Why do we order our numbers from most significant digit and not the least significant?
$begingroup$
Not sure if this belongs here, but it occurred to me that when I add two numbers, I start adding them from the left to the right. Probably because of the simple fact that I read from left to right, or maybe because it feels natural to repeat myself while adding (in the example below): "5+1.. six hundred...4+3..six hundred seventy... 8+3.. six hundred eighty one".
If I were to add 148 and 533, in my head, I'd go:
- 1+5, makes 6
- 4+3, makes 7
- 8+3, makes 11, let's modify that last one and make that 8 instead
Consider flipping it around. Assume were to write it like this: 841 + 335:
- 8+3, makes 11, in other words, 1 and I'm going to add 1 to the next sum
- 4+3 makes 7, but we have 1 from the last sum, so 8
- 1+5 makes 6
Is there any obvious good (historic?) reason to why we write it the way we do? Are the numbers easier to read from left to right because of that's the way it has always been done, or because of some other reason? Do we ever speak of endianness in other situations than for base 2? For languages where you read from right to left, how does one usually mentally add numbers?
math-history significant-figures
asked Dec 13 '18 at 22:15
Daniel OlssonDaniel Olsson
1183
$endgroup$
|
show 5 more comments
$begingroup$
Not sure if this belongs here, but it occurred to me that when I add two numbers, I start adding them from the left to the right. Probably because of the simple fact that I read from left to right, or maybe because it feels natural to repeat myself while adding (in the example below): "5+1.. six hundred...4+3..six hundred seventy... 8+3.. six hundred eighty one".
If I were to add 148 and 533, in my head, I'd go:
- 1+5, makes 6
- 4+3, makes 7
- 8+3, makes 11, let's modify that last one and make that 8 instead
Consider flipping it around. Assume were to write it like this: 841 + 335:
- 8+3, makes 11, in other words, 1 and I'm going to add 1 to the next sum
- 4+3 makes 7, but we have 1 from the last sum, so 8
- 1+5 makes 6
Is there any obvious good (historic?) reason to why we write it the way we do? Are the numbers easier to read from left to right because of that's the way it has always been done, or because of some other reason? Do we ever speak of endianness in other situations than for base 2? For languages where you read from right to left, how does one usually mentally add numbers?
math-history significant-figures
asked Dec 13 '18 at 22:15
Daniel OlssonDaniel Olsson
1183
$endgroup$
$begingroup$
I do the same as you for 148 + 533, but that is a 'nice' example. When it's 987 + 654 it's more difficult imo to do it that way.
$endgroup$
– T. Fo
Dec 13 '18 at 22:19
$begingroup$
Regardless of how we perform addition mentally, in writing, we start adding two numbers by adding their ones' digit and carrying over the excess to the tens' place and so on from right to left. At-least that was how I was taught addition, and just to be clear I have only known languages read left-to-right.
$endgroup$
– Shubham Johri
Dec 13 '18 at 22:19
1
$begingroup$
I doubt it. We read left to right and that a six digit number between 500000 and 599999 is a lot more important than that it end with a 2. So it makes perfect sense that we list them left to right just as we read. As for adding, the only reason we add them "backwords" from least important to most important is that if we need to carry we carry to the next step-- we don't have to then double back to what we did before. But we COULD do it in your direction and there ARE advantages. For one thing if we only care about the mos significant three digits we can stop after three.
$endgroup$
– fleablood
Dec 13 '18 at 22:22
$begingroup$
It's interesting that both you and T. Ford instinctively do it in your head that way. I guess I had it pounded in my head to do it the other way. If I were asked what approximately (within 50) is $148+ 533$ I'd have to force myself to go in your direction which would be a lot mor appropriate.
$endgroup$
– fleablood
Dec 13 '18 at 22:26
1
$begingroup$
You might enjoy "exploding dots". These algorithms allow any value in any column, and do all the carrying at the end: gdaymath.com/courses/exploding-dots
$endgroup$
– Ethan Bolker
Dec 13 '18 at 23:10
|
show 5 more comments
$begingroup$
Not sure if this belongs here, but it occurred to me that when I add two numbers, I start adding them from the left to the right. Probably because of the simple fact that I read from left to right, or maybe because it feels natural to repeat myself while adding (in the example below): "5+1.. six hundred...4+3..six hundred seventy... 8+3.. six hundred eighty one".
If I were to add 148 and 533, in my head, I'd go:
- 1+5, makes 6
- 4+3, makes 7
- 8+3, makes 11, let's modify that last one and make that 8 instead
Consider flipping it around. Assume were to write it like this: 841 + 335:
- 8+3, makes 11, in other words, 1 and I'm going to add 1 to the next sum
- 4+3 makes 7, but we have 1 from the last sum, so 8
- 1+5 makes 6
Is there any obvious good (historic?) reason to why we write it the way we do? Are the numbers easier to read from left to right because of that's the way it has always been done, or because of some other reason? Do we ever speak of endianness in other situations than for base 2? For languages where you read from right to left, how does one usually mentally add numbers?
math-history significant-figures
asked Dec 13 '18 at 22:15
Daniel OlssonDaniel Olsson
1183
$endgroup$
Not sure if this belongs here, but it occurred to me that when I add two numbers, I start adding them from the left to the right. Probably because of the simple fact that I read from left to right, or maybe because it feels natural to repeat myself while adding (in the example below): "5+1.. six hundred...4+3..six hundred seventy... 8+3.. six hundred eighty one".
If I were to add 148 and 533, in my head, I'd go:
- 1+5, makes 6
- 4+3, makes 7
- 8+3, makes 11, let's modify that last one and make that 8 instead
Consider flipping it around. Assume were to write it like this: 841 + 335:
- 8+3, makes 11, in other words, 1 and I'm going to add 1 to the next sum
- 4+3 makes 7, but we have 1 from the last sum, so 8
- 1+5 makes 6
Is there any obvious good (historic?) reason to why we write it the way we do? Are the numbers easier to read from left to right because of that's the way it has always been done, or because of some other reason? Do we ever speak of endianness in other situations than for base 2? For languages where you read from right to left, how does one usually mentally add numbers?
math-history significant-figures
math-history significant-figures
asked Dec 13 '18 at 22:15
Daniel OlssonDaniel Olsson
1183
asked Dec 13 '18 at 22:15
Daniel OlssonDaniel Olsson
1183
edited Dec 14 '18 at 15:26
Daniel Olsson
asked Dec 13 '18 at 22:15
Daniel OlssonDaniel Olsson
1183
asked Dec 13 '18 at 22:15
Daniel OlssonDaniel Olsson
1183
asked Dec 13 '18 at 22:15
Daniel OlssonDaniel Olsson
1183
1183
$begingroup$
I do the same as you for 148 + 533, but that is a 'nice' example. When it's 987 + 654 it's more difficult imo to do it that way.
$endgroup$
– T. Fo
Dec 13 '18 at 22:19
$begingroup$
Regardless of how we perform addition mentally, in writing, we start adding two numbers by adding their ones' digit and carrying over the excess to the tens' place and so on from right to left. At-least that was how I was taught addition, and just to be clear I have only known languages read left-to-right.
$endgroup$
– Shubham Johri
Dec 13 '18 at 22:19
1
$begingroup$
I doubt it. We read left to right and that a six digit number between 500000 and 599999 is a lot more important than that it end with a 2. So it makes perfect sense that we list them left to right just as we read. As for adding, the only reason we add them "backwords" from least important to most important is that if we need to carry we carry to the next step-- we don't have to then double back to what we did before. But we COULD do it in your direction and there ARE advantages. For one thing if we only care about the mos significant three digits we can stop after three.
$endgroup$
– fleablood
Dec 13 '18 at 22:22
$begingroup$
It's interesting that both you and T. Ford instinctively do it in your head that way. I guess I had it pounded in my head to do it the other way. If I were asked what approximately (within 50) is $148+ 533$ I'd have to force myself to go in your direction which would be a lot mor appropriate.
$endgroup$
– fleablood
Dec 13 '18 at 22:26
1
$begingroup$
You might enjoy "exploding dots". These algorithms allow any value in any column, and do all the carrying at the end: gdaymath.com/courses/exploding-dots
$endgroup$
– Ethan Bolker
Dec 13 '18 at 23:10
|
show 5 more comments
$begingroup$
I do the same as you for 148 + 533, but that is a 'nice' example. When it's 987 + 654 it's more difficult imo to do it that way.
$endgroup$
– T. Fo
Dec 13 '18 at 22:19
$begingroup$
Regardless of how we perform addition mentally, in writing, we start adding two numbers by adding their ones' digit and carrying over the excess to the tens' place and so on from right to left. At-least that was how I was taught addition, and just to be clear I have only known languages read left-to-right.
$endgroup$
– Shubham Johri
Dec 13 '18 at 22:19
1
$begingroup$
I doubt it. We read left to right and that a six digit number between 500000 and 599999 is a lot more important than that it end with a 2. So it makes perfect sense that we list them left to right just as we read. As for adding, the only reason we add them "backwords" from least important to most important is that if we need to carry we carry to the next step-- we don't have to then double back to what we did before. But we COULD do it in your direction and there ARE advantages. For one thing if we only care about the mos significant three digits we can stop after three.
$endgroup$
– fleablood
Dec 13 '18 at 22:22
$begingroup$
It's interesting that both you and T. Ford instinctively do it in your head that way. I guess I had it pounded in my head to do it the other way. If I were asked what approximately (within 50) is $148+ 533$ I'd have to force myself to go in your direction which would be a lot mor appropriate.
$endgroup$
– fleablood
Dec 13 '18 at 22:26
1
$begingroup$
You might enjoy "exploding dots". These algorithms allow any value in any column, and do all the carrying at the end: gdaymath.com/courses/exploding-dots
$endgroup$
– Ethan Bolker
Dec 13 '18 at 23:10
$begingroup$
I do the same as you for 148 + 533, but that is a 'nice' example. When it's 987 + 654 it's more difficult imo to do it that way.
$endgroup$
– T. Fo
Dec 13 '18 at 22:19
$begingroup$
I do the same as you for 148 + 533, but that is a 'nice' example. When it's 987 + 654 it's more difficult imo to do it that way.
$endgroup$
– T. Fo
Dec 13 '18 at 22:19
$begingroup$
Regardless of how we perform addition mentally, in writing, we start adding two numbers by adding their ones' digit and carrying over the excess to the tens' place and so on from right to left. At-least that was how I was taught addition, and just to be clear I have only known languages read left-to-right.
$endgroup$
– Shubham Johri
Dec 13 '18 at 22:19
$begingroup$
Regardless of how we perform addition mentally, in writing, we start adding two numbers by adding their ones' digit and carrying over the excess to the tens' place and so on from right to left. At-least that was how I was taught addition, and just to be clear I have only known languages read left-to-right.
$endgroup$
– Shubham Johri
Dec 13 '18 at 22:19
1
1
$begingroup$
I doubt it. We read left to right and that a six digit number between 500000 and 599999 is a lot more important than that it end with a 2. So it makes perfect sense that we list them left to right just as we read. As for adding, the only reason we add them "backwords" from least important to most important is that if we need to carry we carry to the next step-- we don't have to then double back to what we did before. But we COULD do it in your direction and there ARE advantages. For one thing if we only care about the mos significant three digits we can stop after three.
$endgroup$
– fleablood
Dec 13 '18 at 22:22
$begingroup$
I doubt it. We read left to right and that a six digit number between 500000 and 599999 is a lot more important than that it end with a 2. So it makes perfect sense that we list them left to right just as we read. As for adding, the only reason we add them "backwords" from least important to most important is that if we need to carry we carry to the next step-- we don't have to then double back to what we did before. But we COULD do it in your direction and there ARE advantages. For one thing if we only care about the mos significant three digits we can stop after three.
$endgroup$
– fleablood
Dec 13 '18 at 22:22
$begingroup$
It's interesting that both you and T. Ford instinctively do it in your head that way. I guess I had it pounded in my head to do it the other way. If I were asked what approximately (within 50) is $148+ 533$ I'd have to force myself to go in your direction which would be a lot mor appropriate.
$endgroup$
– fleablood
Dec 13 '18 at 22:26
$begingroup$
It's interesting that both you and T. Ford instinctively do it in your head that way. I guess I had it pounded in my head to do it the other way. If I were asked what approximately (within 50) is $148+ 533$ I'd have to force myself to go in your direction which would be a lot mor appropriate.
$endgroup$
– fleablood
Dec 13 '18 at 22:26
1
1
$begingroup$
You might enjoy "exploding dots". These algorithms allow any value in any column, and do all the carrying at the end: gdaymath.com/courses/exploding-dots
$endgroup$
– Ethan Bolker
Dec 13 '18 at 23:10
$begingroup$
You might enjoy "exploding dots". These algorithms allow any value in any column, and do all the carrying at the end: gdaymath.com/courses/exploding-dots
$endgroup$
– Ethan Bolker
Dec 13 '18 at 23:10
|
show 5 more comments
3 Answers
3
active
oldest
votes
$begingroup$
Not a specifically historical reason because I don't know the history, but two plausible practical reasons that come to mind:
- with infinite decimals, there's no last digit to start from
- in everyday situations, the most significant digit is literally the one most significant to us: for example if something costs £$1249.99$ then the $1$ at the beginning is far more important to my ability to afford it than the $9$ at the end. The notation gives us the important information first—it's over £$1000$—then fills in the details.
This contrasts, though, with older phrases like five-and-twenty (and their equivalents still in use in some languages). But maybe that phrase is designed to emphasise twenty. I can't really take that further without making this a linguistics or psychology question.
answered Dec 13 '18 at 22:51
timtfjtimtfj
2,503420
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add a comment |
$begingroup$
When working with pencil and paper the advantage of starting at the ones place is that you don't have to change anything you have already written down because of carries. That is the way I was taught in school. When working in my head I tend to go down from the top as you do. I do the same when multiplying, I multiply the two most significant figures, then each most significant by the other next most, and so on, adding carries as I go. One big advantage is if you just want an approximate answer you can stop at the appropriate time.
answered Dec 13 '18 at 22:22
Ross MillikanRoss Millikan
301k24200375
$endgroup$
add a comment |
$begingroup$
There is a difference between
ordering numbers
and operating on them.
When ordering them,
the basic operation is
(usually) comparing them.
Especially when comparing reals,
there may be no lowest order digit,
so the comparison is best done
from the high order digits.
When operating
(e.g., add or multiply),
we are usually given the complete number
and want to obtain the result,
so it does not matter
if the operation is done
from the high digits
or the low digits.
answered Dec 13 '18 at 22:53
marty cohenmarty cohen
75k549130
$endgroup$
$begingroup$
And if we only want a rough answer, we get that by operating on the first digit or two.
$endgroup$
– timtfj
Dec 13 '18 at 22:57
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Not a specifically historical reason because I don't know the history, but two plausible practical reasons that come to mind:
- with infinite decimals, there's no last digit to start from
- in everyday situations, the most significant digit is literally the one most significant to us: for example if something costs £$1249.99$ then the $1$ at the beginning is far more important to my ability to afford it than the $9$ at the end. The notation gives us the important information first—it's over £$1000$—then fills in the details.
This contrasts, though, with older phrases like five-and-twenty (and their equivalents still in use in some languages). But maybe that phrase is designed to emphasise twenty. I can't really take that further without making this a linguistics or psychology question.
answered Dec 13 '18 at 22:51
timtfjtimtfj
2,503420
$endgroup$
add a comment |
$begingroup$
Not a specifically historical reason because I don't know the history, but two plausible practical reasons that come to mind:
- with infinite decimals, there's no last digit to start from
- in everyday situations, the most significant digit is literally the one most significant to us: for example if something costs £$1249.99$ then the $1$ at the beginning is far more important to my ability to afford it than the $9$ at the end. The notation gives us the important information first—it's over £$1000$—then fills in the details.
This contrasts, though, with older phrases like five-and-twenty (and their equivalents still in use in some languages). But maybe that phrase is designed to emphasise twenty. I can't really take that further without making this a linguistics or psychology question.
answered Dec 13 '18 at 22:51
timtfjtimtfj
2,503420
$endgroup$
add a comment |
$begingroup$
Not a specifically historical reason because I don't know the history, but two plausible practical reasons that come to mind:
- with infinite decimals, there's no last digit to start from
- in everyday situations, the most significant digit is literally the one most significant to us: for example if something costs £$1249.99$ then the $1$ at the beginning is far more important to my ability to afford it than the $9$ at the end. The notation gives us the important information first—it's over £$1000$—then fills in the details.
This contrasts, though, with older phrases like five-and-twenty (and their equivalents still in use in some languages). But maybe that phrase is designed to emphasise twenty. I can't really take that further without making this a linguistics or psychology question.
answered Dec 13 '18 at 22:51
timtfjtimtfj
2,503420
$endgroup$
Not a specifically historical reason because I don't know the history, but two plausible practical reasons that come to mind:
- with infinite decimals, there's no last digit to start from
- in everyday situations, the most significant digit is literally the one most significant to us: for example if something costs £$1249.99$ then the $1$ at the beginning is far more important to my ability to afford it than the $9$ at the end. The notation gives us the important information first—it's over £$1000$—then fills in the details.
This contrasts, though, with older phrases like five-and-twenty (and their equivalents still in use in some languages). But maybe that phrase is designed to emphasise twenty. I can't really take that further without making this a linguistics or psychology question.
answered Dec 13 '18 at 22:51
timtfjtimtfj
2,503420
edited Dec 13 '18 at 23:06
answered Dec 13 '18 at 22:51
timtfjtimtfj
2,503420
answered Dec 13 '18 at 22:51
timtfjtimtfj
2,503420
answered Dec 13 '18 at 22:51
timtfjtimtfj
2,503420
2,503420
add a comment |
add a comment |
$begingroup$
When working with pencil and paper the advantage of starting at the ones place is that you don't have to change anything you have already written down because of carries. That is the way I was taught in school. When working in my head I tend to go down from the top as you do. I do the same when multiplying, I multiply the two most significant figures, then each most significant by the other next most, and so on, adding carries as I go. One big advantage is if you just want an approximate answer you can stop at the appropriate time.
answered Dec 13 '18 at 22:22
Ross MillikanRoss Millikan
301k24200375
$endgroup$
add a comment |
$begingroup$
When working with pencil and paper the advantage of starting at the ones place is that you don't have to change anything you have already written down because of carries. That is the way I was taught in school. When working in my head I tend to go down from the top as you do. I do the same when multiplying, I multiply the two most significant figures, then each most significant by the other next most, and so on, adding carries as I go. One big advantage is if you just want an approximate answer you can stop at the appropriate time.
answered Dec 13 '18 at 22:22
Ross MillikanRoss Millikan
301k24200375
$endgroup$
add a comment |
$begingroup$
When working with pencil and paper the advantage of starting at the ones place is that you don't have to change anything you have already written down because of carries. That is the way I was taught in school. When working in my head I tend to go down from the top as you do. I do the same when multiplying, I multiply the two most significant figures, then each most significant by the other next most, and so on, adding carries as I go. One big advantage is if you just want an approximate answer you can stop at the appropriate time.
answered Dec 13 '18 at 22:22
Ross MillikanRoss Millikan
301k24200375
$endgroup$
When working with pencil and paper the advantage of starting at the ones place is that you don't have to change anything you have already written down because of carries. That is the way I was taught in school. When working in my head I tend to go down from the top as you do. I do the same when multiplying, I multiply the two most significant figures, then each most significant by the other next most, and so on, adding carries as I go. One big advantage is if you just want an approximate answer you can stop at the appropriate time.
answered Dec 13 '18 at 22:22
Ross MillikanRoss Millikan
301k24200375
answered Dec 13 '18 at 22:22
Ross MillikanRoss Millikan
301k24200375
answered Dec 13 '18 at 22:22
Ross MillikanRoss Millikan
301k24200375
answered Dec 13 '18 at 22:22
Ross MillikanRoss Millikan
301k24200375
301k24200375
add a comment |
add a comment |
$begingroup$
There is a difference between
ordering numbers
and operating on them.
When ordering them,
the basic operation is
(usually) comparing them.
Especially when comparing reals,
there may be no lowest order digit,
so the comparison is best done
from the high order digits.
When operating
(e.g., add or multiply),
we are usually given the complete number
and want to obtain the result,
so it does not matter
if the operation is done
from the high digits
or the low digits.
answered Dec 13 '18 at 22:53
marty cohenmarty cohen
75k549130
$endgroup$
$begingroup$
And if we only want a rough answer, we get that by operating on the first digit or two.
$endgroup$
– timtfj
Dec 13 '18 at 22:57
add a comment |
$begingroup$
There is a difference between
ordering numbers
and operating on them.
When ordering them,
the basic operation is
(usually) comparing them.
Especially when comparing reals,
there may be no lowest order digit,
so the comparison is best done
from the high order digits.
When operating
(e.g., add or multiply),
we are usually given the complete number
and want to obtain the result,
so it does not matter
if the operation is done
from the high digits
or the low digits.
answered Dec 13 '18 at 22:53
marty cohenmarty cohen
75k549130
$endgroup$
$begingroup$
And if we only want a rough answer, we get that by operating on the first digit or two.
$endgroup$
– timtfj
Dec 13 '18 at 22:57
add a comment |
$begingroup$
There is a difference between
ordering numbers
and operating on them.
When ordering them,
the basic operation is
(usually) comparing them.
Especially when comparing reals,
there may be no lowest order digit,
so the comparison is best done
from the high order digits.
When operating
(e.g., add or multiply),
we are usually given the complete number
and want to obtain the result,
so it does not matter
if the operation is done
from the high digits
or the low digits.
answered Dec 13 '18 at 22:53
marty cohenmarty cohen
75k549130
$endgroup$
There is a difference between
ordering numbers
and operating on them.
When ordering them,
the basic operation is
(usually) comparing them.
Especially when comparing reals,
there may be no lowest order digit,
so the comparison is best done
from the high order digits.
When operating
(e.g., add or multiply),
we are usually given the complete number
and want to obtain the result,
so it does not matter
if the operation is done
from the high digits
or the low digits.
answered Dec 13 '18 at 22:53
marty cohenmarty cohen
75k549130
answered Dec 13 '18 at 22:53
marty cohenmarty cohen
75k549130
answered Dec 13 '18 at 22:53
marty cohenmarty cohen
75k549130
answered Dec 13 '18 at 22:53
marty cohenmarty cohen
75k549130
75k549130
$begingroup$
And if we only want a rough answer, we get that by operating on the first digit or two.
$endgroup$
– timtfj
Dec 13 '18 at 22:57
add a comment |
$begingroup$
And if we only want a rough answer, we get that by operating on the first digit or two.
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– timtfj
Dec 13 '18 at 22:57
$begingroup$
And if we only want a rough answer, we get that by operating on the first digit or two.
$endgroup$
– timtfj
Dec 13 '18 at 22:57
$begingroup$
And if we only want a rough answer, we get that by operating on the first digit or two.
$endgroup$
– timtfj
Dec 13 '18 at 22:57
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I do the same as you for 148 + 533, but that is a 'nice' example. When it's 987 + 654 it's more difficult imo to do it that way.
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– T. Fo
Dec 13 '18 at 22:19
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Regardless of how we perform addition mentally, in writing, we start adding two numbers by adding their ones' digit and carrying over the excess to the tens' place and so on from right to left. At-least that was how I was taught addition, and just to be clear I have only known languages read left-to-right.
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– Shubham Johri
Dec 13 '18 at 22:19
1
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I doubt it. We read left to right and that a six digit number between 500000 and 599999 is a lot more important than that it end with a 2. So it makes perfect sense that we list them left to right just as we read. As for adding, the only reason we add them "backwords" from least important to most important is that if we need to carry we carry to the next step-- we don't have to then double back to what we did before. But we COULD do it in your direction and there ARE advantages. For one thing if we only care about the mos significant three digits we can stop after three.
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– fleablood
Dec 13 '18 at 22:22
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It's interesting that both you and T. Ford instinctively do it in your head that way. I guess I had it pounded in my head to do it the other way. If I were asked what approximately (within 50) is $148+ 533$ I'd have to force myself to go in your direction which would be a lot mor appropriate.
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– fleablood
Dec 13 '18 at 22:26
1
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You might enjoy "exploding dots". These algorithms allow any value in any column, and do all the carrying at the end: gdaymath.com/courses/exploding-dots
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– Ethan Bolker
Dec 13 '18 at 23:10