What Would Math be Like Without Zero?

[Alanay]

It's something I've been thinking about recently, would math be simpler or way more complicated without 0?

1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21...

I've been trying to do some simple equations without 0, and with small numbers the results are usually the same. But would getting rid of 0 solve any problems, for example dividing 0 by 0. I'm not sure how much this has been thought about before and if there has been any reasoning why this would be a terrible idea so I'd like your guy's opinions on the matter.

 

[jedishrfu]

Here's a book about it:

https://books.google.com/books?id=o...h1nQgw2#v=onepage&q=life without zero&f=false

How would you represent 10 or 101 or 1001...? Its important as a placeholder in our decimal representation of numbers.

 

[fresh_42]

"People first could handle their crop through counting and adding. But as soon as they started borrowing crop, they had to turn to negative numbers to get theirs accounts balanced. Necessary for it was by the way the discovery of zero, which wasn't easy at the times. Why should one count nothing? It took a while."

Eliminating the 0 is nothing else than counting. It has nothing to do with calculation, computing or any modern science any more.
You could ask as well how it would be if we only can count to 5 since we have just 5 fingers. But even the Neanderthals could count further ...

 

[micromass]

How would you represent 10 or 101 or 1001...? Its important as a placeholder in our decimal representation of numbers.

OK, but this is actually irrelevant. You can have 0 as a placeholder and still not accept the existence of 0 by itself. In fact, that is what most ancient civilizations actually did.

 

[rootone]

The Romans had no zero.
(Although I expect some of the soldiers who didn't get paid after losing a battle were well able to grasp the concept)

 

[Svein]

There are at least answers to this:

• The practical: If you have 5 apples and you give away 5, what have you left?
• The mathematical: 0 is needed as the identity element of addition (https://en.wikipedia.org/wiki/Identity_element).

 

[micromass]

• The mathematical: 0 is needed as the identity element of addition (https://en.wikipedia.org/wiki/Identity_element).

OK, true. But what if we do not have an identity element for addition. What would go wrong mathematically?

 

[pwsnafu]

The Romans had no zero.
(Although I expect some of the soldiers who didn't get paid after losing a battle were well able to grasp the concept)

We don't know of any Roman numeral for zero before 725 (which is first use of N for zero we know of). We do know that the word nulla was used for the concept.

 

[256bits]

• The practical: If you have 5 apples and you give away 5, what have you left?

You had as many apples as you had before you went apple picking. Why go through all the trouble of recording no change?

 

[PeroK]

OK, true. But what if we do not have an identity element for addition. What would go wrong mathematically?

You'd lose associativity:

$(2 + 1) - 1 = 3 - 1 = 2$
but
$2 + (1 - 1)$ would de undefined.

We already have to take care that a denominator is not 0, and this would extend to everything. If you have $x-y$ anywhere, you'd have to exclude the case $x = y$.

$cos(\pi/2)$ would be undefined. But, then $tan(\pi/2)$ is not defined and we manage that.

Losing the concept of zeroes of a function would be a major loss, I would say.

I'd prefer to exclude a number like 673, since that doesn't turn up very often. Missing out 673 would be much easier to deal with.

 

[FactChecker]

If I had $5 and owed someone $5, what would my net worth be? How can simple calculations be done without 0? In abstract algebra, the simplest thing above a "set" is the "group". A group requires an "identity" element, which, when added to another number, causes no change. That is 0. Without that it is not possible to do even the simplest math.

 

[Alanay]

If I had $5 and owed someone $5, what would my net worth be? How can simple calculations be done without 0? In abstract algebra, the simplest thing above a "set" is the "group". A group requires an "identity" element, which, when added to another number, causes no change. That is 0. Without that it is not possible to do even the simplest math.

You could use a placeholder or simply not write anything at all. You can do even the simplest of math without 0.

 

[PeroK]

You could use a placeholder or simply not write anything at all. You can do even the simplest of math without 0.

You can do a lot of maths without 673. Even more than you can do without 0.

 

[Svein]

You could use a placeholder or simply not write anything at all. You can do even the simplest of math without 0.

Then how would you detect the difference between "haven't written anything" and zero?

 

[Alanay]

You can do a lot of maths without 673. Even more than you can do without 0.

Okay, sure but that wasn't the topic. Although now I'm curious why 673 is so useless. Does it just not appear much in any equations?

 

[PeroK]

Okay, sure but that wasn't the topic. Although now I'm curious why 673 is so useless. Does it just not appear much in any equations?

Have you ever used the number 673?

 

[Alanay]

Then how would you detect the difference between "haven't written anything" and zero?

You could do something like

net worth:

I know using a placeholder would be better, but wouldn't having no 0 in equations save much room on the whiteboard. We can still define variables with a placeholder or just the word nothing/empty or something more technical if you prefer.

 

[Alanay]

Have you ever used the number 673?

Good point.

 

[Svein]

a placeholder or just the word nothing/empty or something more technical if you prefer.

Yes, of course, we mathematicians are a lazy lot. It is much easier to write "0" than "nothing".

 

[Alanay]

Yes, of course, we mathematicians are a lazy lot. It is much easier to write "0" than "nothing".

I understand. I'm not talking about when defining the value of something with 0. I'm talking about larger numbers and simply not showing anything if it has no value. We could even just use 0 as a placeholder but get rid of 10s, 20s etc.

 

[PeroK]

You could use a placeholder or simply not write anything at all. You can do even the simplest of math without 0.

Here's a thing. You'd have to change all the computers and all the computer systems to stop them using 0. You can't have computers doing things like:

$5 + 0 + 1 + 6 = 12$

If people aren't allowed to use 0. If a computer can add four numbers together, one of which is "0", then why can't I?

You'd have to forbid 0 being entered in a numeric field, as it isn't a number. It would be just as meaningless as entering "a" in a numeric field.

 

[Alanay]

Here's a thing. You'd have to change all the computers and all the computer systems to stop them using 0. You can't have computers doing things like:

$5 + 0 + 1 + 6 = 12$

If people aren't allowed to use 0. If a computer can add four numbers together, one of which is "0", then why can't I?

Because it's almost redundant, what's the point of adding a value of nothing.

 

[fresh_42]

Have you ever used the number 673?

+++ Breaking News +++ Sieve of Eratosthenes actually a sieve +++

 

[Svein]

Because it's almost redundant, what's the point of adding a value of nothing.

But if you have an expression like 5+p+1+6=12 and you want the value for p that satisfies the equation, what would you do? The obvious result is p=0, but if we are not allowed 0, you would either write p= or "p is nothing"...

 

[PeroK]

Because it's almost redundant, what's the point of adding a value of nothing.

Because it's there! That might be the sales from your four salesmen. The second salesman didn't sell anything, so you enter 0. If you can't enter 0, then the program can't simply add the four numbers together.

You really, really don't like 0 do you?

Also, if the figures were profit and loss and they were:

$+3 - 5 + 2 + 7$

Then the program would blow up $+3 - 5 + 2 = -2 + 2 =$ error; invalid arithmetic operation

You simply can't add those four numbers if 0 is not a valid number.

 

[Alanay]

Because it's there! That might be the sales from your four salesmen. The second salesman didn't sell anything, so you enter 0. If you can't enter 0, then the program can't simply add the four numbers together.

You really, really don't like 0 do you?

Also, if the figures were profit and loss and they were:

$+3 - 5 + 2 + 7$

Then the program would blow up $+3 - 5 + 2 = -2 + 2 =$ error; invalid arithmetic operation

You simply can't add those four numbers if 0 is not a valid number.

Again, I understand. This can be resolved with 0 just like before, however getting rid of 0s in decimals and 10s, 20s etc. might make things simpler.

 

[fresh_42]

The finding of zero has been one of the real big achievements of mankind. First time someone in India long ago decided to count "nothing". It's been the seed of all that came afterwards: economics, calculation, mathematics, physics, computing and so on. To debate a world without zero is like debating a world without fire or a wheel! It would make a lot of more sense to discuss a world without electricity! At least this could theoretically happen.

 

[micromass]

But if you have an expression like 5+p+1+6=12 and you want the value for p that satisfies the equation, what would you do? The obvious result is p=0, but if we are not allowed 0, you would either write p= or "p is nothing"...

I don't see that as a problem. Ancient mathematicians looked at such equations and just said they had no solution.

I mean, the proposal of the OP is clearly: what happens if only do math in the structure $\mathbb{R}^+\setminus \{0\}$. This is not problematic a priori, ancient mathematicians used to do this. So I think what he is really asking is: is there a theorem or result that uses only notions from $\mathbb{R}^+\setminus\{0\}$, but which can only (or more easily) be solved by accepting the existence of $0$ and even negative numbers.

 

[PeroK]

I don't see that as a problem. Ancient mathematicians looked at such equations and just said they had no solution.

I mean, the proposal of the OP is clearly: what happens if only do math in the structure $\mathbb{R}^+\setminus \{0\}$. This is not problematic a priori, ancient mathematicians used to do this. So I think what he is really asking is: is there a theorem or result that uses only notions from $\mathbb{R}^+\setminus\{0\}$, but which can only (or more easily) be solved by accepting the existence of $0$ and even negative numbers.

The remainder theorem? But, I guess if you don't think 0 has any purpose, then you're not interested in the zeroes of a polynomial function. Or of any function.

I'm no expert in what the ancients did mathematically although they obviously had amazing ability at numerical calculations, plain geometry and trig.

But, the vast ocean of modern mathematics was wholly unknown to them. Calculus, algebraic geometry, algebra itself. You can't do any of that unless you know that adding any two numbers produces a number. And that requires 0.

Not to mention IT. You need 0+0 = 0, however pointless that appears to the OP.

 

[micromass]

But, I guess if you don't think 0 has any purpose, then you're not interested in the zeroes of a polynomial function. Or of any function.

Oh, but you are. The ancients did not have zero, but were able to formulate zeroes of polynomial equations without using zero. For example $x^2 - x - 1=0$ was written as $x^2 = x + 1$ obviously. But the disadvantage of this approach is that the solution to this is very different from the solution to a slightly different equation $x^2 + x = 1$. The concept of 0 and negative numbers allows us to write all those special cases into one general case $\alpha x^2 + \beta x + \gamma = 0$ and give a general solution.

Why would we be interested in zeroes of polynomial equations? Because those have a very geometric meaning. Quadratic functions correspond to parabolas and linear functions with lines. The ancient already solved third-degree equations by using conic sections to their advantage. But it seems especially the concept of zero and negative numbers that was missing to implement a full link between geometry and algebra. This link was advanced by Fermat and Descartes and has revolutionized mathematics.

 

[aikismos]

We do modular math without zero every day.

..., 1 o'clock, 2 o'clock, ... , 11 o'clock, 12 o'clock, ...

 

[fresh_42]

We do modular math without zero every day.

..., 1 o'clock, 2 o'clock, ... , 11 o'clock, 12 o'clock, ...

But we do even more by using the zero: light on - light off - radio on - radio off - pc on - pc off - ...

 

[aikismos]

But we do even more by using the zero: light on - light off - radio on - radio off - pc on - pc off - ...

Well, the question of the nature of 'use' and 'more' is debatable. But it bears reminding ourselves that the use of 0 to represent off is arbitrary. You can use a perfectly isomorphic Boolean algebra using 1 and 2 instead... :D

 

[FactChecker]

You could use a placeholder

Any "placeholder" with the same properties as 0 is just using a different symbol for the same thing.

or simply not write anything at all.

Have I written a zero after this sentence or not?

You can do even the simplest of math without 0.

Really? What is 1-1? That is pretty simple math that can not be answered without 0 because 0 is the answer.

 

[Alanay]

Any "placeholder" with the same properties as 0 is just using a different symbol for the same thing.
Have I written a zero after this sentence or not?
Really? What is 1-1? That is pretty simple math that can not be answered without 0 because 0 is the answer.

Calculating 1-1 does not require you to write it down. You can still calculate at least the simplest of math without 0 obviously, but since you say "Really?" it seams you don't believe that. Try 1+1 or 11+21...