August 11, 2008
If you’re new here, you may want to subscribe to my RSS feed. Thanks for visiting! How do you symbolize different bases? Is there a way that mathematicians write “base 2″ for example, without having to write out the words?
There are several ways that bases can be symbolized. The two most common are simply to subscript the number of the base to the right and down of the number, like this:
That lets you know that we are dealing with the number 101 (base two), not the number 101 (base ten). 101 in base two would be 5 in base ten.
Sometimes the base is written out as a word in the subscript, like:
Depending on the context, one may be more convenient than the other, but both are accepted. It is probably best to use the written out word in subscript, because there are other uses for a subscripted number to the right of a number in math. Using the written out word, as in:
makes unambiguously clear that you are only talking about a base.
Anyone care to have a shot at what 423six would be in base 10? Leave it in a comment.
Tags: base notation , how to write bases
August 10, 2008
This short lesson is a continuation of the posts at:
In those lessons, we talked about what bases are, what they’re used for, and how to change numbers from base 10 to base 2 (easy!)
It’s even easier to change numbers from base 2 into base 10.
When you read a number in base 2, you simply have to add the columns together that have a 1 in them, and ignore the columns with a 0 in them.
In the number 111(base 2) there is a 1 in the fours, twos, and ones columns. Simply add 4, 2, and 1 to get the base 10 value, which is 7.
The number 10110 has a 1 in the sixteens column, another in the fours column, and another in the twos column. So add 16 + 4 + 2, to give you 22, base 10.
You have to admit that’s pretty easy.
What would the number 110101 (base 2) be, in base 10?
Answer it in a comment, if you like.
Tags: convert base 2 to base 10
August 9, 2008
A number is a concept that we have for some value. For example, hold out four fingers. You can conceive of the number four, you know how many are there. That is the number - more or less the concept you have in your mind.
A numeral is a name or a symbol for that concept. The symbol may be a 4 (in base 10) or a 100 (in base 2) or IV (if you are using Roman numerals) or |||| if you are using tally marks, etc. All of those symbols look different on paper. But the concept in your mind remains the same.
So a number may be expressed many different ways, using different numerals. But a numeral will always represent the same number, as long as you know what system (base, Roman, tally, etc.) you are using.
It might help to think of it like languages. For example, a “book” is a word for that thing you read, with many pages. In German, it’s a “Buch,” in French it’s a “libre,” in Spanish it’s a “libro,” and in Vulcan it’s, well, I don’t know what it is in Vulcan, but you get the picture. They are all different words for the same Idea. The book is the actual object, but “book,” “Buch,” “livre” and “libro” are simply words, or names for the object.
So you might consider numbers to be the Ideas, and numerals to be the names for the Ideas.
Of course, as always, there are more in-depth ways to look at this issue, but the above should give you a good, working basis to explore further, if you wish.
I hope this gave you something to think about,
Your pal,
Brian
Tags: numbers , numerals
August 7, 2008
A reader recently sent in this problem:
Help, how do you solve this???
The area of a rectangle is 624cm2.
The base is 8 less than 5 times the height.
What is the perimeter?
I can not find out how to do it on line. I have a number of similar problems to solve
thanks.
- A. Reader.
Professor Homunculus replies:
Hi, Reader,
I must say that it is a good thing that it is not taught online. Specific problems should never be shown online. That would be show-and-tell, not teaching.
What you need to learn is the concepts behind the problems, then you’ll be able to crack all problems that are similar. One of the concepts is a very interesting and important part of algebra.
Here’s a way to start figuring out how to solve your problem:
(more…)
Tags: area , base , geometry , height , perimeter , rectangles , word problems quadratic equation
July 27, 2008
Photo credits: two unknown and one from monkeymucker
Edited by Brian
I recently read a post at:
dolcevitaacademy, which talked a bit about something that is close to my heart, which is: should you focus on teaching concrete skills first, or rather on concepts? (I’ve blogged about that on the post about Math Skills v. Math Concepts.)
This seems to be about what the “math wars” are about.
If you don’t know what the math wars are, here’s the quick and dirty on them:
At some point, around the early sixties, American educators decided that what we needed to improve our math education was a new way to teach math. This was probably due to our getting our butts kicked by the Soviets in the Space Race for awhile in the late fifties.
Some genius came up with “The New Math,” which was basically a somewhat new way to teach math; it had nothing to do with any kind of actually new math . I mean, two plus two still equalled four.
Since then, “traditionalist” and “constructivists” have been sniping at each other about “which way is the best way.” (I like to snipe at both.)
(more…)
Tags: math education , math wars , new math
July 22, 2008
I’ve got lots more to post about bases, but right now I’m faced with a paradox - Hintikka’s Paradox, to be precise.
Hintikka’s Paradox comes from Deontic Logic, a form of Modal Logic. I first read about it in Raymond Smullyan’s “Alice in Puzzleland’ (a brilliant book about logic, and Alice in Wonderland, that is worth looking into.)
In the introduction to “… Puzzleland,” Smullyan describes Hintikaka’s Paradox this way:
“Is it proper to call morally wrong something a person cannot do? Hintikka has a notorious arguent designed to show it is wrong to try to do something impossible. There is now a large literature on this strange question…”
I’ve yet to encounter much of that literature, and boy, I have looked. I probably wouldn’t have understood most of it, anyway.
But it boils down to this, Hintikka’s Paradox implies that, “What is not possible is positively forbidden.”
It’s important not to approach this from a “common sense” frame of mind. Common sense is usually neither, and is often a disadvantage when approaching counter-intuitive material. So try to keep an open mind.
(more…)
Tags: deontic logic , Hintikkas Paradox , logic , modal logic
July 3, 2008
Continued from the post about ”What is a Base?”:
The same person wrote a follow up comment:
“you are not pretending i’m stupid!!!!! Okay is a base the number you can multiply by?????
“example: base two is 2,4,6,8,10,12,14,16,18 ?????????i don’t know what you mean!”
Yeah, keep trying to convince me that you’re stupid. From your grammar and your tone, you’re starting to make some headway.
But I generally don’t believe a child can be stupid. Misguided, full of anxiety about themselves and the world, OK, but stupid is reserved for adults (where a lot of people make up for lost time).
Maybe I wasn’t clear enough, so let me try again. A base is a way to write a number using place value (columns). The amount of digits you decide to use in the columns determines the number of the base. If you use ten digits per column, the number will be in base 10. If you use three digits per column (the digits 0, 1 and 2), the number will be in base 3. You will understand this better as you read on. (more…)
Tags: convert base 10 to base 2 , what do we use bases for
June 30, 2008
photo by lsiegert
A curious reader asked this question:
What is a base?? I’m sorry but I’m in the sixth grade and never heard of a base and then all of the sudden it’s in my homework. Will you please explain to me in easy fifth or fourth grade words what a base is? Pretend I’m stupid or something!
Professor Homunculus replies:
Well, that’s going to be hard to pretend, because you are obviously smart enough to ask for help. You also did a good job expressing your question, so here goes:
Bases are different ways to express numbers. Like languages are different ways to express thoughts. You could say, “butterfly” in English, or “mariposa” in Spanish, “papillion,” in French, or “schmetterling,” in German, but they would all mean the same thing, just different names for it.
You can write the number 11 in base ten, or as 21 in base five, or as A in base eleven, and they all stand for the same amount.
Just as in different languages, there are specific times you need to use different bases. That is a little hard to understand, right now, I know, but first you must learn how to translate into different bases, before you can understand anything about them.
Fortunately, it is much, much easier to learn how to translate from base to base than from language to language.
A base is the amount of digits we use to represent our numbers with.
We normally use what it called the base ten system. As you know, we normally use only ten digits - 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 to make up all of our numbers. After 9, we have to start a new column (called the “tens” column, because it tell us how many tens we have).
(more…)
Tags: base 10 , base 2 , bases , what are bases , what is a base
June 27, 2008
Hey, you droogs,
There was an interesting post on the Whallah! blog about an article in the Associated Press, concerning the education of math teachers in public schools.
Apparently the National Council on Teacher Quality has done a comprehensive study to come to the conclusion that everyone who is not an “expert” has known for years: Teachers are not being taught math adequately, and generally fail to teach it well to their students. (Do tell…)
Isn’t it funny that the “establishment” will never admit that? It takes an expensive academic “study” to show what is already known, yet Universities (in general) will not do anything about the way they teach teacher how to teach math. They will try some new, expensive methods that some textbook company has lobbied for, of course. But they won’t try anything that might actually work.
(more…)
Tags: math ed , math education , math pedagogy , multiplication , public schools
June 19, 2008
A mom recently wrote in to ask this question about standard and expanded notation.
“How do you know when you are writing in standard form, expanded form? For example, is the expanded for of 30,048
30000 + 40 + 8 ?
Or for 29,486, the expanded form = 20000 + 9000 + 400 + 80 + 6 ?”
Professor Homunculus replies:
Precisely! Oddly enough, there is no “standard” for “standard.” What I mean is, for 30,048 the standard form could also be considered:
(more…)
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