Discussing Numbers: The Number Line, Types of Numbers, Sets, Inequalities, and Intervals October 23, 2006Posted by LearningNerd in Arithmetic, Math, Mondays, Number Theory, Pre-Algebra, Science.
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I said I’d start with the basics, so here they are!
By the way, this post includes three videos. They’re a bit long and not particularly entertaining, but they’re useful if you’d like to take a break from reading.
It all starts with the number line, which, as the name implies, is simply a line marked with numbers. Math teachers love to use number lines to teach everything from arithmetic to calculus, and though I remember the number line as it appeared on many repetitive worksheets, I realize that it can actually be useful. Like an outline, the number line is a great way to visualize the abstract ideas you’re working with: smaller numbers on the left, bigger numbers on the right. For an example, check out this nifty tool that compares two numbers using a number line.
A number system is a set of numbers that can be used in arithmetic operations (like addition and subtraction). As we all learned in school a long, long time ago, there are different types of numbers like positive numbers, negative numbers, even numbers and odd numbers. (Those types of numbers can be listed in sets, but they aren’t considered number systems. I looked for a formal explanation of this, but I’ve yet to find one.)
There are several types of numbers or number systems commonly taught in math classes:
- natural numbers (also called counting numbers)
See Number Types for a basic overview, and/or watch this video:
I’ll be learning more about irrational and complex numbers later; they aren’t important until you get to more advanced mathematics.
Like I already mentioned, a set is just a group of things considered as a whole. When discussing numbers, mathematicians save time by using set notation. The whole thing’s pretty simple, but it uses some symbols that you don’t see too often. See Set Notation for a review, and be sure to bookmark the page for reference. The following video also explains set notation:
Just for fun, here’s an interesting math problem I came across: “Which is greater, the number of rational numbers between 0 and 1 or the number of rational numbers between 0 and 2?” See the answer and explanation here. (By the way, I got it wrong. )
The inequality signs (like < and >) are used to compare numbers. They’re pretty straightforward; 13 is less than 27 (13 < 27), two is greater than negative three (2 > -3), and so on. Rest assured, it does get more complicated.
For kids or anyone who wants some practice, here are a couple interactive games and quizzes:
An interval is basically just a set of numbers containing every number between two numbers. As with sets, intervals have their own notation; see Interval Notation for a summary of the basics and remember to bookmark the page for reference.
The following video reviews inequalities, intervals, and set notation:
As a final review, check out this interactive example of Set and Interval Notation, where you can move an interval on a number line and see how the notation changes.
Overview of Mathematical Terms and Notation October 16, 2006Posted by LearningNerd in Math, Mondays, Science.
As Denning Miller explains in Miller’s Popular Mathematics,
…there is no vocabulary that is easier to acquire than the mathematical one. There are no tenses, no genders, no conjugations, no declensions, and only a minimum of grammar. Imagine how pleasant it would be, after you had learned a few words of Spanish, to begin talking and expressing yourself in the language. You can do that in mathematics…
But mathematicians have a knack for avoiding plain English wherever possible. Instead of saying “Find the answer to this problem,” they’ll say, “Evaluate this expression.” So, before I start relearning math, I thought I should avoid future confusion by reviewing this mathematical gibberish.
It’s interesting to note how each math term relates to its plain English counterpart; for example, “to evaluate” means to judge the value of something (like appraising the value of a gold coin), and in math it means to find the numerical value of something. Value, another math term, is also a common word with overlapping definitions — see value. As you’ll notice, all the terms and concepts in math are just specific versions of abstract ideas that apply to every field.
- Number n. - An abstract entity representing a count or measurement.
- Variable n. – A quantity that can be unknown and/or subject to change, usually represented by a letter like x.
- Set n. – A collection of distinct things considered as a whole.
- Equation n. – A statement that two things are the same, as in 2=2 or 2+2=4.
- Operation n. – A process that combines two values to produce a third, like addition or division.
- Expression n. – A group of numbers, variables, and operators (like + or -). I think of an expression as the mathematical version of a phrase in English, whereas an equation is like a complete sentence. (Compare the expression “two plus three” to the equation “Two plus three equals five.”)
- Function n. - An operation that creates a relationship in which one variable depends on another and each input can have only one output. (Don’t worry about this one until calculus.)
- Value n. – A quantity; an output of an operation or a function.
- Evaluate v. – To find the value of (an expression). (Evaluate 2+2 to get 4.)
- Simplify v. – To reduce (an expression) to only the necessary parts. (Simplify 2x+3x to get 5x.)
- Algorithm n. – A step-by-step set of rules used to solve a problem.
- Axiom n. – A self-evident truth or a statement that is assumed to be true, which is used as a basis to prove other statements. Also called a postulate.
- Proof n. – A logical demonstration that a statement is true; the process of establishing a theorem (see below).
- Theorem n. – A statement that has been proven true, deduced from certain axioms using a proof.
- Earliest Known Uses of Some of the Words of Mathematics - some interesting etymologies of math terms.
- Mathematical Notation: Past and Future – History – a detailed essay with some interesting photos.
- Earliest Uses of Various Mathematical Symbols – some interesting trivia.
- The History of Mathematical Symbols – has sections on each major symbol.
- History of Mathematical Notation – Wikipedia’s shorter overview.
- Mathwords – a dictionary of math terms with lots of pictures and examples.
- A Maths Dictionary for Kids – has interactive examples for each term.
- Glossary of Mathematical Terms – some more advanced explanations of terms.
- Table of Mathematical Symbols – on Wikipedia.
- Roman Letters used in Mathematics – on Wikipedia.
- Greek Letters Used in Mathematics – on Wikipedia.
- Mathematical Jargon – on Wikipedia.
How to Enjoy Learning Math October 9, 2006Posted by LearningNerd in Math, Mondays, Science.
I don’t need to explain why people should know math. You know why. But how can we convince ourselves that math is actually interesting? Since I’m going to relearn mathematics on my own, I thought I should start by examining the reasons why I and so many others don’t love math, in order to get some ideas about how to make it more interesting. I asked a few questions on the sci.math Usenet group and on 43 Things. A few mathematicians responded, one of whom has been studying math since the year I was born!
Unfortunately, the more common question is “Why do people like math?” As explained by these responders, math simply isn’t taught in an interesting way — and our cultural stigma against it certainly doesn’t help, either.
Same reason a lot of people hate spinach or broccoli. They’ve been forced to have it because, they are told, “it’s good for you”, but they can neither see nor perceive those benefits, and it is being served by mostly uninspired cooks who don’t much care for its taste either. — See full post.
I learned much better on my own than in high school which was a detriment to my learning. — See full post.
Some have had a math teacher in the past who was either incompetent or who hated math…. However, I think the biggest reason why a lot of people hate math is because of a prevailing culture in which it’s “cool” not to know any math. — See full post.
I don’t think math is really the problem most people have, I think how it’s TAUGHT is what people have a problem with! — See full post.
But no-one ever mentioned anything interesting about maths at school. I never learnt about sacred geometry, I learnt to ‘use a compass’; I never learnt aout the concept of ‘relativity’, I learnt greater-than and lesser-than signs. It didn’t work for me. I don’t think the teachers were very inspired or the syllabus was holding them back or something. — See full post.
Like many of the responders, I didn’t like math because of school. Memorization, repetitive worksheets, stressing about grades, cramming for finals — no wonder so many people hate math! But you have to take responsibility for your own education; only you can make yourself interested in a subject. Here are a few ways to help yourself enjoy learning math:
Know what math really is. Math isn’t about arithmetic or memorizing formulas. It’s about problem-solving, deducing truth, and exploring the concepts of change, quantity and structure. Take a glance at Wikipedia’s entry on Mathematics for more details — it’s actually an interesting read.
Learn to apply math. Like I said, math is all about problem-solving; so, use it to solve your problems! This is what makes math a creative subject as well as a logical one. Sure, applied mathematics is quite advanced, but identifying a use for it will motivate you to work your way up. Math classes generally teach students the formulas before giving any examples of how they can be applied, but perhaps it’d be more interesting if you started with a real-world problem that requires math and worked backwards, finding the formulas that you would need. I might try this myself.
Explore the history of math. This won’t help everyone, of course. “Ew, math and history? Are you trying to kill me?” I don’t like history much, either, but it adds another dimension to the study of mathematics. Take a break from math problems and do some reading. Try learning the origin of a formula before you learn how to use it. Who discovered it? How? Why? What I hated most about math class was not having time to answer those kinds of questions. The teacher had to finish the lesson plan every day, and I had to spend my time on homework every night. That’s why I highly recommend learning math (or any other subject) on your own; you can leave room for yourself to stray off topic.
Relate math to other subjects. Math relates to music, art, architecture, business, science, and even philosophy. If you’re interested in any of those, then learn about them while you learn about math. I’m hoping to write posts about all of those subjects in the future, and I plan to find out for myself just how well math can help with them. In the meantime, here are a couple interesting websites: Wolfram Tones: An Experiment in a New Kind of Music (interactive) and A Logarithmic Image Transformation (at least scroll through to see the cool images).
Surround yourself with math enthusiasts. Sure, you sit with a bunch of other students in math class, but the students rarely talk to each other about math. Most of them don’t want to be there, and it shows. Get inspired by joining a group of people that are passionate about math — people who truly want to learn and discuss it. If you want a study partner, feel free to email me!
Hang in for the long haul. I was surprised at how many math enthusiasts said that learning math shouldn’t be fun. They brought up a good point, though: it can’t all be fun. As one responder said here, enjoying math is like enjoying chess; it becomes more interesting over time. Like learning a new language or learning to play an instrument, mathematics doesn’t become truly rewarding until you’ve made a long-term commitment to it.
So, tell me: what do you think of math? Love it? Hate it? How do you think we could make it more interesting?