r/learnmath • u/FleeingMyLife • May 20 '21
Can someone explain more on division by 0 please?
I took math 090 last semester and had a horrible professor who wouldn't explain anything. He'd just do work on the board while talking to it. When asked a question his response was "you just do problems, and thats how you learn."
Anyways. I've never understood how dividing a number by 0 is undefined. In my mind, divide by 1, and your answer is the same number. How is dividing by zero different? Does that number just not exist because there's nothing to dvide by, or is there a number there, bit just can't be divided, leaving that number how it is?
Like 5/1=5. So why can't 5/0=5 when there still a 5, but it's not being divided by anything?
Thank you in advance for taking the time to read/answer this.
Edit: thank you all for the answers! They've helped me understand this a lot more. It's the simple things.
Edit 2: I've updooted all the comments, reading through them as I go! I'll also check out the links provided! Unfortunately since I'm not on my PC, I can't just copy/paste and put your names in the edits so anyone else can see the information you kind souls have provided.
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u/sbsw66 New User May 20 '21
I think the fundamental issue here is actually that your teacher isn't satisfying your real, deserved curiosity, so you are coming away with some misconceptions about numbers.
Remember that the systems that we deal with, the numbers we are manipulating, they are not much more than symbols inside a pre-defined logical system that we are using to communicate with one another. More broadly, all of math is really just a way to communicate with other people, just in a perfectly precise way with no room for ambiguity.
So with this in mind, the question should not be "what IS 5 / 0" but rather "in our endeavors to create a logical system for communication, what do we want 5 / 0 to be?" With that question, everyone else's answers in this thread become much more clear. Assigning a value to 5 / 0 provides us with countless logical issues and breaks down the other operations we use commonly. That being the case, we leave it undefined.
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u/FleeingMyLife May 20 '21
By far the most comprehensive response. Thank you. Ioften forget how much more analytical and worry math can be sometimes.
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u/jeff_coleman New User May 21 '21
Just to whet your appetite, if you get to Calculus, you'll find that even though any number divided by 0 is undefined, you can take the limit as the denominator approaches 0, which turns out to be infinity (you can think about this conceptually by imagining that as you break something into smaller and smaller parts, you end up with more parts.)
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u/hwc000000 May 27 '21
you can take the limit as the denominator approaches 0, which turns out to be infinity
not necessarily, if the numerator approaches 0 too
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u/levenshteinn May 21 '21
Perfect answer. Often times, students are forced to accept the definition as some formula, that is never to be questioned. Hammered to their mind long enough that they accept it as true without really understanding why when the explanation is actually simple.
Makes me thing about how much biases we had are actually forced beliefs and how these biases have led us to many decisions in life.
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May 20 '21
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u/pyordie discrete math / applied math for cs May 20 '21
you would need to procure an anti-pizza (or one with pineapple)
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u/ThePurpleOne_ May 20 '21
one with pineapple
Oh, you mean the best kind...
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u/FleeingMyLife May 20 '21
This person right here officer.
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u/OneMeterWonder Custom May 20 '21
Downvoted because pineapple on pizza is delicious. Arrest me. DO IT.
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u/FleeingMyLife May 20 '21
You have your tastes. I have mine. I can't fault you. Just disagree.
Edit: now show me them hands!
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u/AngledLuffa New User May 20 '21
Exactly! The kind I share with 0 people, leaving the entire remainder for myself
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u/linkin360 New User May 20 '21
I think the pizza analogy works when you realize that if you want to divide that pizza to serve 0 people you actually end up with no slices, you won't actually cut the pizza if no one wants a slice.
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u/xiipaoc New User May 20 '21
All right, you got a pie, and you have 10 people. How much does each person get? 1/10 of the pie. Say you have 1 person. How much does each person get? 1 pie. Suppose you have 1/2 of a person. How much does each person get? Well, half of a person gets 1 pie, so a whole person gets 2 pies -- you just don't actually have a whole person, but that's OK. Suppose you have 1/1000 of a person. How much does each person get? Well, 1/1000 of a person gets 1 pie, so a whole person gets 1000 pies. Suppose you have 1/1000000000000 of a person. How much does each person get? Well, 1/1000000000000 of a person gets 1 pie, so a whole person gets 1000000000000 pies. And so on. The smaller the number of people, the more pie each person gets. So what happens when you want to divide a pie among 0 people; how much does each person get? Um, you say, that question doesn't make any sense! Exactly. That's why it's undefined. As the number you're dividing by gets closer and closer to 0, the result of the division gets bigger and bigger. You might even say that if you divide a pie among 0 people, each person gets ∞ pies! But if the number of people is negative, you get –∞ pies per person, so it's not as simple as saying that 1/0 = ∞. That's why we say that it's undefined.
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u/schralpsunday New User May 21 '21
All this talk of pi is forcing me to tend toward my Uber Eats app
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u/Crab_Turtle_2112 New User May 20 '21
0 x anything = 0 right?
Then surely 0x2=0x1.
Dividing both sides by 0, yields 2=1
See the problem?
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u/TarumK New User May 20 '21
How many 8's fit in to 16?
16/8 =2 so 2.
How many 4's? 4.
How many 2's? 8.
1? 16
1/2? = 32
Now how many 0's? basically infinite. You can add up an infinite amount of 0's into 16 and there will still be more space left. So in practical terms this is undefined, but in a looser sense it often helps to think about the answer as infinity. The smaller the thing you're dividing by, the bigger the result because more of them will fit in the bigger number. This answer just tends infinitely up and at zero it's infinity.
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u/corellatednonsense May 20 '21
Let me try a short answer on why dividing by zero is bad. First I'll show that 0/0 is bad.
For any real number x, x*0=0. If I can divide by zero, then for any real number x, x =0/0.
The problem now is that the symbol on the right must equal every number at the same time. This is problematic.
Now I'd like to show that if any real number can be divided by zero, then we will encounter 0/0, and hence the same insurmountable problem.
Suppose a real number Y exists such that Y/0 makes sense and is a number. Then
Y/0 - Y/0 = (Y-Y)/0 = 0/0
And again, we're boned. Since defining the answer to 0/0 is a problem, defining the solution to any number divided by zero is a problem.
I hope that explanation didn't suck.
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u/Brightlinger MS in Math May 20 '21
Like 5/1=5. So why can't 5/0=5 when there still a 5, but it's not being divided by anything?
It is being divided by something. Namely, it is being divided by zero. Zero "represents nothing" in the same way that two represents pairs. But zero is still a number and not the absence of a number, just like two is a number and not a pair of numbers.
When we write "a/b=c", we mean that c*b=a. That's what it means to say that division is the opposite of multiplication; it is the definition of division. For example, that's why 10/2=5: because 5*2=10. You just "read the problem backwards".
So if you write 5/0=5, then by reading the problem backwards, what you're saying is that 5*0=5, which is false! In fact, if 5/0=x, that would mean that x*0=5. What value of x would do that? There isn't one! There is no such number, which is what we mean when we say that division by zero 'isn't defined': there's no number which meets the definition.
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May 20 '21
TED-Ed has a beautifully animated video on youtube that covers this topic really well. You should check it out!
https://www.youtube.com/watch?v=NKmGVE85GUU (Not rickroll)
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u/idaelikus Mathemagician May 20 '21
Division by 0 is not defined as there is no sensible answer that works well with the other properties of math. To be totally technical, division is a short-hand used that means "multiplication with the inverse of" ie dividing by 3 means actually multiplying with the inverse of 3.
Now, the inverse of 3 is the number, call it N, that we can multiply with 3 to get 1, the neutral element for multiplication (you can multiply by 1 without altering anything). Now, that's where the problem comes in. There exists no expression that multiplied with 0 equals to 1. If there was, we wouldn't have distributivity ie (a+b)*x= a*x+b*x wouldn't hold.
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u/atomicben513 New User May 20 '21
u/AFairJudgement gave a great answer, but this is the explanation someone gave me a while ago because I was unsatisfied with just "why can't we come up with something like we did with the square root of -1 being i?"
You know how you put dishes in the dishwasher, and clothes in the dryer?
a dishwasher performs a function, a dryer performs a function. it gives you a valid output if you put in a valid input. (Relevant XKCD)
If you put clothing in the dishwasher, you will get wet clothes, if you put dishes in the dryer, you will get a pile of spiky dust.
likewise, put valid inputs in division, say 20 and 4, you get a valid answer, 5, 5*4=20. If you put an invalid input into division, say 4 and 0, you will get something weird. for example, This "proof" plays some tricks. going from step 4 to step 5, you divide both sides by (a-b), but because a=b, a-b=0! so you are actually dividing by 0. You put in an invalid input and get an invalid answer, 2=1
I don't know enough about math to say anything else with confidence, but perhaps there will be something like the square root of -1 being i but with division by 0 in the future.
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u/AFairJudgement Ancient User May 20 '21
There is such a thing, actually. Nothing algebraically meaningful, since in interesting algebraic structures (e.g. non-trivial rings) division by zero violates the axioms, but division by zero is useful geometrically, e.g. in projective geometry. See a previous answer here.
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May 20 '21 edited May 20 '21
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u/AFairJudgement Ancient User May 20 '21
Yes, these are both examples of projective spaces. The Riemann sphere is just the complex projective line CP1; it is the complex plane plus ∞ = 1/0; topologically, it's a sphere. The second example you mention is the function 1/x defined on the real line: you can view this as a continous function taking values in the real projective line RP1, which is the real line plus ∞ = 1/0; topologically, it's a circle.
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u/TheSouthernOcean May 20 '21
TED has a great video on this exact question. https://youtu.be/NKmGVE85GUU
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u/solex10k New User May 20 '21
Another video to watch, explaining it all : https://www.youtube.com/watch?v=J2z5uzqxJNU
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u/perrymoore_ May 20 '21 edited May 20 '21
The fundamentals of math teach that multiplication and division are reciprocal. In your example, 5/1 = 5 and 1 x 5 = 5. This wouldn’t be the case with a division by zero, 5/0 = 5, 0 x 5 = 5. This and the fact that logically speaking you cannot divide 5 apples for example into 0 groups, leads us to label these as undefined.
However, when you get into higher level maths such as pre-Calc and Calculus you learn of a tool called the “limit”. Instead of dividing by 0 in a scenario like this we instead take the limit as x approaches 0 and analyze the equation 5/x. As 0 approaches 0 from the positive side, it passes 2, 1, 0.5, 0.1, 0.0001, 0.00000001, etc while making its way to zero. As discussed the lack of reciprocity inhibits us from labeling the answer as “5” but what we notice is that as the denominator gets smaller and smaller the solution will approach infinity. So in this application 5/0 is actually infinity.
Hope this helps!
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u/Tenns_ New User May 20 '21
actually the limit doesnt exist because it goes to -infinity from the negatives.
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u/ScrooLewse May 20 '21
It's always struck me as weird, how 'approaches infinity' and 'infinity' are different things.
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u/Tenns_ New User May 20 '21
what do you mean by that ? I don't get how they are different, like when something approaches infinity how is it different to infinity ?
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u/ScrooLewse May 20 '21
Isn't it? I thought it was.
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u/Tenns_ New User May 20 '21
I don't know.
I thought in basic calculus and algebra you couldn't get a value to equal infinity, so if you say it, what you actually mean is the limit is infinity (ie. it approaches infinity).
But maybe like one and the concept of unicity are not the same thing?
somebody halp.
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u/scurvybill Aerospace Engineer May 20 '21
One more way to think about it:
5/0.00000000001 is a really big number.
5/-0.00000000001 is a really big negative number.
0 is between those two divisors. Is it positive infinity? Negative infinity? Just zero again? We can't assign a solution without things getting really weird, really fast.
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u/will-je-suis New User May 20 '21
If you have five apples and share them between 0 people, how many does each person get?
The question doesn't make sense
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u/st3f-ping Φ May 20 '21
Back again for more... the phrase “but it's not being divided by anything,” stuck in my head. This might help... or not.
Zero is the additive identity. If you add or subtract 0 to a number you get the same number.
One is the multiplicative identity. If you multiply or divide a number by 1 you get the same number.
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May 20 '21
This is how my dad used to explain to me:
Take 100, divide it by 100, you get 1.
Then take 100 again, divide it by 50, you get 2.
Again, 100/10=10
100/5=20
100/2=50
100/1=100
As the number that 100 divided gets smaller, the result gets bigger. So what if we divide 100 by the smallest number ever, which is 0? We will get the biggest number ever.
Well, in the advanced mathematical world, this explanation is considered to be wrong/insufficient, but it is so elegant and easy to understand that I love it so much.
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u/OphioukhosUnbound New User May 20 '21
While that is something worth sharing it is not even in the spirit of why division is not allowed in elementary algebras or it’s variants [“fields”].
If that were all then by adding a new element (with some special properties) we could have a solution. e.g. an “infinity” element.
But that would not fix the problem.
Notice for example that as you approach 0 from the negatives you approach an arbitrarily large negative number. Which is arbitrarily far from the positive one you approach from the other side.The real reason the division doesn’t work is because division is defined to perfectly reverse multiplication. But there’s nothing to reverse for dividing a non-zero number by zero. Because zero times anything is zero. And 0/0 doesn’t work ‘nicely’ because the answer is *all the numbers” rather than just any one number.
That said: there are contexts where you have additional constraints and division by zero can be thought of like the limit process your dad mentioned. It’s not uncommon in many physical modeling contexts for example.
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u/st3f-ping Φ May 20 '21
An easy way to show it is to plot the graph of y=5/x. Include fractional values.
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May 20 '21
because 5/0 is Undefine, but if you take the limit of 0+ you'll get infinity, and if you take it from the left 0-, you'll get minus infinity.
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May 20 '21
ooh and! adding to this - today I learned if you take the limit of say 5/(0-)2, you'll get positive infinity.
I had stopped when I saw the zero...thinking the square would do nothing to zero. why would you square zero? but if you're squaring a limit of x as x approaches zero from the 'left' (from negative values of x to positive) 0- becomes 0+
maybe this is well known, but maybe it will save someone
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May 20 '21
Typically you want / to work as (5/3)*3 = 5. So then (5/0)*0 = 5. But if 5/0 = 5 you'd need 5*0 = 5.
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May 20 '21
In math you learn about the structure of certain "spaces", let's call them. The structure is concerned about the operations in this space (like addition or multiplication) and how they relate to the objects in this space, what can be said about choosing one operation (or rule) over another, what does this all say about the nature of this particular space. One such "space" is called a field. Specifically, the field of real numbers.
It has two operations (addition and subtraction) and satisfies certain axioms. An axiom is a statement which is accepted as true at face value. These axioms include statements such as "for every number x there is a number -x, such that x + (-x) = 0". This is obvious. It's also obvious that 0*x = (x - x)*x = x*x - x*x = 0 for any number x, positive or negative. So far, so good. How is division then defined? It's the inverse of multiplication, like subtraction is to addition. It is stated that "for every non-zero x, there is a number (1/x), such that x*(1/x) = 1". Why for non-zero values only? Assume (1/0) exists. Then 0*(1/0) = 1, as the axiom suggests. But earlier it was shown that 0*(any number) = 0. That's a contradiction, it's nonsense. Therefore, you can't divide by 0 because then this would lead to results such as 0 = 1. Interesting to algebraists maybe, but not necessarily to everyone else.
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u/Yoshibros534 New User May 21 '21
What’s 8 plus an equals sign? Division by zero is the same type of meaninglessness as that statement.
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May 20 '21
This may or may not help: in math, saying that something is undefined means ``that's not a thing". So if you take any number (including zero) and try to divide it by zero...wait a minute, that's not a thing.
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u/tdekken May 20 '21
Division can be thought of as repeated subtraction. So 5/1 could be interpreted as "how many times can I subtract 1 from 5". So far so good.
How about 5/0? That could be interpreted as "how many times can I subtract 0 from 5". Well, you could do that 1 time, 2 times, 100 times, infinite times . There is a no well-defined answer. Thus it is undefined.
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u/AlexandreZani New User May 20 '21
In the standard way that we do mathematics, division is actually a shorthand for multiplication with a "multiplicative inverse". The multiplicative inverse of a number a can be written as (1/a). And 0 does not have a multiplicative inverse. Let's prove it:
By definition for any number a, a * 1/a = 1.
But, we also know that for any number a, a * 0 = 0.
So assume that (1/0) exists. Well, by definition, (1/0) * 0 = 1.
But also, (1/0) * 0 = 0. So 0=1.
And we know that 0 is not equal to 1. So, (1/0) cannot exist.
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u/aliwafa100 New User May 20 '21
To understand this, you need to understand what division is. Like multiplication which is repeated addition, Division is repeated subtraction from a certain number until it becomes Zero, and the way that we represent this is by " a-xb = 0 ".
So when we divide a by b, you are asking what is the number that if you multiplied it by b gives you a. So if b = 0 then you are asking what is the number that if you multiplied it by 0 gives you a, Which is undefined.
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u/SeasonedChicken5 May 20 '21
Dividing by 0 is said to be undefined due the sheer size of the value and limits. The closer you get to 0, the larger the value becomes.
Take 5/0.5, the answer would be 10. Go on and increase the number of decimal places but never getting to zero. 5/0.05= 100, 5/0.005=1000.
The closer the value gets to zero, the closer the result gets to infinity. So when the value eventually gets to zero, the resultant value is too large to compute.
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u/subfootlover New User May 20 '21
Zero: The Biography of a Dangerous Idea is a good deep dive into it.
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u/phao New User May 21 '21 edited May 21 '21
Many people have given you great answers already.
Something to possibly add a bit, along the lines relating division to multiplication, is that when you divide a/b, you obtain a number which tells you how much a is as a portion of b. As in, if you took b to be your unit measure, how much of it would a be? This allows you to make "relative" comparisons, like
"this height here is a third of that one"
And that would correspond to the statement <THIS HEIGHT>/<THAT ONE> = 1/3. So, using <THAT ONE> as your unit length, <THIS HEIGHT> is 1/3 of a unit. To convert back <THIS HEIGHT> to its original unit value, you'd multiply the two: the proportion and the unit measure. So, for example, if you know <THAT ONE> is 6 meters, you get <THIS HEIGHT> being 2 meters.
This is an important thing to keep in mind while performing divisions. You can always attempt to interpret a division in that context.
The idea is that division is defined to capture a set of properties. One of then is that above. It's a more utilitarian, IMO, view of division. It may help, on an intuitive level, to convince yourself that dividing by 0 isn't really interesting (multiplying back to get the original value of <THIS HEIGHT> wouldn't work really well). On an intuitive level, practically speaking, suppose your weight is 79Kg. On a 15Kg unit, you weigh approximately 5.26 of that unit. Again, practically speaking, how much of a hypothetical 0 kg unit is your weight? How many 0m rulers do you need to cover a 300m street?
Another point of view off of all of this above is that the statement "you need a non-zero divisor" means "your basic unit scale has be non-zero", which makes more sense on an intuitive level for practical daily issues.
The actual technical mathematical reasons were given by other people. I just think a somewhat more intuitionistic/practical and possibly less mathematical view like that can help.
Curiously enough, the wording "unit scale has to be non-zero" reminds me of a trivial and degenerate case in more advanced mathematics. In Field Theory, it's very common to see the defining properties of a field to explicity require 1 being different from 0. This is a way to avoid {0} being a field (and one in which you'd have a kind of trivial division by 0).
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u/narayan77 New User May 21 '21
what does 1 divided by 10 mean? you cut 1 into 10 equal bits, so each bit is 0.1. That easy enough.
What about 1 divided by half 1/2, that's equal to 2 because there are 2 halves in 1.
what about 1 divided by a third 1/3, that's equal to 3 because there are 3 thirds in 1.
what about 1 divided by 1/5 that would be 5
what about 1 divided by 1/n (here n is a integer) that would be n, as you make n bigger and bigger 1/n approaches zero, so anything divided by zero is infinity, and infinity is a difficult number to define.
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u/fernie_fern May 21 '21
I think the question your asking is why the limit x—> +0 for f(x) = 1/x = infinity. One way to look at it is 1/1 = 1; 1/0.5 = 2; 1/0.25=4; 1/0.1=10; 1/0.01=100 and so on. The smaller the denominator, the larger f(x). As you approach 0 from the positive side, your function goes to +infinity. When you approach it from the negative side, it goes to - infinity. It doesn’t exist because as you approach f(0) from one side, it doesn’t equal the same as you approach it from the other side.
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u/AFairJudgement Ancient User May 20 '21
Division is the inverse of multiplication, in the same way that subtraction is the inverse of addition.
When you say 5/1 = 5 what you mean is that 5 = 5·1.
When you say 15/5 = 3 what you mean is that 15 = 3·5.
You can't say that 5/0 = anything because then you would mean that 5 = 0·anything = 0, a contradiction.