Mixing
result of multiplication having zeroes after the decimal
$begingroup$
Given the multiplication $3.25 times 0.4$,
the primary school students learn that
- we multiply the digit 4 to the number 325 which result in 1300
- we count and then add the number of decimal place existing in 3.25 and 0.4 corresponding to 3
- we move the decimal point 3 times to the right which corresponding to 1.300
My main question regards those two zeroes at the back and how to justify it.
Is this incorrect to simplify the number to 1.3? or the opposite is this incorrect to leave the answer as $1.300$?
My approach to this would be to say that if one keep the answer as $1.300$, this number can be interpreted as the result of an approximation such as , for example, $1.2999 approx 1.300$ correct to 4 significant figure
approximation computational-mathematics
edited Jan 24 at 22:55
Bernard
123k741116
asked Jan 24 at 22:51
gegugegu
829423
$endgroup$
|
show 1 more comment
$begingroup$
Given the multiplication $3.25 times 0.4$,
the primary school students learn that
- we multiply the digit 4 to the number 325 which result in 1300
- we count and then add the number of decimal place existing in 3.25 and 0.4 corresponding to 3
- we move the decimal point 3 times to the right which corresponding to 1.300
My main question regards those two zeroes at the back and how to justify it.
Is this incorrect to simplify the number to 1.3? or the opposite is this incorrect to leave the answer as $1.300$?
My approach to this would be to say that if one keep the answer as $1.300$, this number can be interpreted as the result of an approximation such as , for example, $1.2999 approx 1.300$ correct to 4 significant figure
approximation computational-mathematics
edited Jan 24 at 22:55
Bernard
123k741116
asked Jan 24 at 22:51
gegugegu
829423
$endgroup$
1
$begingroup$
Personally, I'd write it as $1.3$.
$endgroup$
– Bernard
Jan 24 at 22:56
$begingroup$
depends on the context. Are sig figs important? or is succinctness preferable?
$endgroup$
– Chase Ryan Taylor
Jan 24 at 22:56
1
$begingroup$
If the answer is supposed to be exact, it doesn't matter, but if the numbers come from some measurement, then keeping the extra zeroes is important.
$endgroup$
– herb steinberg
Jan 24 at 22:57
1
$begingroup$
$1.3 = 1 + frac 3{10}$ and $1.300 = 1 + frac 3{10} + frac 0{100} + frac 0{100}$. And $1 + frac 3{10} = 1 + frac 3{10} + frac 0{100} + frac 0{100}$ so they are exactly the same thing. Or $3.25 times .4= 325times frac 1{100}times 4times frac 1{10} = 1300times frac 1{1000} = 13timesfrac 1 {10} = 1.3$ We don't need those zeros as they factor out. That we got them when we multiplied $4times 325$ isn't relevant as we don't need them to hold any places that aren't being used.
$endgroup$
– fleablood
Jan 25 at 0:02
1
$begingroup$
I think you should ask this at Mathematics Educators Stack Exchange, where you'll get answers which take into account the likely understanding of primary school children and the possible pitfalls of teaching one approach or another. That's as important as the mathematics really—for instance you don't want to accidentally train them to think multiplying several approximate values on their calculators gives a result accurate to all the decimal places that magically appear!
$endgroup$
– timtfj
Jan 25 at 11:44
|
show 1 more comment
$begingroup$
Given the multiplication $3.25 times 0.4$,
the primary school students learn that
- we multiply the digit 4 to the number 325 which result in 1300
- we count and then add the number of decimal place existing in 3.25 and 0.4 corresponding to 3
- we move the decimal point 3 times to the right which corresponding to 1.300
My main question regards those two zeroes at the back and how to justify it.
Is this incorrect to simplify the number to 1.3? or the opposite is this incorrect to leave the answer as $1.300$?
My approach to this would be to say that if one keep the answer as $1.300$, this number can be interpreted as the result of an approximation such as , for example, $1.2999 approx 1.300$ correct to 4 significant figure
approximation computational-mathematics
edited Jan 24 at 22:55
Bernard
123k741116
asked Jan 24 at 22:51
gegugegu
829423
$endgroup$
Given the multiplication $3.25 times 0.4$,
the primary school students learn that
- we multiply the digit 4 to the number 325 which result in 1300
- we count and then add the number of decimal place existing in 3.25 and 0.4 corresponding to 3
- we move the decimal point 3 times to the right which corresponding to 1.300
My main question regards those two zeroes at the back and how to justify it.
Is this incorrect to simplify the number to 1.3? or the opposite is this incorrect to leave the answer as $1.300$?
My approach to this would be to say that if one keep the answer as $1.300$, this number can be interpreted as the result of an approximation such as , for example, $1.2999 approx 1.300$ correct to 4 significant figure
approximation computational-mathematics
approximation computational-mathematics
edited Jan 24 at 22:55
Bernard
123k741116
asked Jan 24 at 22:51
gegugegu
829423
edited Jan 24 at 22:55
Bernard
123k741116
asked Jan 24 at 22:51
gegugegu
829423
edited Jan 24 at 22:55
Bernard
123k741116
edited Jan 24 at 22:55
Bernard
123k741116
edited Jan 24 at 22:55
Bernard
123k741116
123k741116
asked Jan 24 at 22:51
gegugegu
829423
asked Jan 24 at 22:51
gegugegu
829423
asked Jan 24 at 22:51
gegugegu
829423
829423
1
$begingroup$
Personally, I'd write it as $1.3$.
$endgroup$
– Bernard
Jan 24 at 22:56
$begingroup$
depends on the context. Are sig figs important? or is succinctness preferable?
$endgroup$
– Chase Ryan Taylor
Jan 24 at 22:56
1
$begingroup$
If the answer is supposed to be exact, it doesn't matter, but if the numbers come from some measurement, then keeping the extra zeroes is important.
$endgroup$
– herb steinberg
Jan 24 at 22:57
1
$begingroup$
$1.3 = 1 + frac 3{10}$ and $1.300 = 1 + frac 3{10} + frac 0{100} + frac 0{100}$. And $1 + frac 3{10} = 1 + frac 3{10} + frac 0{100} + frac 0{100}$ so they are exactly the same thing. Or $3.25 times .4= 325times frac 1{100}times 4times frac 1{10} = 1300times frac 1{1000} = 13timesfrac 1 {10} = 1.3$ We don't need those zeros as they factor out. That we got them when we multiplied $4times 325$ isn't relevant as we don't need them to hold any places that aren't being used.
$endgroup$
– fleablood
Jan 25 at 0:02
1
$begingroup$
I think you should ask this at Mathematics Educators Stack Exchange, where you'll get answers which take into account the likely understanding of primary school children and the possible pitfalls of teaching one approach or another. That's as important as the mathematics really—for instance you don't want to accidentally train them to think multiplying several approximate values on their calculators gives a result accurate to all the decimal places that magically appear!
$endgroup$
– timtfj
Jan 25 at 11:44
|
show 1 more comment
1
$begingroup$
Personally, I'd write it as $1.3$.
$endgroup$
– Bernard
Jan 24 at 22:56
$begingroup$
depends on the context. Are sig figs important? or is succinctness preferable?
$endgroup$
– Chase Ryan Taylor
Jan 24 at 22:56
1
$begingroup$
If the answer is supposed to be exact, it doesn't matter, but if the numbers come from some measurement, then keeping the extra zeroes is important.
$endgroup$
– herb steinberg
Jan 24 at 22:57
1
$begingroup$
$1.3 = 1 + frac 3{10}$ and $1.300 = 1 + frac 3{10} + frac 0{100} + frac 0{100}$. And $1 + frac 3{10} = 1 + frac 3{10} + frac 0{100} + frac 0{100}$ so they are exactly the same thing. Or $3.25 times .4= 325times frac 1{100}times 4times frac 1{10} = 1300times frac 1{1000} = 13timesfrac 1 {10} = 1.3$ We don't need those zeros as they factor out. That we got them when we multiplied $4times 325$ isn't relevant as we don't need them to hold any places that aren't being used.
$endgroup$
– fleablood
Jan 25 at 0:02
1
$begingroup$
I think you should ask this at Mathematics Educators Stack Exchange, where you'll get answers which take into account the likely understanding of primary school children and the possible pitfalls of teaching one approach or another. That's as important as the mathematics really—for instance you don't want to accidentally train them to think multiplying several approximate values on their calculators gives a result accurate to all the decimal places that magically appear!
$endgroup$
– timtfj
Jan 25 at 11:44
1
1
$begingroup$
Personally, I'd write it as $1.3$.
$endgroup$
– Bernard
Jan 24 at 22:56
$begingroup$
Personally, I'd write it as $1.3$.
$endgroup$
– Bernard
Jan 24 at 22:56
$begingroup$
depends on the context. Are sig figs important? or is succinctness preferable?
$endgroup$
– Chase Ryan Taylor
Jan 24 at 22:56
$begingroup$
depends on the context. Are sig figs important? or is succinctness preferable?
$endgroup$
– Chase Ryan Taylor
Jan 24 at 22:56
1
1
$begingroup$
If the answer is supposed to be exact, it doesn't matter, but if the numbers come from some measurement, then keeping the extra zeroes is important.
$endgroup$
– herb steinberg
Jan 24 at 22:57
$begingroup$
If the answer is supposed to be exact, it doesn't matter, but if the numbers come from some measurement, then keeping the extra zeroes is important.
$endgroup$
– herb steinberg
Jan 24 at 22:57
1
1
$begingroup$
$1.3 = 1 + frac 3{10}$ and $1.300 = 1 + frac 3{10} + frac 0{100} + frac 0{100}$. And $1 + frac 3{10} = 1 + frac 3{10} + frac 0{100} + frac 0{100}$ so they are exactly the same thing. Or $3.25 times .4= 325times frac 1{100}times 4times frac 1{10} = 1300times frac 1{1000} = 13timesfrac 1 {10} = 1.3$ We don't need those zeros as they factor out. That we got them when we multiplied $4times 325$ isn't relevant as we don't need them to hold any places that aren't being used.
$endgroup$
– fleablood
Jan 25 at 0:02
$begingroup$
$1.3 = 1 + frac 3{10}$ and $1.300 = 1 + frac 3{10} + frac 0{100} + frac 0{100}$. And $1 + frac 3{10} = 1 + frac 3{10} + frac 0{100} + frac 0{100}$ so they are exactly the same thing. Or $3.25 times .4= 325times frac 1{100}times 4times frac 1{10} = 1300times frac 1{1000} = 13timesfrac 1 {10} = 1.3$ We don't need those zeros as they factor out. That we got them when we multiplied $4times 325$ isn't relevant as we don't need them to hold any places that aren't being used.
$endgroup$
– fleablood
Jan 25 at 0:02
1
1
$begingroup$
I think you should ask this at Mathematics Educators Stack Exchange, where you'll get answers which take into account the likely understanding of primary school children and the possible pitfalls of teaching one approach or another. That's as important as the mathematics really—for instance you don't want to accidentally train them to think multiplying several approximate values on their calculators gives a result accurate to all the decimal places that magically appear!
$endgroup$
– timtfj
Jan 25 at 11:44
$begingroup$
I think you should ask this at Mathematics Educators Stack Exchange, where you'll get answers which take into account the likely understanding of primary school children and the possible pitfalls of teaching one approach or another. That's as important as the mathematics really—for instance you don't want to accidentally train them to think multiplying several approximate values on their calculators gives a result accurate to all the decimal places that magically appear!
$endgroup$
– timtfj
Jan 25 at 11:44
|
show 1 more comment
3 Answers
3
active
oldest
votes
$begingroup$
I think there are two separate issues here:
- the multiplication of two numbers, treated as precise
- the real-world validity of the level of precision.
Treated purely as arithmetic, it's obviously correct to say that the zeros can be discarded without affecting the result.
If you were multiplying two physical measurements, then the number of significant figures you kept would express the precision you believed the result to have, so it would be reasonable to either round the result to the same number of significant figures as the original numbers, or make an actual calculation of the expected error.
The thing with multiplication is that it normally produces an unrealistic number of significant figures.
So I think you try to teach them:
$3.25 × 0.4$ is $1.3$
- keeping the zeros in, or keeping lots of decimal places, means we're saying that the numbers we're using really are that accurate.
I think it's useful for them to know about the second one, even though they probably won't actually use it until much later. (I don't think I was actually introduced to the concept of significant figures until I was doing science subjects at secondary school.)
answered Jan 24 at 23:17
timtfjtimtfj
2,468420
$endgroup$
$begingroup$
Thx for your input. Actually, the context of this question was that in one class (first year of high school) almost all the students were showing me as result: 1.300 with no measurement involved.
$endgroup$
– gegu
Jan 25 at 18:08
add a comment |
$begingroup$
Decimal is only shorthand.
$3.25$ is shorthand for $3 + frac 2{10} + frac 5{100}$ and $.4$ is shorthand for $frac 4{10}$.
So $3.25 times .4 = (3 + frac 2{10} + frac 5{100})(frac 4{10}) = $
$frac {12}{10} + frac 8{100} + frac {20}{1000} =$
$frac {10}{10} + frac {2}{10} + frac 8{100} + frac 2{100}=$
$1 + frac {2}{10} + frac {10}{100} =$
$1 + frac 2{10} + frac 1{10} = $
$1 + frac 3{10} = $
$1.3$.
But $1 + frac 3{10} = 1 + frac 3{10} + frac 0{100} +frac 0{1000} + frac 0{10000}$
So it's equally correct to write this as $1.3000$
Or $1 + frac 3{10} = 0times 1000 + 0times 100 + 0times 10 + 1 + frac 3{10} + frac 0{100}$ so it would be okay (albeit weird) to write it as $001.30$.
We don't write those leading $0$s in the beginning because ... we don't have to and it'd be really weird to. Also we intuitively start at the biggest part of a number and work our way to the smaller so we don't usually think about all those (non-existent) powers of $10$ before we start.
Likewise we don't write all the trailing $0$s at the end because we don't have to. However because we started at the biggest part and worked to the smallest we often do think about those (non-existent) negative powers of $10$ after we end.
In fact $1.3$ is actually $1.30000000000.....$ with an infinite number of $0$ after the end. We don't write them because... we don't need to. (And we can't.)
I'd even argue that $1.3$ is actually $......00000001.3000000.....$ but we don't worry about any of the zeros that aren't saving a place between digits that are "doing work".
answered Jan 24 at 23:55
fleabloodfleablood
72.7k22788
$endgroup$
$begingroup$
thx for the idea of using the formulation of the expanded form to explain that it is the same number.
$endgroup$
– gegu
Jan 25 at 18:18
add a comment |
$begingroup$
Keep in mind, if you are going to introduce significant digits to grade school students, you need to better understand them yourself. 0.4 has 1 significant digit. To think that 3.25×0.4 = 1.300 is to think that the number of significant digits in a result is the sum of the two operands' significant digits. The truth is that it is the lesser of the 2.
In my opinion, the concept is best left for high schoolers, and your kids should just use 1.3 as the answer. That said, there's no harm in asking peers how they do it. I am in a high school, as an in-house math tutor, and often have to defer to how a teacher handles a topic or how the department does, regardless of my own preference.
answered Jan 27 at 16:24
JoeTaxpayerJoeTaxpayer
2,20121326
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3086468%2fresult-of-multiplication-having-zeroes-after-the-decimal%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I think there are two separate issues here:
- the multiplication of two numbers, treated as precise
- the real-world validity of the level of precision.
Treated purely as arithmetic, it's obviously correct to say that the zeros can be discarded without affecting the result.
If you were multiplying two physical measurements, then the number of significant figures you kept would express the precision you believed the result to have, so it would be reasonable to either round the result to the same number of significant figures as the original numbers, or make an actual calculation of the expected error.
The thing with multiplication is that it normally produces an unrealistic number of significant figures.
So I think you try to teach them:
$3.25 × 0.4$ is $1.3$
- keeping the zeros in, or keeping lots of decimal places, means we're saying that the numbers we're using really are that accurate.
I think it's useful for them to know about the second one, even though they probably won't actually use it until much later. (I don't think I was actually introduced to the concept of significant figures until I was doing science subjects at secondary school.)
answered Jan 24 at 23:17
timtfjtimtfj
2,468420
$endgroup$
$begingroup$
Thx for your input. Actually, the context of this question was that in one class (first year of high school) almost all the students were showing me as result: 1.300 with no measurement involved.
$endgroup$
– gegu
Jan 25 at 18:08
add a comment |
$begingroup$
I think there are two separate issues here:
- the multiplication of two numbers, treated as precise
- the real-world validity of the level of precision.
Treated purely as arithmetic, it's obviously correct to say that the zeros can be discarded without affecting the result.
If you were multiplying two physical measurements, then the number of significant figures you kept would express the precision you believed the result to have, so it would be reasonable to either round the result to the same number of significant figures as the original numbers, or make an actual calculation of the expected error.
The thing with multiplication is that it normally produces an unrealistic number of significant figures.
So I think you try to teach them:
$3.25 × 0.4$ is $1.3$
- keeping the zeros in, or keeping lots of decimal places, means we're saying that the numbers we're using really are that accurate.
I think it's useful for them to know about the second one, even though they probably won't actually use it until much later. (I don't think I was actually introduced to the concept of significant figures until I was doing science subjects at secondary school.)
answered Jan 24 at 23:17
timtfjtimtfj
2,468420
$endgroup$
$begingroup$
Thx for your input. Actually, the context of this question was that in one class (first year of high school) almost all the students were showing me as result: 1.300 with no measurement involved.
$endgroup$
– gegu
Jan 25 at 18:08
add a comment |
$begingroup$
I think there are two separate issues here:
- the multiplication of two numbers, treated as precise
- the real-world validity of the level of precision.
Treated purely as arithmetic, it's obviously correct to say that the zeros can be discarded without affecting the result.
If you were multiplying two physical measurements, then the number of significant figures you kept would express the precision you believed the result to have, so it would be reasonable to either round the result to the same number of significant figures as the original numbers, or make an actual calculation of the expected error.
The thing with multiplication is that it normally produces an unrealistic number of significant figures.
So I think you try to teach them:
$3.25 × 0.4$ is $1.3$
- keeping the zeros in, or keeping lots of decimal places, means we're saying that the numbers we're using really are that accurate.
I think it's useful for them to know about the second one, even though they probably won't actually use it until much later. (I don't think I was actually introduced to the concept of significant figures until I was doing science subjects at secondary school.)
answered Jan 24 at 23:17
timtfjtimtfj
2,468420
$endgroup$
I think there are two separate issues here:
- the multiplication of two numbers, treated as precise
- the real-world validity of the level of precision.
Treated purely as arithmetic, it's obviously correct to say that the zeros can be discarded without affecting the result.
If you were multiplying two physical measurements, then the number of significant figures you kept would express the precision you believed the result to have, so it would be reasonable to either round the result to the same number of significant figures as the original numbers, or make an actual calculation of the expected error.
The thing with multiplication is that it normally produces an unrealistic number of significant figures.
So I think you try to teach them:
$3.25 × 0.4$ is $1.3$
- keeping the zeros in, or keeping lots of decimal places, means we're saying that the numbers we're using really are that accurate.
I think it's useful for them to know about the second one, even though they probably won't actually use it until much later. (I don't think I was actually introduced to the concept of significant figures until I was doing science subjects at secondary school.)
answered Jan 24 at 23:17
timtfjtimtfj
2,468420
edited Jan 24 at 23:25
answered Jan 24 at 23:17
timtfjtimtfj
2,468420
answered Jan 24 at 23:17
timtfjtimtfj
2,468420
answered Jan 24 at 23:17
timtfjtimtfj
2,468420
2,468420
$begingroup$
Thx for your input. Actually, the context of this question was that in one class (first year of high school) almost all the students were showing me as result: 1.300 with no measurement involved.
$endgroup$
– gegu
Jan 25 at 18:08
add a comment |
$begingroup$
Thx for your input. Actually, the context of this question was that in one class (first year of high school) almost all the students were showing me as result: 1.300 with no measurement involved.
$endgroup$
– gegu
Jan 25 at 18:08
$begingroup$
Thx for your input. Actually, the context of this question was that in one class (first year of high school) almost all the students were showing me as result: 1.300 with no measurement involved.
$endgroup$
– gegu
Jan 25 at 18:08
$begingroup$
Thx for your input. Actually, the context of this question was that in one class (first year of high school) almost all the students were showing me as result: 1.300 with no measurement involved.
$endgroup$
– gegu
Jan 25 at 18:08
add a comment |
$begingroup$
Decimal is only shorthand.
$3.25$ is shorthand for $3 + frac 2{10} + frac 5{100}$ and $.4$ is shorthand for $frac 4{10}$.
So $3.25 times .4 = (3 + frac 2{10} + frac 5{100})(frac 4{10}) = $
$frac {12}{10} + frac 8{100} + frac {20}{1000} =$
$frac {10}{10} + frac {2}{10} + frac 8{100} + frac 2{100}=$
$1 + frac {2}{10} + frac {10}{100} =$
$1 + frac 2{10} + frac 1{10} = $
$1 + frac 3{10} = $
$1.3$.
But $1 + frac 3{10} = 1 + frac 3{10} + frac 0{100} +frac 0{1000} + frac 0{10000}$
So it's equally correct to write this as $1.3000$
Or $1 + frac 3{10} = 0times 1000 + 0times 100 + 0times 10 + 1 + frac 3{10} + frac 0{100}$ so it would be okay (albeit weird) to write it as $001.30$.
We don't write those leading $0$s in the beginning because ... we don't have to and it'd be really weird to. Also we intuitively start at the biggest part of a number and work our way to the smaller so we don't usually think about all those (non-existent) powers of $10$ before we start.
Likewise we don't write all the trailing $0$s at the end because we don't have to. However because we started at the biggest part and worked to the smallest we often do think about those (non-existent) negative powers of $10$ after we end.
In fact $1.3$ is actually $1.30000000000.....$ with an infinite number of $0$ after the end. We don't write them because... we don't need to. (And we can't.)
I'd even argue that $1.3$ is actually $......00000001.3000000.....$ but we don't worry about any of the zeros that aren't saving a place between digits that are "doing work".
answered Jan 24 at 23:55
fleabloodfleablood
72.7k22788
$endgroup$
$begingroup$
thx for the idea of using the formulation of the expanded form to explain that it is the same number.
$endgroup$
– gegu
Jan 25 at 18:18
add a comment |
$begingroup$
Decimal is only shorthand.
$3.25$ is shorthand for $3 + frac 2{10} + frac 5{100}$ and $.4$ is shorthand for $frac 4{10}$.
So $3.25 times .4 = (3 + frac 2{10} + frac 5{100})(frac 4{10}) = $
$frac {12}{10} + frac 8{100} + frac {20}{1000} =$
$frac {10}{10} + frac {2}{10} + frac 8{100} + frac 2{100}=$
$1 + frac {2}{10} + frac {10}{100} =$
$1 + frac 2{10} + frac 1{10} = $
$1 + frac 3{10} = $
$1.3$.
But $1 + frac 3{10} = 1 + frac 3{10} + frac 0{100} +frac 0{1000} + frac 0{10000}$
So it's equally correct to write this as $1.3000$
Or $1 + frac 3{10} = 0times 1000 + 0times 100 + 0times 10 + 1 + frac 3{10} + frac 0{100}$ so it would be okay (albeit weird) to write it as $001.30$.
We don't write those leading $0$s in the beginning because ... we don't have to and it'd be really weird to. Also we intuitively start at the biggest part of a number and work our way to the smaller so we don't usually think about all those (non-existent) powers of $10$ before we start.
Likewise we don't write all the trailing $0$s at the end because we don't have to. However because we started at the biggest part and worked to the smallest we often do think about those (non-existent) negative powers of $10$ after we end.
In fact $1.3$ is actually $1.30000000000.....$ with an infinite number of $0$ after the end. We don't write them because... we don't need to. (And we can't.)
I'd even argue that $1.3$ is actually $......00000001.3000000.....$ but we don't worry about any of the zeros that aren't saving a place between digits that are "doing work".
answered Jan 24 at 23:55
fleabloodfleablood
72.7k22788
$endgroup$
$begingroup$
thx for the idea of using the formulation of the expanded form to explain that it is the same number.
$endgroup$
– gegu
Jan 25 at 18:18
add a comment |
$begingroup$
Decimal is only shorthand.
$3.25$ is shorthand for $3 + frac 2{10} + frac 5{100}$ and $.4$ is shorthand for $frac 4{10}$.
So $3.25 times .4 = (3 + frac 2{10} + frac 5{100})(frac 4{10}) = $
$frac {12}{10} + frac 8{100} + frac {20}{1000} =$
$frac {10}{10} + frac {2}{10} + frac 8{100} + frac 2{100}=$
$1 + frac {2}{10} + frac {10}{100} =$
$1 + frac 2{10} + frac 1{10} = $
$1 + frac 3{10} = $
$1.3$.
But $1 + frac 3{10} = 1 + frac 3{10} + frac 0{100} +frac 0{1000} + frac 0{10000}$
So it's equally correct to write this as $1.3000$
Or $1 + frac 3{10} = 0times 1000 + 0times 100 + 0times 10 + 1 + frac 3{10} + frac 0{100}$ so it would be okay (albeit weird) to write it as $001.30$.
We don't write those leading $0$s in the beginning because ... we don't have to and it'd be really weird to. Also we intuitively start at the biggest part of a number and work our way to the smaller so we don't usually think about all those (non-existent) powers of $10$ before we start.
Likewise we don't write all the trailing $0$s at the end because we don't have to. However because we started at the biggest part and worked to the smallest we often do think about those (non-existent) negative powers of $10$ after we end.
In fact $1.3$ is actually $1.30000000000.....$ with an infinite number of $0$ after the end. We don't write them because... we don't need to. (And we can't.)
I'd even argue that $1.3$ is actually $......00000001.3000000.....$ but we don't worry about any of the zeros that aren't saving a place between digits that are "doing work".
answered Jan 24 at 23:55
fleabloodfleablood
72.7k22788
$endgroup$
Decimal is only shorthand.
$3.25$ is shorthand for $3 + frac 2{10} + frac 5{100}$ and $.4$ is shorthand for $frac 4{10}$.
So $3.25 times .4 = (3 + frac 2{10} + frac 5{100})(frac 4{10}) = $
$frac {12}{10} + frac 8{100} + frac {20}{1000} =$
$frac {10}{10} + frac {2}{10} + frac 8{100} + frac 2{100}=$
$1 + frac {2}{10} + frac {10}{100} =$
$1 + frac 2{10} + frac 1{10} = $
$1 + frac 3{10} = $
$1.3$.
But $1 + frac 3{10} = 1 + frac 3{10} + frac 0{100} +frac 0{1000} + frac 0{10000}$
So it's equally correct to write this as $1.3000$
Or $1 + frac 3{10} = 0times 1000 + 0times 100 + 0times 10 + 1 + frac 3{10} + frac 0{100}$ so it would be okay (albeit weird) to write it as $001.30$.
We don't write those leading $0$s in the beginning because ... we don't have to and it'd be really weird to. Also we intuitively start at the biggest part of a number and work our way to the smaller so we don't usually think about all those (non-existent) powers of $10$ before we start.
Likewise we don't write all the trailing $0$s at the end because we don't have to. However because we started at the biggest part and worked to the smallest we often do think about those (non-existent) negative powers of $10$ after we end.
In fact $1.3$ is actually $1.30000000000.....$ with an infinite number of $0$ after the end. We don't write them because... we don't need to. (And we can't.)
I'd even argue that $1.3$ is actually $......00000001.3000000.....$ but we don't worry about any of the zeros that aren't saving a place between digits that are "doing work".
answered Jan 24 at 23:55
fleabloodfleablood
72.7k22788
answered Jan 24 at 23:55
fleabloodfleablood
72.7k22788
answered Jan 24 at 23:55
fleabloodfleablood
72.7k22788
answered Jan 24 at 23:55
fleabloodfleablood
72.7k22788
72.7k22788
$begingroup$
thx for the idea of using the formulation of the expanded form to explain that it is the same number.
$endgroup$
– gegu
Jan 25 at 18:18
add a comment |
$begingroup$
thx for the idea of using the formulation of the expanded form to explain that it is the same number.
$endgroup$
– gegu
Jan 25 at 18:18
$begingroup$
thx for the idea of using the formulation of the expanded form to explain that it is the same number.
$endgroup$
– gegu
Jan 25 at 18:18
$begingroup$
thx for the idea of using the formulation of the expanded form to explain that it is the same number.
$endgroup$
– gegu
Jan 25 at 18:18
add a comment |
$begingroup$
Keep in mind, if you are going to introduce significant digits to grade school students, you need to better understand them yourself. 0.4 has 1 significant digit. To think that 3.25×0.4 = 1.300 is to think that the number of significant digits in a result is the sum of the two operands' significant digits. The truth is that it is the lesser of the 2.
In my opinion, the concept is best left for high schoolers, and your kids should just use 1.3 as the answer. That said, there's no harm in asking peers how they do it. I am in a high school, as an in-house math tutor, and often have to defer to how a teacher handles a topic or how the department does, regardless of my own preference.
answered Jan 27 at 16:24
JoeTaxpayerJoeTaxpayer
2,20121326
$endgroup$
add a comment |
$begingroup$
Keep in mind, if you are going to introduce significant digits to grade school students, you need to better understand them yourself. 0.4 has 1 significant digit. To think that 3.25×0.4 = 1.300 is to think that the number of significant digits in a result is the sum of the two operands' significant digits. The truth is that it is the lesser of the 2.
In my opinion, the concept is best left for high schoolers, and your kids should just use 1.3 as the answer. That said, there's no harm in asking peers how they do it. I am in a high school, as an in-house math tutor, and often have to defer to how a teacher handles a topic or how the department does, regardless of my own preference.
answered Jan 27 at 16:24
JoeTaxpayerJoeTaxpayer
2,20121326
$endgroup$
add a comment |
$begingroup$
Keep in mind, if you are going to introduce significant digits to grade school students, you need to better understand them yourself. 0.4 has 1 significant digit. To think that 3.25×0.4 = 1.300 is to think that the number of significant digits in a result is the sum of the two operands' significant digits. The truth is that it is the lesser of the 2.
In my opinion, the concept is best left for high schoolers, and your kids should just use 1.3 as the answer. That said, there's no harm in asking peers how they do it. I am in a high school, as an in-house math tutor, and often have to defer to how a teacher handles a topic or how the department does, regardless of my own preference.
answered Jan 27 at 16:24
JoeTaxpayerJoeTaxpayer
2,20121326
$endgroup$
Keep in mind, if you are going to introduce significant digits to grade school students, you need to better understand them yourself. 0.4 has 1 significant digit. To think that 3.25×0.4 = 1.300 is to think that the number of significant digits in a result is the sum of the two operands' significant digits. The truth is that it is the lesser of the 2.
In my opinion, the concept is best left for high schoolers, and your kids should just use 1.3 as the answer. That said, there's no harm in asking peers how they do it. I am in a high school, as an in-house math tutor, and often have to defer to how a teacher handles a topic or how the department does, regardless of my own preference.
answered Jan 27 at 16:24
JoeTaxpayerJoeTaxpayer
2,20121326
answered Jan 27 at 16:24
JoeTaxpayerJoeTaxpayer
2,20121326
answered Jan 27 at 16:24
JoeTaxpayerJoeTaxpayer
2,20121326
answered Jan 27 at 16:24
JoeTaxpayerJoeTaxpayer
2,20121326
2,20121326
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3086468%2fresult-of-multiplication-having-zeroes-after-the-decimal%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
Personally, I'd write it as $1.3$.
$endgroup$
– Bernard
Jan 24 at 22:56
$begingroup$
depends on the context. Are sig figs important? or is succinctness preferable?
$endgroup$
– Chase Ryan Taylor
Jan 24 at 22:56
1
$begingroup$
If the answer is supposed to be exact, it doesn't matter, but if the numbers come from some measurement, then keeping the extra zeroes is important.
$endgroup$
– herb steinberg
Jan 24 at 22:57
1
$begingroup$
$1.3 = 1 + frac 3{10}$ and $1.300 = 1 + frac 3{10} + frac 0{100} + frac 0{100}$. And $1 + frac 3{10} = 1 + frac 3{10} + frac 0{100} + frac 0{100}$ so they are exactly the same thing. Or $3.25 times .4= 325times frac 1{100}times 4times frac 1{10} = 1300times frac 1{1000} = 13timesfrac 1 {10} = 1.3$ We don't need those zeros as they factor out. That we got them when we multiplied $4times 325$ isn't relevant as we don't need them to hold any places that aren't being used.
$endgroup$
– fleablood
Jan 25 at 0:02
1
$begingroup$
I think you should ask this at Mathematics Educators Stack Exchange, where you'll get answers which take into account the likely understanding of primary school children and the possible pitfalls of teaching one approach or another. That's as important as the mathematics really—for instance you don't want to accidentally train them to think multiplying several approximate values on their calculators gives a result accurate to all the decimal places that magically appear!
$endgroup$
– timtfj
Jan 25 at 11:44