dance26.htm

[johngreene]

So on this page, what's Rd?
http://www.interq.or.jp/japan/se-inoue/e_dance26.htm

They define is as Rd = Rc + Re

So the sum of the collector resistor and the emitter resistor.

They say to first calculate Rd with this formula:

Rd = (Vcc/Ic)*(1/2)

To make the operating point the center of the load line.

Then you pick the Gain you want and derive the collector and emitter resistor you need to satisfy these conditions.

--john

[R.G.]

I beleive it was Wittgenstein who argued that psychology commints a semantic fallacy. Once discovered, any psychological phenomenon is given a name, and in the next generations of scientists (the ones that learn from those who made the initial discovery), this name is what signifies the original phenomena. What Wittgenstein was concerned with was the fact that with time, we loose the original understanding of the phenomenon, and start using the name given to it to describe it.

This strikes me as reaching for something deep and missing. It amounts to damning all of human conceptualization and symbolization to nothing more than labeling.

There would be some truth there if the characteristics of the underlying phenomena were not taught along with the label. However, where we use a label as merely a handle, a form of shorthand for the underlying phenomena, and understand the basis for the label, the label is merely a convenient shorthand.

I believe he (she?) may have been onto something like the origin of buzz words. The tendency of humans who have no deeper understanding to toss around labels with no understanding of the phenomena matches. And to the extent that using a label or symbol for a phenomena makes it easy for the posers to carry this off, it's true. However, I believe that it is not necessarily true that naming inhibits understanding. The original namers also had a lot of MISconceptions about the phenomena mixed into the concepts as well. Presumably later learners get less of the original misconceptions, too. So you could argue the case from either side.

In fact, I think it is possible to argue that the human capacity for gathering up a number of facets of underlying phenomena into one label, then gathering up a number of first order labels into second order labels, then those into higher order labels leaves the human mind room to grow. Otherwise, if the human mind has a finite capacity, learning all of the first-order phenomena would fill it up eventually.

The increase in human knowledge leads one to believe that either (a) human storage is infinite (!?) or (b) multitiered conceptualizations lets us manipulate knowledge more efficiently, moving only the "pointers" until we have to expand the pointer into the understanding of the underlying phenomena - that is, indexing helps.

[Nasse]

http://www.arachnoid.com/lutusp/calculator.html

My favorite online calculator (RPN rocks, no = button or parentheses needed;)

[RaceDriver205]

Has anyone considered what might happen, if someone managed to divide the square root of negative one by zero?