College Algebra - Lecture 2 - Language of mathematics

now we're going to talk about another subject we've looked at numbers now we're going to look at something called the map the language of mathematics and this is a topic that really needs a little bit of discussion because when you learn mathematics you have to realize you're really learning a new language and to become fluent in it you ought to have some idea of how it operates so the first thing we're going to look at here is learning to read mathematics so here are a few facts about mathematics you need to know mathematics and you notice when I write it I'm writing it with a capital M just like you'd write English or German or French or Spanish with a capital letter think of it as a language for the time being all right mathematics first thing you need to note is that it is primarily a written language now this is very much unlike all the natural languages which start out as spoken languages primarily mathematics is a written language what is it designed to do well it is designed to discuss easily the following kinds of things one basic objects and what I mean by that are basic mathematical objects mental objects if you like what kind of objects am I talking about well I'm talking about numbers as you've already seen sets what will later talk about functions etc it is a language that is designed to discuss basic objects easily what else is it designed to do it is designed to discuss operations on these objects so operations on those objects now that is extremely important that is really what algebra is about what do I mean by operations well let me give you some indication addition subtraction multiplication division and that all happens of numbers but those are not the only kind of operations composition of functions when we study functions later in the course you will find out what that means but that's an operation you can perform on functions what else how about Union and intersection of sets sets we've talked about a little bit we didn't discuss you need an intersection but those are operations on sets etc so mathematics allows us to discuss with ease and in a written form basic objects and operations on those objects now what else can we say about the mathematical language mathematics again with a capital M mathematics has a new language feature now this is a feature that the natural languages that we all know don't have this is a new language feature and you can write it in several different ways you can say it is the notion of placeholders or another form is to say dummy variables placeholders is the one I like for the time being and let me show you exactly what I mean by writing down an example of a statement in the language of mathematics that illustrates this aspect example in mathematics you can write down the following statement 3 times X plus 4 equals 3x plus 12 now that should be easy to see that if I multiply 3 times X I get the 3 X 3 times the 4 I get the 12 so this statement is true but notice the following other statements 3 times a plus 4 equals 3 a plus 12 or even more generally three times box plus 4 equals 3 times box plus 12 in mathematics these are all the same right what doesn't matter what the letter is or whether I use another symbol there it says the same statement in each case now you try that with a natural language go in choose a word or a sentence and change the letters or change the words it is unlikely that you would end up with a sentence of the same meaning so mathematics is very different in that respect now what does it take to learn to read mathematics effort well you already knew that practice and I hope you also had a sense of that imitation well you know what this is sounding like it's sounding like exactly the kind of thing they teach you when you're learning another language well mathematics is a language so you might expect that that's going to be the case what kind of words are used to indicate this what are key words that you might hear that suggests that there's a language involved key phrases organized work your instructors are always saying make your work organized the phrase standard forgive my spelling their standard form we have standard forms for certain expressions in mathematics as we will discuss later in the course another word another phrase complete mathematical sentences now once you start thinking of mathematics as a language to say complete mathematical sentences is not a nonsense statement we will talk more about this in a minute because sentences have certain aspects to them they have nouns in them they have pronouns perhaps they have verbs in them so these are phrases that describe the kind of work that needs to be put in to learn mathematics now let's go ahead and look at some of the symbols and some of the words that are used in this language called mathematics so some symbols of algebra and we'll look at a couple of words also so some symbols and words of algebra some of these you've known for a long time but I'll make a few remarks as we go you know that symbol from grade school from elementary school this stands for the addition of two numbers you also know this symbol which stands for subtraction if it's an operation you also know that it stands for negation because we've seen on the real number line this is an indicator to go left in this course we will use a dot for multiplication multiplication and on this topic let me put here do not use the X which is the elementary school version of the multiplication symbol the reason is simple it looks like the variable X and it has actually survived in many places sometimes in scientific notation and sometimes in other places you'll see the X occur but we will tend to use the multiplication or we will put two numbers or variables next to each other to indicate multiplication all right let's move on division you'll see I wrote the word before I wrote down any notation here's the way you want to write division if you're writing by hand as much as possible a line that's horizontal and then you will put something on the top and something on the bottom and why don't you go ahead and call that the top and the bottom if you actually look through the history of our notation and the words we use you'll find out that when we call this a numerator on top or a denominator on the bottom that just means top and bottom so there's no reason we can't use that terminology here now division also has another way of being written with a forward slash now that's often written when you have to write things all in a single line but when you're doing your work at by yourself on paper you're going to want to write it horizontally or vertically rather as much as possible and just for the record do not use the old notation for division like so all right we will not need that alright there are some other symbols that are used and we'll explain them more later in the course but let me just introduce them here so you'll will have seen them there is this symbol sometimes with a number in front these are often called radicals and radical of course is a word meaning root and that's what these are about now where did this come from well let me give you at least a pseudo history of where this may have come from it originally looked like that and what did that come from that probably was the result of writing down the letter R R for radical or root the R becomes transformed into something that looks like this and then finally into something like this now you may notice here that I've done something that seems a little unusual to you I put a little tail I call these tails on the end of my root symbols that tells me where the root symbol ends so I'd like you to do that as much as possible you will see me do it throughout the course so it should be fairly common by then what other sorts of symbols that we see very often well there are the grouping symbols parentheses square brackets see we've already seen these we've also seen the curly brackets or braces these might be called grouping symbols they allow us to talk about a bunch of things and interval perhaps open or closed a set of objects etc there are some other symbols we will encounter I'm not going to give you an entire list now but there are a couple of things that we'll see we've already seen or will see this symbol and it goes the other way also remember this is the greater than or less than symbol it doesn't matter which is which remember that the big side is the larger number and the small side where the point is is where the small number goes so it doesn't matter what orientation it is as long as that the point is a small number and the other side is the big number so one might write 2 is greater than 3 but you can also write that 3 is less than 2 except that that's not true and that would make you unhappy wouldn't it so let me show you another notation this is where you put a slash through when something's not true all right what I meant to say is 2 is less than 3 or 3 is greater than 2 there I'm redeemed all right let me give you a couple of words of mathematics that we use a lot and tend to be sloppy about these students seem to be sloppy about terms and factors these are very simple ideas terms are things which are added and factors are things which are multiplied that's it that's all there is to it use the words correctly as you would in any language and you'll be better off because people will understand you and you will understand what it is you written back to symbols here's a famous one I've already used it all throughout this course it's the equal symbol equals or equal to sometimes we say equivalent to but those are two ways of phrasing the equal sign there is another important symbol that is becoming more and more common and I want you to get used to it it's like the equal sign except that there are two wavy lines what does this mean well this doesn't mean equals equal lines or nice straight lines when they're wavy like this what we mean is approximately approximately equal to these are very important when you have calculators around and we will use this throughout the course let me show you an example of how this might be used I might say that square root of 2 is approximately 1 point 4 1 that's the wrong answer if it were an equal sign but as an approximation it's perfectly good I might also say that pi is approximately 3.14 again if that were an equal sign it would be wrong but as an approximation it is perfectly good and so on now here are two pieces of notation that are vital to understanding mathematics and students don't see it early enough so I'm introducing it here this is an arrow going from left to right it will be used in theorems and means implies it also means the phrase if dot dot dot then where if would appear on the left of the arrow and the then would follow the arrow if you have an arrow going from left to right and also one from right to left then you can put them together into a double arrow this is usually referred to as double implication it sometimes is stated verbally or written as if and only if so when you see this you should translate it as to a double arrow there's a symbol for that that's become common iff and you might also say is equivalent to so this says that two statements are equivalent now in a moment I will show you how these are used in actual statements of mathematics but let me continue on with my list and say a few words about letter conventions letter conventions why do mathematicians choose letters to stand for certain kinds of things well these conventions are due to descartes so i won't claim any originality for them these have been around quite a while here's the letter conventions as a rule of course they're not always adhere to these are only conventions these are not absolute laws ABC numbers toward the beginning of the alphabet usually stand for constants when we're doing mathematics algebra the numbers in the middle like i j k l m m and so on these often stand for integers in fact they often stand for positive integers numbers like 1 2 3 4 etc and finally at the end those variable letters XY and z these stand for variables so these are the unknowns the x y&z that are living underneath my head here XY & z these are often used for the unknowns for variables that we would like to discover in the process of solving our equations so a B and C stand for constants i j k l m and n those middle numbers stand for integers and finally the x y&z stand for variables all right well let's go back to my list and we'll talk about the next aspect of the language of mathematics nouns pronouns and the main verb of algebra if we're talking about mathematics is a language there have to be these parts of speech so let's talk about them first let's talk about nouns what would nouns be in mathematics nouns are equal to number expressions because nouns are things that always have the same meaning for example five is a noun in mathematics four plus twelve point three is a number when the addition is done and that's also going to be a noun three squared for example or something more complicated three times seven minus 9 plus a half etc all of these are nouns they have one and only one meaning even though a little effort might be needed to write them out as a single number okay what else did I suggest pronouns pronouns what would be pronouns well these are things that stand in for nouns and can be replaced by nouns so these are going to be what we call variable expressions see these work just exactly like pronouns work for example 3x plus five is a perfectly good mathematical pronoun if you put in a variable value for the X then you get a noun here's another one a times B put in values for a and B it becomes a noun as it stands it's a pronoun u squared minus V squared etc all of these are legitimate mathematical pronouns if you fill them in they become downs now let's come to the main verb of mathematics there's one main verb I think you know what it is the main verb is equals and with this verb I can now define a sentence a specific kind of sentence definition and equation one of the most important sentences we talk about is a declarative sentence remembering your grammar is a declarative sentence in which the verb as I said above is equals that's what an equation is it is a sentence sentence and the verb is equals so when you write down equations we want to be careful to write out complete mathematical sentences we want it all to make sense so you'll see the notation that I use in the way I express things the way I organize my work and the way I use standard forms all of these will be because I'm trying to preserve my sentences all right let's go back to my list and look at my final topic for the language of mathematics theorems corollaries lemmas and all that what are all of those and why do mathematicians use all those odd words well let's see here me write these down theorems corollaries lemmas propositions there's another one that's used and so on what are all of those these are simply statements of fact in mathematics that's all they are they're a way of writing down a fact why are all these different words used well for historical reasons is there any different meaning to them that we ought to discern well a little bit theorems usually mean some kind of fairly major result corollary is a theorem that follows immediately upon another theorem so it's a trailor if you like a lemma is often a smaller result the lemma or proposition are usually smaller results that don't really warrant being called theorems there's their tiny results there's still facts they're just a little bit less important so there's a kind of hierarchy for the use of these words this is not hard and fast but I think that if you realize these all are just statements of facts that none of these words will bother you all right what do these facts look like there are two main forms that these theorems corollaries lemmas etc come in so two main forms of what I've called up there generically facts what are the two main forms the left to right arrow form and the double arrow form I will give you one illustration of each one in other words I will write down a very simple theorem that indicates a left to right argument and one that indicates a back and forth statement these are both statements so here's an example this is if you like we'll call it a theorem very simple result if remember the arrow can be rewritten in English as if and then it then follows if a and B are odd you remember what odd numbers are a lot of natural numbers then a plus B is even now there is the theorem and notice it's a one directional arrow if this statement holds then this statement follows and this is not a hard theorem this is one you know already if you take two odd numbers and add them together like adding three and five you get eight which is an even number that will always work also notice that this doesn't work in the reverse direction suppose you add two numbers and get an even number that does not mean that a and B are even for example if I take two plus six those are both even numbers they add up to eight which is an even number but neither 2 nor 6 are odd so the theorem does not reverse so this is a one directional result now if you see this when you see theorems this will help make clear how to use these results you must have the if in order to draw the Venn conclusion now here's an example of the other form example here's another theorem I guess I should have called it a corollary or something just to vary the name of it let's call this a theorem and here's how this will go for any numbers you can assume I mean real numbers here X a and B the following two statements are equivalent x plus a equals B there is one statement that is true this is equivalent to the statement that X is equal to B minus a now this is an example of a double arrow theorem and again this is easy enough for you to see why it's true I don't actually have to produce a proof X plus a equals B is the same as this because if I started at the top if I subtracted a from both sides on the left I would have X on the right I would have B minus a that's what this says and reversing it if I start with X equal B minus a if I add a to both sides on the left I get X plus a on the right I get B which is what the top says so these are in fact equivalent statements so I can start at either place and go to the other that's what the double arrows mean so there's an example of the double arrow theorem the one before which I'll return to here is an example of a single arrow theorem these are the only two kinds of theorems there are so when you're studying mathematics and you come across theorems lemmas corollaries propositions etc you realize there are only these two different kinds this one and the double arrow kind all right well that's all I have to say about the language of mathematics you should spend some time trying to become familiar with the symbols and the suggestions I made about nouns and verbs etc and then after this we will move on with basics

Channel: UMKC

Views: 155,073

Rating: 4.9111109 out of 5

Keywords: Basic objects, place holders, dummy variables, algebra, terms, factors, equals, approximately equal to, implies, if...then, double implication, if and only if, equivalent to, Letter conventions, Descartes, constants, number expressions, variable expressions, equation, theorems, corollaries, lemmas, propositions.

Id: N7xVAmn10Xg

Length: 27min 9sec (1629 seconds)

Published: Mon May 04 2009

Reddit Comments

[u/chastric]

Does anyone know of linguistic work on the "translation" between written math language and natural languages? When I studied math, I always thought it was intriguing how operations and relations became verbalized, and that arbitrary symbols could be ascribed a part of speech (and not always nouns, or even typical open-class categories for that matter).

[u/Mesonit]

Very interesting! I'm going to watch the rest of the series now.