Another possible reason: calculators give decimals (at least, ordinary, non-scientific calculators do, and some scientific calculators also do by default). It’s easier to put some numbers into a calculator and then give the answer than to simplify expressions involving fractions.
Another advantage of decimals is that it’s easier (once you get denominators higher than 3 or 4) to tell at a glance whether, for instance, .41813 is greater or less than .46275, than to tell whether 143/342 is greater or less than 354/765—and easier to tell that they’re both fairly close together, and between 2/5 and 1/2.
If only they knew how to calculate a fraction in the calculator…!
For real. Seriously. I can’t count how many times I’ve been asked where the fraction button is on the calculator, because they’re trying to get that beloved decimal out of 7/9, but… there’s… just… no… fraction… button.
When I point out that the button they’re looking for is just above the times (x) button, they look at me like I’ve just now landed from the fourth planet of Rigel.
I’ve worked with TI-84s in the past that have a specific fraction button, but the ones at my current school don’t have them.
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You don’t need high end anything, lots of regular scientific calculators manipulate fractions directly(as long as they don’t involve symbolic variables of course, you do need a CAS for those). I have a Casio fx-991(with neither graphics nor programming, so quite simple) and it shows you fractions by default unless there are irrational numbers involved or they get very big, and you can switch between the representations. If you type 0.1+0.1, it answers 1/5.
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Tell them to use long division. When they figure out that long division is division, they can use their calculator.
Some calculators have a fraction button – and frankly, they are the ones I prefer. They also have them on the SAT approved calculators now, so it’s not a big challenge to get kids to use them. I love that. Since I hate doing fractions in my head. 🙂
It could be that decimals are better for ordering purposes, as you can instantly know that 0.333 > 0.234, for example. On the other hand, fractions are better for algebraic purposes, as you can instantly know that 1/3 is the inverse of 3.
Why not “Speak roots!”? After all you get answers like sqrt(57)+5 when solving quadratic equations. Ain’t it obvious? Simple language is easier to understand, even if it is less exact.
Reminds of that old joke: A group in a balloon asked a man on ground where they are. After some thinking the man replied: “You are in a balloon!”. One guy in the balloon then said: “Must be a mathematician. The answer is 100% correct, but useless.”
But seriously, pure geometry or algebra is of course not useless. It is only so that you need to take extreme care to apply the answers to the real world. And that already starts in schools. So fractions are nice, but decimal numbers are more useful, even if you lose accuracy.
I give you a problem: The top of a tower appears at 20° above the horizon. Moving towards the tower 100m it is 24°. How high is the tower? To solve this, you clearly need pure math. To solve it correctly (taking care of errors in measurement), you need applied math, which is of course only a more involved application of pure math. But you will be surprised about the range of possible answers.
I have an opinion on how a lot of our current math malaise is based on we math teachers allowing mathematics to become seen as the tool of the other fields, rather than a thing of beauty in its own right. sqrt(2) and pi are exact, but you can’t cut a log or measure the height of a ball’s bounce to pi, so 1.41 and 3.14 are “close enough” for the real world.
Some day soon I’ll rant at much more depth than this. 😀
If you need a number to be an exact answer then it should be left in its non-decimal form but if you plug that into any computer program it will round it to a certain specification that the computer determines or the user if the number is an infinite decimal. Most applications that we use are not perfect and requires a tolerance of a certain amount.
The imitation of what a Blackberry was in 1999 cell phone that I use has a calculator that won’t let me input a number larger than 99999999. Not when it comes to arithmetic that is :-)=
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If you’re solving quadratic equations in closed form, speaking roots is the right thing to do, up to the point where you actually need a numerical value. That way you don’t lose precision until you have to.
It’s the same thing with retaining values like pi and e as symbolic values, when they come up. Closed-form exact solutions are precious things! It’s best not to throw them away until you have to, because, say, you need to cut a piece of wood to the length you’ve calculated.
I took an entire class as an undergraduate where they laid out in careful detail *exactly* how computers can get math incorrect. It’s amazing, really, how in some cases little approximations can become big errors. And it mostly stems from the fact that computers use binary to do all their math.
How did we get around this? By doing algebra and calculus without substitutions, of course! It’s impressive how much improvement you can get from a machine when you “just” rearrange the math to happen in a different order.
Not to mention the insights you can get from an algebraic or fractional solution that just doesn’t come up in decimal. Seriously, with fractions, you get 6*(1/3) = 2. Pulling the same stunt with decimal notation gives you 1.9999999…, which *is* the same thing, but is a terrible way to write it.
And actually, it’s not solely because of the binary. It’s more because we humans insist on clinging to decimal, when computers have to function in a base that’s an exponent of 2. The base conversions are a major source of error.
If we’d just switch to octal, we could talk to computers much more reliably.
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Well, I honestly think that octal vs. decimal doesn’t have much to do with my point. Any choice of base will have the same problems as I described, and as described in the original post. The real problem is the lack of flexibility in how students understand numbers. That this extends to computers is just another example of how this lack of flexibility can hurt us.
Computer are ridiculously good at converting between bases (it’s exactly the kind of problem they are well-suited for). I don’t think base 10 vs. base 8/16/2 has been a serious problem in computing for a few years now. It’s mistaking the representation for the number that is the real source of trouble.
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JavaScript is particularly awful for rounding error (even worse than Microsoft!). Go here and try 3.1 * 3.68, for instance. :p
Looks entirely normal to me! All floating point systems I know have have this particular weakness. It comes from the fact that 0.1 is non-terminating in binary, and cannot be represented exactly. Thus, you get a very *small* error. Many languages hide that error from you when they print the answer, but that’s not the same as not having the error in the first place. JavaScript is just more honest!
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You wrote, “And it mostly stems from the fact that computers use binary to do all their math.”
I was responding to that. The reason computers generate errors we don’t expect has much more to do with base conversion errors than with computers using binary. As you point out, the mistakes that can be attributed to same-base approximations (e.g., 1/3 + 1/3 + 1/3) are going to be an issue regardless of the base. But converting from decimal to binary, doing some math, and then converting back to decimal creates another cycle for introducing error, and that’s where the cases that confuse the end user come from (like the JavaScript error I presented, or Microsoft Excel’s famous gaffe). This is because fractions that can be represented as a terminating value in one base will infinitely repeat in another: one fifth, for instance, is 0.2 in decimal but 0.0011(0011) in binary. We humans expect more error from 1/3 than from 1/5, but both are infinitely repeating in binary.
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It is true that some of the errors which come from doing computer arithmetic is from base conversions. No argument there. Mostly this is a good argument for either teaching students how to deal with fractions rather than relying on a calculator, or teaching computers how to use fractions. (Both have been done.) Computers do make it rather easy, nowadays, to make a mess out of simple things.
Even with that all said, though, we would still have problems with precision for two reasons, even if we thought in octal or hexadecimal:
(a) every base has fractions which become repeating, er, decimals. All we have done by changing base is move the problem around. Fractions actually solve the problem in all integer bases, and are very useful for solving those problems in any situation.
(b) Irrational numbers of all stripes cause truncation errors in any base. Relying on computers or “basal” representations, rather than math, will always be a problem as long as our representations are finite. That’s what the post is really getting at.
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Reblogged this on mhorley and commented:
Love this. My students do this all the time. I love the “door” analogy!
I spend time with decimals every day! My kids jump on their calculators and start spitting out decimals like little machine guns! When they first started using these mind numbing devices, they thought the numbers stopped when the screen ended. (Bad news for the irrational numbers of the world!) at least now they know there is more! I regularly ask them to turn their decimals into fractions. I also talk about what fractions are… I can see I need to add what decimals ‘really’ are. I love the door analogy, I mean the ‘tiktiktiktik’ thingy! I also love your posts. They make me all light hearted and giggly over math. Thank you!
For me, decimals always felt as though they were the answer. Leaving fractions, operators and functions in an answer felt lazy. It gave me a great deal of joy to discover that, while decimals give a better idea of position on the number line, leaving a number in “part-solved” equation form is often cleaner and more useful.
Exact calculation with fractions is not practical because of the rapid growth in the size of the numerator and denominator. So decimals are preferred for practical computation. However fractions are superior for theory.
Modern devices such as cell phones are perfectly capable of running Scheme or Common Lisp interpreters. In these languages, (/ 10 6), meaning “10 divided by 6”, results in the fraction 5/3. If you want inexact results, start off with inexact arguments: (/ 10.0 6) or (/ 10 6.0) will return 1.6666666666666667. In Scheme, the standard “inexact->exact” and “exact->inexact” functions will do conversions: (inexact->exact 3.141592653589793) evaluates to 884279719003555/281474976710656.
I have to admit, I have a certain fondness for bull-headedly pushing through problems in strange and awkward ways and then figuring out how that could actually be equivalent to doing it the “right” way. But it doesn’t mean you ought to do it that way every time.
Decimals allow you to *describe* numbers that can’t be written as fractions, but any number that can be written down *exactly* using decimals can be written as a fraction.
The number described by 0.101001000100001000001… is probably Σ_{k=1}^{inf} 10^-T(k), where T(n) is the triangular number sequence (https://oeis.org/A000217), but that number is irrational and hence can’t be written exactly as either a decimal or a fraction.
But any terminating or repeating decimal is rational, and hence can be written as a fraction.
One week you write about infinite series that converge to the so elegant pi, and the nest you complain about your students representing simple fractions as infinite series of decimals. Make up your mind!
One advantage of decimals is that harder to compare 17/91 to 7/29. Which one is bigger. I don’t know, cross multiply. But. 0.187 to 0.241, well that is a snap.
We look at too many screens with decimal representation.
There is an illusion of sophistication to spit out all of these decimals even thought it is in every way less accurate.
You should drop the term “decimal” since this discussion has nothing to do with base TEN. In any integer base whatsoever you have a similar positional representation of numbers. There is NOTHING special about base TEN (I don’t use “10” because that is the value of the base in every single base!!!).
Hahahahahahahaha! I have been on a crusade about this for ages: “Fractions are your friends!” (said in my best “Finding Nemo” shark voice). On the very day this showed up in my email, I had kids solving systems of equations and they told me .3 for 1/3 ! I stopped everything, showed them your post and for the rest of the week, any time someone tried giving a decimal instead of a fraction, the rest of the class would say, “tiktiktiktiktik……”. It was hysterical!
I don’t teach a classroom of children, but this doesn’t bother me so much, and you could talk about calculus and approaching limits and how it would never quite equal the same. Look at our add an opportunity. 😉
Here’s good “mnemonic” Decimals=Devils Fractions=Friends.
My guess is that you teach algebra or higher math, so your students might feel recursive to elementary school, where they learned arithmetic. Arithmetic is easier with a place-value system (basically decimals).
Another good mnemonic: PEMDAS: Please Email My Dad A Shark
Another possible reason: calculators give decimals (at least, ordinary, non-scientific calculators do, and some scientific calculators also do by default). It’s easier to put some numbers into a calculator and then give the answer than to simplify expressions involving fractions.
Another advantage of decimals is that it’s easier (once you get denominators higher than 3 or 4) to tell at a glance whether, for instance, .41813 is greater or less than .46275, than to tell whether 143/342 is greater or less than 354/765—and easier to tell that they’re both fairly close together, and between 2/5 and 1/2.
If only they knew how to calculate a fraction in the calculator…!
For real. Seriously. I can’t count how many times I’ve been asked where the fraction button is on the calculator, because they’re trying to get that beloved decimal out of 7/9, but… there’s… just… no… fraction… button.
When I point out that the button they’re looking for is just above the times (x) button, they look at me like I’ve just now landed from the fourth planet of Rigel.
Wait until they discover computer algebra systems that actually can manipulate fractions directly. Some high-end calculators have them.
I’ve worked with TI-84s in the past that have a specific fraction button, but the ones at my current school don’t have them.
You don’t need high end anything, lots of regular scientific calculators manipulate fractions directly(as long as they don’t involve symbolic variables of course, you do need a CAS for those). I have a Casio fx-991(with neither graphics nor programming, so quite simple) and it shows you fractions by default unless there are irrational numbers involved or they get very big, and you can switch between the representations. If you type 0.1+0.1, it answers 1/5.
Tell them to use long division. When they figure out that long division is division, they can use their calculator.
Some calculators have a fraction button – and frankly, they are the ones I prefer. They also have them on the SAT approved calculators now, so it’s not a big challenge to get kids to use them. I love that. Since I hate doing fractions in my head. 🙂
It could be that decimals are better for ordering purposes, as you can instantly know that 0.333 > 0.234, for example. On the other hand, fractions are better for algebraic purposes, as you can instantly know that 1/3 is the inverse of 3.
Why not “Speak roots!”? After all you get answers like sqrt(57)+5 when solving quadratic equations. Ain’t it obvious? Simple language is easier to understand, even if it is less exact.
You mean “more exact”. The answer sqrt(2) is exact, whereas a 7 digit decimal aproximation is just an approximation.
Reminds of that old joke: A group in a balloon asked a man on ground where they are. After some thinking the man replied: “You are in a balloon!”. One guy in the balloon then said: “Must be a mathematician. The answer is 100% correct, but useless.”
But seriously, pure geometry or algebra is of course not useless. It is only so that you need to take extreme care to apply the answers to the real world. And that already starts in schools. So fractions are nice, but decimal numbers are more useful, even if you lose accuracy.
I give you a problem: The top of a tower appears at 20° above the horizon. Moving towards the tower 100m it is 24°. How high is the tower? To solve this, you clearly need pure math. To solve it correctly (taking care of errors in measurement), you need applied math, which is of course only a more involved application of pure math. But you will be surprised about the range of possible answers.
Unless I’ve misunderstood you, and you think that decimals are “simpler” than roots, with which I disagree.
I have an opinion on how a lot of our current math malaise is based on we math teachers allowing mathematics to become seen as the tool of the other fields, rather than a thing of beauty in its own right. sqrt(2) and pi are exact, but you can’t cut a log or measure the height of a ball’s bounce to pi, so 1.41 and 3.14 are “close enough” for the real world.
Some day soon I’ll rant at much more depth than this. 😀
If you need a number to be an exact answer then it should be left in its non-decimal form but if you plug that into any computer program it will round it to a certain specification that the computer determines or the user if the number is an infinite decimal. Most applications that we use are not perfect and requires a tolerance of a certain amount.
The imitation of what a Blackberry was in 1999 cell phone that I use has a calculator that won’t let me input a number larger than 99999999. Not when it comes to arithmetic that is :-)=
If you’re solving quadratic equations in closed form, speaking roots is the right thing to do, up to the point where you actually need a numerical value. That way you don’t lose precision until you have to.
It’s the same thing with retaining values like pi and e as symbolic values, when they come up. Closed-form exact solutions are precious things! It’s best not to throw them away until you have to, because, say, you need to cut a piece of wood to the length you’ve calculated.
This. So much this.
I took an entire class as an undergraduate where they laid out in careful detail *exactly* how computers can get math incorrect. It’s amazing, really, how in some cases little approximations can become big errors. And it mostly stems from the fact that computers use binary to do all their math.
How did we get around this? By doing algebra and calculus without substitutions, of course! It’s impressive how much improvement you can get from a machine when you “just” rearrange the math to happen in a different order.
Not to mention the insights you can get from an algebraic or fractional solution that just doesn’t come up in decimal. Seriously, with fractions, you get 6*(1/3) = 2. Pulling the same stunt with decimal notation gives you 1.9999999…, which *is* the same thing, but is a terrible way to write it.
And actually, it’s not solely because of the binary. It’s more because we humans insist on clinging to decimal, when computers have to function in a base that’s an exponent of 2. The base conversions are a major source of error.
If we’d just switch to octal, we could talk to computers much more reliably.
Well, I honestly think that octal vs. decimal doesn’t have much to do with my point. Any choice of base will have the same problems as I described, and as described in the original post. The real problem is the lack of flexibility in how students understand numbers. That this extends to computers is just another example of how this lack of flexibility can hurt us.
Computer are ridiculously good at converting between bases (it’s exactly the kind of problem they are well-suited for). I don’t think base 10 vs. base 8/16/2 has been a serious problem in computing for a few years now. It’s mistaking the representation for the number that is the real source of trouble.
JavaScript is particularly awful for rounding error (even worse than Microsoft!). Go here and try 3.1 * 3.68, for instance. :p
http://www.anaesthetist.com/mnm/javascript/calc.htm
Looks entirely normal to me! All floating point systems I know have have this particular weakness. It comes from the fact that 0.1 is non-terminating in binary, and cannot be represented exactly. Thus, you get a very *small* error. Many languages hide that error from you when they print the answer, but that’s not the same as not having the error in the first place. JavaScript is just more honest!
You wrote, “And it mostly stems from the fact that computers use binary to do all their math.”
I was responding to that. The reason computers generate errors we don’t expect has much more to do with base conversion errors than with computers using binary. As you point out, the mistakes that can be attributed to same-base approximations (e.g., 1/3 + 1/3 + 1/3) are going to be an issue regardless of the base. But converting from decimal to binary, doing some math, and then converting back to decimal creates another cycle for introducing error, and that’s where the cases that confuse the end user come from (like the JavaScript error I presented, or Microsoft Excel’s famous gaffe). This is because fractions that can be represented as a terminating value in one base will infinitely repeat in another: one fifth, for instance, is 0.2 in decimal but 0.0011(0011) in binary. We humans expect more error from 1/3 than from 1/5, but both are infinitely repeating in binary.
It is true that some of the errors which come from doing computer arithmetic is from base conversions. No argument there. Mostly this is a good argument for either teaching students how to deal with fractions rather than relying on a calculator, or teaching computers how to use fractions. (Both have been done.) Computers do make it rather easy, nowadays, to make a mess out of simple things.
Even with that all said, though, we would still have problems with precision for two reasons, even if we thought in octal or hexadecimal:
(a) every base has fractions which become repeating, er, decimals. All we have done by changing base is move the problem around. Fractions actually solve the problem in all integer bases, and are very useful for solving those problems in any situation.
(b) Irrational numbers of all stripes cause truncation errors in any base. Relying on computers or “basal” representations, rather than math, will always be a problem as long as our representations are finite. That’s what the post is really getting at.
Reblogged this on mhorley and commented:
Love this. My students do this all the time. I love the “door” analogy!
I spend time with decimals every day! My kids jump on their calculators and start spitting out decimals like little machine guns! When they first started using these mind numbing devices, they thought the numbers stopped when the screen ended. (Bad news for the irrational numbers of the world!) at least now they know there is more! I regularly ask them to turn their decimals into fractions. I also talk about what fractions are… I can see I need to add what decimals ‘really’ are. I love the door analogy, I mean the ‘tiktiktiktik’ thingy! I also love your posts. They make me all light hearted and giggly over math. Thank you!
For me, decimals always felt as though they were the answer. Leaving fractions, operators and functions in an answer felt lazy. It gave me a great deal of joy to discover that, while decimals give a better idea of position on the number line, leaving a number in “part-solved” equation form is often cleaner and more useful.
“So you say that 0.3333333333333333333333333333333333333333…. is more ‘numerical’ than 1/3. Ok, now calculate one third of 6”
And then try to figure out how to grade it when they come back with 1.9999999999999999… and point out that this actually is equal to 2.
That’s of course if they don’t come back with 1.8181818181818…
Exact calculation with fractions is not practical because of the rapid growth in the size of the numerator and denominator. So decimals are preferred for practical computation. However fractions are superior for theory.
The whole point of the post is that decimals require *infinitely long* expressions. 1.0 / 3.0 requires infinite space (in both decimal and binary).
Reblogged this on tglennb and commented:
I’ve always like fractions. No surprise there.
Modern devices such as cell phones are perfectly capable of running Scheme or Common Lisp interpreters. In these languages, (/ 10 6), meaning “10 divided by 6”, results in the fraction 5/3. If you want inexact results, start off with inexact arguments: (/ 10.0 6) or (/ 10 6.0) will return 1.6666666666666667. In Scheme, the standard “inexact->exact” and “exact->inexact” functions will do conversions: (inexact->exact 3.141592653589793) evaluates to 884279719003555/281474976710656.
I have to admit, I have a certain fondness for bull-headedly pushing through problems in strange and awkward ways and then figuring out how that could actually be equivalent to doing it the “right” way. But it doesn’t mean you ought to do it that way every time.
But then I can’t say .101001000100001000001…
Thanks, this is an important point: decimals allow you to write down numbers that can’t be written as fractions, so fractions are not general enough.
Decimals allow you to *describe* numbers that can’t be written as fractions, but any number that can be written down *exactly* using decimals can be written as a fraction.
The number described by 0.101001000100001000001… is probably Σ_{k=1}^{inf} 10^-T(k), where T(n) is the triangular number sequence (https://oeis.org/A000217), but that number is irrational and hence can’t be written exactly as either a decimal or a fraction.
But any terminating or repeating decimal is rational, and hence can be written as a fraction.
Sure you can. It’s $sum_{n=1}^(infty) 10^(\frac{-n(n+1)}{2})$. It’s an infinite series either way; that’s provably the best you can do.
One week you write about infinite series that converge to the so elegant pi, and the nest you complain about your students representing simple fractions as infinite series of decimals. Make up your mind!
One advantage of decimals is that harder to compare 17/91 to 7/29. Which one is bigger. I don’t know, cross multiply. But. 0.187 to 0.241, well that is a snap.
We look at too many screens with decimal representation.
There is an illusion of sophistication to spit out all of these decimals even thought it is in every way less accurate.
Please, tell me that’s a joke … 🙂
My favourite way of saying “one third” is “33 and one third percent” !!!
That gets the hearer thereof so close to *exactly comprehending the speaker exactly* that it suits the purpose, eh?
You should drop the term “decimal” since this discussion has nothing to do with base TEN. In any integer base whatsoever you have a similar positional representation of numbers. There is NOTHING special about base TEN (I don’t use “10” because that is the value of the base in every single base!!!).
Hahahahahahahaha! I have been on a crusade about this for ages: “Fractions are your friends!” (said in my best “Finding Nemo” shark voice). On the very day this showed up in my email, I had kids solving systems of equations and they told me .3 for 1/3 ! I stopped everything, showed them your post and for the rest of the week, any time someone tried giving a decimal instead of a fraction, the rest of the class would say, “tiktiktiktiktik……”. It was hysterical!
And hysterically proves that What Goes Around Sometimes Comes Around Very Nicely Indeed.
Very concise, kudos!
I don’t teach a classroom of children, but this doesn’t bother me so much, and you could talk about calculus and approaching limits and how it would never quite equal the same. Look at our add an opportunity. 😉
Here’s good “mnemonic” Decimals=Devils Fractions=Friends.
My guess is that you teach algebra or higher math, so your students might feel recursive to elementary school, where they learned arithmetic. Arithmetic is easier with a place-value system (basically decimals).
Another good mnemonic: PEMDAS: Please Email My Dad A Shark
PEMDAS is evil, and any mnemonic that needs a mnemonic is a bad mnemonic.
Is there anyway I could get this cartoon on a poster or something??? Would love to hang it in my classroom!!
so good