How are different bases pronounced?

Feynmanfan

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I'm unsure how different bases are usually read aloud. Is 215 = 156 read as "twenty one base five equals fifteen base six"? Or "two fives and one equals one six and five"?
 
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Yeah, it's an interesting question, and not one that I think has a clear answer.

I would probably read it as twenty-one (base 5), even though twenty-one literally means "two tens and one." I have never heard anyone say "two fives and one", nor would I read it that way mentally.

One alternative might be to read it as "two one", i.e. just list the digits in order. This is what I do in binary (base 2). So I would read 102 as "one zero" rather than as "ten."
 
I'm unsure how different bases are usually read aloud. Is 215 = 156 read as "twenty one base five equals fifteen base six"? Or "two fives and one equals one six and five"?

The "official" rule is that other than base ten, you just read out the digits: "two one base five".

If I say "twenty-one base five", I know I'll still be understood, but it's technically wrong.

But you'd better not read A016 as "A-ty"! That's definitely "A zero base sixteen".
 
I'm unsure how different bases are usually read aloud. Is 215 = 156 read as "twenty one base five equals fifteen base six"? Or "two fives and one equals one six and five"?


\(\displaystyle 21_5\) is read "two one base five" & is equal to \(\displaystyle 2\cdot 5^1+1\cdot 5^0\) base ten.

\(\displaystyle 15_6\) is read "one five base six" & is equal to \(\displaystyle 1\cdot 6^1+5\cdot 6^0\) base ten.
 
Thanks for the replies.
The "official" rule is that other than base ten, you just read out the digits: "two one base five".

If I say "twenty-one base five", I know I'll still be understood, but it's technically wrong.

Hi Dr Peterson. I would argue that saying "two fives and one" for 215 is best practice for introducing the base concept to children, since it's self-explanatory and systematic. For those who already understand it of course it doesn't matter. It's a very simple concept after all.

This is only useful for two digit numbers but the point is, why not use the simplest and clearest terms whenever possible? Teaching the base concept to eight/nine year olds was I believe tried in France at one stage but quickly abandoned because of the confusion it caused. I think there would have been much better results if they had used the appropriate terms. It's the same mistake which is made in teaching base-ten counting, which is the use of confusing unsystematic terms in preference to self-explanatory systematic terms. Which is to say, every English-speaking child must learn that 11 is "eleven", but why should they not be told at the same time that it means "ten and one", or better, "a ten and one" or "one ten and one", which are the correct and self-explanatory base-ten terms?

That's OT so I'll stop now. I'm not sure this is the right forum for discussions on teaching practices.
 
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I would argue that saying "two fives and one" for 215 is best practice for introducing the base concept to children, since it's self-explanatory and systematic. For those who already understand it of course it doesn't matter. It's a very simple concept after all.
How would say \(\displaystyle 20011_3\) to that same child?
 
Thanks for the replies.

Hi Dr Peterson. I would argue that saying "two fives and one" for 215 is best practice for introducing the base concept to children, since it's self-explanatory and systematic. For those who already understand it of course it doesn't matter. It's a very simple concept after all.

This is only useful for two digit numbers but the point is, why not use the simplest and clearest terms whenever possible? Teaching the base concept to eight/nine year olds was I believe tried in France at one stage but quickly abandoned because of the confusion it caused. I think there would have been much better results if they had used the appropriate terms. It's the same mistake which is made in teaching base-ten counting, which is the use of confusing unsystematic terms in preference to self-explanatory systematic terms. Which is to say, every English-speaking child must learn that 11 is "eleven", but why should they not be told at the same time that it means "ten and one", or better, "a ten and one" or "one ten and one", which are the correct and self-explanatory base-ten terms?

That's OT so I'll stop now. I'm not sure this is the right forum for discussions on teaching practices.

Your question was not about introducing the topic to children, but about what is correct, or how they are "usually" read.

Moreover, "two fives and one" is not a way of reading, but a way to describe the meaning. Each has its place. I would say that in the same contexts in which I might say "two tens and one" in place of "twenty-one" -- that is, only in discussing what "21" means, not in reading it aloud. Yes, it's a good idea in some situations, but it isn't the answer to "how is it usually read aloud".
 
How would say \(\displaystyle 20011_3\) to that same child?

Hi there, obviously not a serious question, and in any case I have already answered it.

Here's one for you: can u say instantly if that number is even or odd? If so, who did learn that from? If not, then read my other thread.
 
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Your question was not about introducing the topic to children, but about what is correct, or how they are "usually" read.

Moreover, "two fives and one" is not a way of reading, but a way to describe the meaning. Each has its place. I would say that in the same contexts in which I might say "two tens and one" in place of "twenty-one" -- that is, only in discussing what "21" means, not in reading it aloud. Yes, it's a good idea in some situations, but it isn't the answer to "how is it usually read aloud".

Yes, it doesn't take me long to climb onto my hobby horse, which is the lamentable ways elementary math is being taught, starting with the essential skill of counting. I should have perhaps simply expressed my gratitude for the information and moved on. I was checking to see if what I suspected was correct, which is that the usual terms are not the proper ones, and that children are not being taught how to count properly in any base, except for Chinese, Japanese, Korean and Welsh-speaking children who have the benefit of being taught base-ten counting, in addition to various traditional forms.

When reading numbers in vernacular English 11 is read as eleven. When reading numbers using base-ten, 11 is read as "one ten and one", or "a ten and one", or "ten and one". As you point out, that explains the numbers, which is of course the reason they should be used. But it only matters at the very beginning. Telling seven and eight year olds that eleven can also be called ten and one is way too late. They should learn that as soon as they can count to ten.

It might rankle to learn that you've been doing something wrong for years, but I can only offer the defence that it's the simple truth. Since the counting sequence in base-ten counting is ".... nine, ten, ten and one" etc, the counting sequence in base-five must be ".... four, five, five and one," and so on. Of course other words may be used, but for beginners that must be the best and clearest way to go

I really hope my style, or lack of, doesn't set you against me. As I see it, math education produces perhaps nine math-averse school graduates for every math-confident one. There is massive room for improvement, and the most glaring example is counting, which can be fixed overnight, for free. Any Korean five-year old can teach younger sibs and playmates how to count to a hundred in a few minutes, and they themselves probably learnt it the same way, from other children, and that's because they have to proper tools to do it.

In contrast I read something recently about a new initiative which might see US children being able to count to 20 by the time they start school. Now that's just sad.

All it takes is for pre-school teachers to teach base-ten counting, which requires no permission from parents, no training, no books and no more instruction than could be contained in a couple of tweets. Frankly I wouldn't waste time on also teaching them the vernacular sequence: better for them to learn that for themselves, which they will easily do, and by so doing they will own it. Better to get out of the way.

And yet the considered opinion of experts, such as Stanislas Dehaene in "The Number Sense", is that introducing base-ten counting is impossible. That's because they wrongly presume that doing so would mean abolishing the vernacular counting sequence, which would indeed be impossible.But as the Welsh example shows, base-ten counting can be introduced without any opposition or trouble, and Welsh-speakers today freely use their traditional counting numbers and base-ten, depending on the situation.

Hmm, here I am again, on a soap-box, riding my hobby horse OT. Better stop now or I'll start getting the dreaded TLDR.
 
Your question was not about introducing the topic to children, but about what is correct, or how they are "usually" read.

Moreover, "two fives and one" is not a way of reading, but a way to describe the meaning. Each has its place. I would say that in the same contexts in which I might say "two tens and one" in place of "twenty-one" -- that is, only in discussing what "21" means, not in reading it aloud. Yes, it's a good idea in some situations, but it isn't the answer to "how is it usually read aloud".

On reflection I realized that you've got this back to front: when we read 14 as "fourteen" we are in fact reading the figures, but backwards,from right to left, and of course that's how we still teach children to read figures. It's obviously a bad thing to do, it's totally misleading and it is the kind of thing which destroys the trust of young students, when they later discover that numbers should be read from left to right.

When we read 11 and 12 as eleven and twelve we are not reading the figures at all, we are using abbreviations which are short for "ten and eleven" and "ten and two elve" which translated into English mean "ten and one left" and "ten and two left" So the words eleven and twelve contain no intelligible reference to the actual figures at all. Similarly someone who habitually reads the word "thousand" as "a grand" is not reading verbalising the word, or the letters, but substituting a synonym.

Saying "two tens and one" is in fact reading the figures perfectly using the decimal place value system. You can point to the actual figures and spell it out. "Twenty-one" is a distorted version which disguises the true meaning by expressing "ten" in pig-latin: "Two en-ty one".

Thanks again for your input.
 
On reflection I realized that you've got this back to front: when we read 14 as "fourteen" we are in fact reading the figures, but backwards,from right to left, and of course that's how we still teach children to read figures. It's obviously a bad thing to do, it's totally misleading and it is the kind of thing which destroys the trust of young students, when they later discover that numbers should be read from left to right.

When we read 11 and 12 as eleven and twelve we are not reading the figures at all, we are using abbreviations which are short for "ten and eleven" and "ten and two elve" which translated into English mean "ten and one left" and "ten and two left" So the words eleven and twelve contain no intelligible reference to the actual figures at all. Similarly someone who habitually reads the word "thousand" as "a grand" is not reading verbalising the word, or the letters, but substituting a synonym.

Saying "two tens and one" is in fact reading the figures perfectly using the decimal place value system. You can point to the actual figures and spell it out. "Twenty-one" is a distorted version which disguises the true meaning by expressing "ten" in pig-latin: "Two en-ty one".

Thanks again for your input.

I don't think you're looking for input at all.
 
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