You will see additional examples that can help you gain a better understanding of these new concepts. Variables, Expressions, and Equations. Before learning about real numbers and the aspects that make up real numbers, you will first learn about the real number line. The study of mathematics requires the use of several collections of numbers. The real number line allows you to visually display graph the numbers you are interested in.
A line is composed of infinitely many points. To each point you can associate a unique number, and with each number, you can associate a particular point.
Coordinate The number associated with a point on the number line is called the coordinate of the point. The point on a number line that is associated with a particular number is called the graph of that number. Now that you have created a number line, it is time see how points on a number line are defined.
A real number is any number that is the coordinate of a point on the real number line. Real numbers whose graphs are to the right of 0 are called positive real numbers, or more simply, positive numbers. Real numbers whose graphs appear to the left of 0 are called negative real numbers, or negative numbers. Watch the video for a simple explanation of positive and negative numbers on a real number line. What Are Positive and Negative Numbers?
The set of real numbers has many subsets. Below is a diagram of real numbers. You will see the terms natural, whole, integers, rational, and irrational numbers which are sets of real numbers. The letter N is the symbol used to represent natural numbers. Natural numbers are also known as counting numbers, and they begin with the number 1 and continue to infinity never ending , which is represented by three dots The letter W is the symbol used to represent whole numbers.
Whole numbers are counting numbers from 0 to infinity. The letter Z is the symbol used to represent integers. An integer can be 0, a positive number to infinity, or a negative number to negative infinity.
What Is an Integer? The letter Q is the symbol that is used to represent rational numbers. Rational numbers are sometimes called fractions. They are numbers that can be written as the quotient of two integers. They have decimal representations that either terminate or do not terminate but contain a repeating block of digits. Some examples are below:. Non-terminating, but repeating.
You will notice a great many points on the number line that are not discussed. These numbers make up the irrational numbers. They will be examined in detail in an algebra course.
Now it is your turn to practice. Try answering the following questions on your own and then select Check Answers to see how well you did. If you look at functional programming languages like F or Lisp , the single letter variable paradigm is quite common for many of the reasons mentioned in the other answers. Take the functions:. These are common and accepted ways to write your functional functions.
Now imperative programming by far the majority of programming is more like a list of instructions, what I would compare in mathematics to a proof.
If you remember back to 'doing proofs,' you state your reason for making each step along the way; in some cases rather verbosely. I am a retired software engineer and a math person. I am going to answer your question based on my own personal experience and observation.
The coding style in your first example is unacceptable today but was common in old days. The programmers deal with real life problems. The reasons programmers use long variable names are mostly for readability and easy to debug. Programmers are asked to use long names because they were having difficulty understanding code written by others and debugging code written in short names.
I was once shown a piece of code written by myself and could not figure out what I was doing. On the other hand, mathematicians mostly deal with abstractions. The formula in your example generally will be present with the explanation you provided. Later on in the literature, the author can use t, which is the number of years, wherever it is needed without repeated explanations.
Most mathematicians follow certain conventions, say they usually use x for an unknown, c for a constant without explanation. This sometimes will give the readers some trouble understanding the text if unusual conventions were used. Also, the reader will have to get used to the notation the author adopts.
Sometimes it's a pain. In either case, there is a compiler there. In the programming case, you have a compiler to help generating machine code for the computer to understand. In the math case, your brain is the compiler to generate the knowledge. I don't think that substituting single-letter variables with more letters would do much positive in most mathematics, because most theorems and such are short enough so that remembering a few variables isn't a problem.
And if you forget, it is just to look at the start of the theorem. Also, there is conventions for what variable-letters to use in different circumstances. And last, it is a way to make it easier to separate words from variables. In programming, the ability to have more letters for one variable, is a huge advantage because a program often consists of thousands of lines, and hundreds of variables. Its only confusing and unproffesional. Even if you try to follow this path, there can be intermediate steps in solving an equation in which these words can not be interpreted in any sensible physical or informal way.
For instance it can be convenient to introduce imaginary numbers even when solving equations with real solutions. The point is that we usually want to be as concise as possible, in other words, we want to be capable of being understood and on the same time do not need to write lots of things.
In truth, you'll never see in the middle of a serious text of math some equations with letters appearing from nowhere without explanations: we always have the definitions first. The point is that there are some conventions that we follow: things that always have the same name, and that our audience is supposed to know about.
But if we do not assume even those basic things from our audience, we define them as well. Now, I'll give you an example that you'll agree that writting everything like you say will make us even unable to proceed with the theory only a masochist would proceed with something like that.
This is an equation from differential geometry:. Now imagine if for each of the letters we write the word it means. An equation like this one, that after making the definitions is at same time compact and understandable, would become a real mess and people would get lazzy to write down equations like that writing every word. So, we do use letters because we gain compactness, we gain simplicity and we do not waste too much time writing, but the benefits of that are only achievable if we combine with clear definitions and organized line of thought.
If so, when someone reaches an equation like that will look at it and say: "well, that's easy to read, the author defined all those terms properly! If we saw something like the above, then the alphabet implicitly used in mathematics would consist of words. So, then the individual letters of that alphabet basically require another language to get understood in the first place. But, of course, you're wanting those symbols also to represent English words which have meaning also.
Thus, you need two alphabets working together at once, as well as two languages working together at once, and it's not at all clear that you'll avoid ambiguity in principle when you do this.
You will have character strings which consist of a word in one language and a letter of the alphabet in the other language at the same time. On the other hand the more common way just requires one alphabet and one language. You just need "v", "d" and "t" as variables in your alphabet But, then we wouldn't seek to use "6", "7", and "8" to denote numbers, because then we would have two languages working simultaneously. First of all, if mathematicians used more-than-one-letter variables, they might confuse them for multiplication.
Second, mathematicians like to be as accurate in as little words as possible. And finally, there are 26 letters in the alphabet. It's not like one equation will use them all, anyways. Much of formal math was developed before there were computers. Thus computation was done by hand, a very repetitive process.
Much literature was written with this in mind. Since pure math is not an applied field, it generally does not throw away models used in the past, the body of relevant work extends back quite far. If the field were to decide that they want to change notation, there would be countless books and journal articles to update to the new notation.
While it may be simple enough to update digital documents, a large number of document stored in analog or raster formats, would likely never get updated. This body of knowledge would likely become inaccessible to future generations and thus lost to history. It is unlikely that the field will change notation unless there is a really really good reason to do so. For example the adoption of Arabic numerals was a vas improvement over Roman numerals and happened at a time when there was a much smaller body of pre-existing literature.
Computer science on the other hand is a new, applied field, a good deal of the advancement of the field has been after the creation of the electronic computer.
The field focuses on elimination of redundant work and aims to build working systems. I'd say partially because mathematical notation has much bigger vocabulary of "symbols" than source code notation. That's the LaTeX notation for the mathematical notation I first mentioned.
Either way, what's important is the concept of sum , regardless of how long or short it's needed to write that concept. If I'm doing a piece of maths, I want to fit as much of it as possible on a single sheet of paper so I can see it all at once.
I've often wondered whether this as why people who are good at maths often have tiny handwriting. I have to get pretty creative within my publications not to exhaust all symbols within the one publication - including capitalization, other fonts and greek letters.
I am not at all able to stay consistent within my entire body of work precisely because there are not enough letters. So, I actually use - at times - symbols with more than one letter.
Thanks to that I have the iron rule to always use a dot for multiplication. However, I try to avoid it, because of many of the mentioned reasons. It also helps to think hard before introducing a new symbol.
This leads me to think whether I really need this new symbol or whether there is an underlying principle that allows me to abstract multiple concepts into a single concepts. This makes me a better researcher and actually leads to new publications by itself. You could say the publications write themselves :-P. However, most publications in general do not need that many symbols and even I do not even nearly need as many symbols as I do while programming.
Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Letters are only used in math when using variables, or to represent unknown values. DragonSlayer Jul 16, Who gave me -2 points?!?
DragonSlayer Jul 18, Some guidelines for question askers. Again a number puzzle. Multiply in writing. Loads of fun printable number and logic puzzles.
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