I think there's a legitimate quibble here. Zero is not so much a number as a place-holder, as it does not designate an amount but rather an absence. 10 is a number, and an even number to be sure. It is composed of the numerals 1 and 0. The zero is a convenient way of denoting that we have one ten and only one ten (10) of something, as opposed, for example, to one ten and one of something (11). Notation for ten without the zero is awkward (1_ is easily confused for 1). But the numeral 0 itself never denotes a number of anything. If there a convention in higher mathematics for regarding zero as even, I am unaware of it (just like the entire rest of anything approximating higher mathematics -- it's only accessible to me on a pop-science level). We've got to have a mathematician around here somewhere though, so what's the real story?
Dude, there is no quibble. Zero is a number. And it's even.
Take two numbers. If the first is even, and the other is even, the result adding or subtracting them is even. If the first is odd and the other is odd, the result adding or subtracting them is even. If they are an odd and an even number, the result adding or subtracting them is odd. So, first of all, since an even number plus zero is even, and an odd number plus zero is odd, zero is an even number. Also, since zero plus zero is zero, then zero is an even number.
My math friend's response: Zero is definitely a number, just like one and two are. It is the quantity of a void, just like twelve is the quantity of a dozen. It is an even number, without a doubt. Absolutely. No question. None. Zero.
But zero is not like one and two. When was the last time you were counting anything and started with zero?
Also, though sconstant makes a certain kind of sense, I'd suggest that "adding zero" isn't really mathematical operation from a practical point of view. It's not strictly nonsensical, at a certain level of abstraction, but it never happens.
"I've added zero to three." "Have you? How can I tell?" "No, look: I've three apples. I add zero apples. I still have three apples." "But where are the zero apples?" "What do you mean where? Zero apples is no apples." "Exactly."
Another way to come at the issue -- and this may help me more than any number of sworn statements from mathematicians that zero is a number and an even one -- might be to ask: When is it mathematically relevant that zero be even rather than odd? (I do not doubt that it may be, but as a non-mathematician I'm having a hard time thinking of an example.)
In Europe they essentially start with zero when counting the floors in a building: http://en.wikipedia.org/wiki/Storey (our 2nd floor being their first)
I would suggest that counting and math are two separate but related things. Zero, it can be argued is the most mathematically important number--without zero there is no calculus, etc. This is a great read even if you are not mathematically inclined: http://www.amazon.com/Zero-The-Biography-Dangerous-Idea/dp/0140296476/ref=sr_1_2?ie=UTF8&qid=1354810219&sr=8-2&keywords=history+of+zero
A number is an abstract concept that can be used to represent a quantity. So, for instance, the number 3 has no intrinsic real-world value unless it has some unit attached: "3 apples"; "3 inches"; "3 dollars"; "3 meters per second per second of acceleration". The abstract number gains real-world significance because we can add units.
But 0 works the same way: it is an abstract concept that, when we put units in place, has real-world significance. So just as I can, in the real world, have 3 apples or move my car 3 inches, I can have 0 apples or move my car 0 inches.
So if I start with 3 apples and take 2 apples away, how many apples do I have left? 1. If I then take 1 more away, how many do I have left? You wouldn't say there's no answer there; you would say the answer is the number 0.
The same thing is true of negative numbers, by the way. If I have a bank account with 3 dollars, and the bank takes away 5 dollars due to a fee, I have –2 dollars left. That's a number, representing a quantity of a real thing because I have added units, just like 0 dollars.
Counting "from" zero makes sense, particularly in non-integer counting situations where fractions of things matter for precision -- counting increments of time, for example.
Maybe you have another example. I like the time example because it allows me to suggest that counting from zero isn't counting with zero. It's like saying "go". It is the beat before anything is counted. Ready, set, go. Is go odd or even? (No.)
I honestly don't mean to be tiresome. I'm just not persuaded and I'm quite enjoying thinking about all these responses.
As a mathematician, I have difficulty wrapping my head around the fact that there is any question zero is even. We've established, though, that not everyone thinks in terms of numbers, so I can accept that this occurs. I can't properly wrap a present to save my life, and I'm a terrible bowler, things that the w and my father, respectively, find mind-boggling. I realize I'm just echoing Roger, but let's get two arguments out of the way.
1. Zero isn't a number, because...
It's a number. It is a concept of quantification, of measurement. You can have zero apples, you can have zero gallons of gas in your car, you can gain zero pounds. The numeral 0 might be used as a placeholder, but that just means it represents 0 of whatever goes in that place. The "absence of [whatever]" does exist, it's an empty set (or an empty interval, depending on whether you're going discrete or continuous). You can still measure it, though: it has measure zero.
2. Dividing zero by two is a special case, which makes it difficult to determine whether zero is even or odd.
I think the confusion with this one comes from the fact that you can't divide BY zero (not that it hasn't been tried: http://throwingthings.blogspot.com/2010/04/gerald-lambeau-grigori-perelman-and-me.html). At any rate, though, you can think of it this way: a number is even if it is equal to 2*n, where n is some integer. Zero is an integer, 2*0 = 0, thus 0 is even.
No, he really means that they count starting with zero. In many computer languages, if one creates an array, the first index of that array is 0. In C, it's because declaring the array really just stores the memory address A of the first element, and to get to element n, you add n*(the size of an element) to A. So, the first element is located at A + 0.
Maybe think of it in terms of Kevin Bacon. How many degrees away from Kevin Bacon is Kevin Bacon? It's gotta be a number.
Thank you! Defining terms this way does indeed, if accepted, position zero as an even number.
When though, is it operationally important for you, as a mathematician, that zero is even rather than odd? Or that it is a "number" in the sense that you don't just realize that x in the case before you is zero and therefore of no moment in your calculations?
Please! -- I get that the notion I'm pushing is gobsmackingly out of synch with accepted mathematical conventions. As a matter of temperament, however, that doesn't settle the question for me. I'm desperate for a "why". -- I need on concrete (or abstract theoretical) example in something like plain English. Otherwise I'm going to have trouble sleeping at least until Anne57's book arrives.
For even-ness, say I am looking at a set of consecutive integers, and I want to do some operation on each one depending if that one is even or odd. Or, I may have some program that's processing in parallel, and I'm choosing to send the even numbers to server A to be processed, and the odd numbers to server B. If my set includes 0, and it's important to me that I'm sending the same number of things to each server, it's important to me that 0 is even; if it's odd, there will be more odd things than even.
For the concept of it as a "number", if you're doing just about anything in abstract algebra, you have the concept of the "additive identity" -- some element e in your set such that, if you add* it to any element, you just get that element back. Regular arithmetic with real numbers is the obvious example; 0 plus any other number is just that number. Another example is modular arithmetic: a number is equivalent to r (mod n) if it's equal to q*n + r, where r is between 0 and n-1 (in other words, if you divide by n, you get remainder r). The additive identity is still 0: if a number is r (mod n), adding 0 (mod n) is equivalent to adding some multiple of n: The remainder will still be r, so 0 + r = r (mod n). If you're actually trying to do something with these constructs, it's important that 0 be an actual element of your set, not just an abstraction.
I hope those help; coming up with good, easy-to-understand examples is not my strongest suit.
*"adding" in theoretical algebra can be defined in some very strange ways, but that's another story.
One example of why the difference matters: We can state a series of rules about how integers work that don't work unless 0 is even. So, for example, if you add or subtract two integers, you can state whether the result will be even or odd depending on what the two inputs are. If you add or subtract two even numbers or two odd numbers, the result is always even; if you add or subtract one even and one odd number, the answer is always odd:
E + E = E: 2 + 4 = 6
E – E = E: 4 – 2 = 2
O + O = E: 5 + 3 = 8
O – O = E: 5 – 3 = 2
E + O = O: 4 + 5 = 9
E – O = O: 6 – 3 = 3
O – E = O: 7 – 2 = 5
Plug in any even and odd numbers you want and this works. But this only works if 0 is even, since an even number ± 0 is even and an odd number ± 0 is odd. So if 0 is not even, these rules break unless you fold in all sorts of exceptions.
Likewise with multiplication: the product of two integers is even unless they are both odd. So:
E × E = E: 6 × 4 = 24
O × E = E: 5 × 2 = 10
O × O = O: 3 × 7 = 21
This pattern works for any integers you plug in. But it doesn't work unless 0 is even, since any number multiplied by 0 is 0. Otherwise, 4 × 0 = 0 would break the rule.
Another, somewhat minor example: The number line alternates between even and odd numbers. E.g., 1, 2, 3, 4, 5 = O, E, O, E, O. This is equally true for negative numbers: –5, –4, –3, –2, –1 = O, E, O, E, O. But the number line spans 0, so if it isn't even, you have a disruption in the pattern: –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5 = O, E, O, E, O, E, O, E, O, E, O.
(I just realized this is basically the same thing sconstant said, but hopefully having it spelled out in this much detail will help.)
You're begging the question. It is not the case that we have "no number" of apples. There is a supply of apples in the universe, and we have have a number of them, and we can count the number of apples that we have. We can count them easily, in this case, because that number is zero.
Isn't the real story here that the BBC agreed with something Bloomberg did?
ReplyDeleteI reject the supposed mathematicians answer. Zero is the lack of numbers, hence is neither odd nor even.
ReplyDeletePlease tell me you're kidding.
ReplyDeleteI think there's a legitimate quibble here. Zero is not so much a number as a place-holder, as it does not designate an amount but rather an absence. 10 is a number, and an even number to be sure. It is composed of the numerals 1 and 0. The zero is a convenient way of denoting that we have one ten and only one ten (10) of something, as opposed, for example, to one ten and one of something (11). Notation for ten without the zero is awkward (1_ is easily confused for 1). But the numeral 0 itself never denotes a number of anything. If there a convention in higher mathematics for regarding zero as even, I am unaware of it (just like the entire rest of anything approximating higher mathematics -- it's only accessible to me on a pop-science level). We've got to have a mathematician around here somewhere though, so what's the real story?
ReplyDeleteI've contacted our consulting mathematician. Yes, we have one.
ReplyDeleteNot at all. Why would I be? Zero is nothing, the absence of numbers (hence the name).
ReplyDeleteDude, there is no quibble. Zero is a number. And it's even.
ReplyDeleteTake two numbers. If the first is even, and the other is even, the result adding or subtracting them is even. If the first is odd and the other is odd, the result adding or subtracting them is even. If they are an odd and an even number, the result adding or subtracting them is odd. So, first of all, since an even number plus zero is even, and an odd number plus zero is odd, zero is an even number. Also, since zero plus zero is zero, then zero is an even number.
My math friend's response: Zero is definitely a number, just like one and two are. It is the quantity of a void, just like twelve is the quantity of a dozen. It is an even number, without a doubt. Absolutely. No question. None. Zero.
ReplyDeleteBut zero is not like one and two. When was the last time you were counting anything and started with zero?
ReplyDeleteAlso, though sconstant makes a certain kind of sense, I'd suggest that "adding zero" isn't really mathematical operation from a practical point of view. It's not strictly nonsensical, at a certain level of abstraction, but it never happens.
"I've added zero to three."
"Have you? How can I tell?"
"No, look: I've three apples. I add zero apples. I still have three apples."
"But where are the zero apples?"
"What do you mean where? Zero apples is no apples."
"Exactly."
Another way to come at the issue -- and this may help me more than any number of sworn statements from mathematicians that zero is a number and an even one -- might be to ask: When is it mathematically relevant that zero be even rather than odd? (I do not doubt that it may be, but as a non-mathematician I'm having a hard time thinking of an example.)
In Europe they essentially start with zero when counting the floors in a building: http://en.wikipedia.org/wiki/Storey (our 2nd floor being their first)
ReplyDeleteI would suggest that counting and math are two separate but related things. Zero, it can be argued is the most mathematically important number--without zero there is no calculus, etc. This is a great read even if you are not mathematically inclined: http://www.amazon.com/Zero-The-Biography-Dangerous-Idea/dp/0140296476/ref=sr_1_2?ie=UTF8&qid=1354810219&sr=8-2&keywords=history+of+zero
Computer scientists count from zero all the time.
ReplyDeleteA number is an abstract concept that can be used to represent a quantity. So, for instance, the number 3 has no intrinsic real-world value unless it has some unit attached: "3 apples"; "3 inches"; "3 dollars"; "3 meters per second per second of acceleration". The abstract number gains real-world significance because we can add units.
ReplyDeleteBut 0 works the same way: it is an abstract concept that, when we put units in place, has real-world significance. So just as I can, in the real world, have 3 apples or move my car 3 inches, I can have 0 apples or move my car 0 inches.
So if I start with 3 apples and take 2 apples away, how many apples do I have left? 1. If I then take 1 more away, how many do I have left? You wouldn't say there's no answer there; you would say the answer is the number 0.
The same thing is true of negative numbers, by the way. If I have a bank account with 3 dollars, and the bank takes away 5 dollars due to a fee, I have –2 dollars left. That's a number, representing a quantity of a real thing because I have added units, just like 0 dollars.
Counting "from" zero makes sense, particularly in non-integer counting situations where fractions of things matter for precision -- counting increments of time, for example.
ReplyDeleteMaybe you have another example. I like the time example because it allows me to suggest that counting from zero isn't counting with zero. It's like saying "go". It is the beat before anything is counted. Ready, set, go. Is go odd or even? (No.)
I honestly don't mean to be tiresome. I'm just not persuaded and I'm quite enjoying thinking about all these responses.
Totally getting that book. Thank you.
ReplyDeleteSo you have no apples, or no number of apples, so zero apples. So zero is not a number of anything.
ReplyDeleteExcept that there is nothing there. No quantity to count, so no number.
ReplyDeleteAs a mathematician, I have difficulty wrapping my head around the fact that there is any question zero is even. We've established, though, that not everyone thinks in terms of numbers, so I can accept that this occurs. I can't properly wrap a present to save my life, and I'm a terrible bowler, things that the w and my father, respectively, find mind-boggling. I realize I'm just echoing Roger, but let's get two arguments out of the way.
ReplyDelete1. Zero isn't a number, because...
It's a number. It is a concept of quantification, of measurement. You can have zero apples, you can have zero gallons of gas in your car, you can gain zero pounds. The numeral 0 might be used as a placeholder, but that just means it represents 0 of whatever goes in that place. The "absence of [whatever]" does exist, it's an empty set (or an empty interval, depending on whether you're going discrete or continuous). You can still measure it, though: it has measure zero.
2. Dividing zero by two is a special case, which makes it difficult to determine whether zero is even or odd.
I think the confusion with this one comes from the fact that you can't divide BY zero (not that it hasn't been tried: http://throwingthings.blogspot.com/2010/04/gerald-lambeau-grigori-perelman-and-me.html). At any rate, though, you can think of it this way: a number is even if it is equal to 2*n, where n is some integer. Zero is an integer, 2*0 = 0, thus 0 is even.
Is –1 a number?
ReplyDeleteComputer scientists count from zero even when counting integer units. Here's a good explanation of why.
ReplyDeleteNo, he really means that they count starting with zero. In many computer languages, if one creates an array, the first index of that array is 0. In C, it's because declaring the array really just stores the memory address A of the first element, and to get to element n, you add n*(the size of an element) to A. So, the first element is located at A + 0.
ReplyDeleteMaybe think of it in terms of Kevin Bacon. How many degrees away from Kevin Bacon is Kevin Bacon? It's gotta be a number.
Thank you! Defining terms this way does indeed, if accepted, position zero as an even number.
ReplyDeleteWhen though, is it operationally important for you, as a mathematician, that zero is even rather than odd? Or that it is a "number" in the sense that you don't just realize that x in the case before you is zero and therefore of no moment in your calculations?
Please! -- I get that the notion I'm pushing is gobsmackingly out of synch with accepted mathematical conventions. As a matter of temperament, however, that doesn't settle the question for me. I'm desperate for a "why". -- I need on concrete (or abstract theoretical) example in something like plain English. Otherwise I'm going to have trouble sleeping at least until Anne57's book arrives.
For even-ness, say I am looking at a set of consecutive integers, and I want to do some operation on each one depending if that one is even or odd. Or, I may have some program that's processing in parallel, and I'm choosing to send the even numbers to server A to be processed, and the odd numbers to server B. If my set includes 0, and it's important to me that I'm sending the same number of things to each server, it's important to me that 0 is even; if it's odd, there will be more odd things than even.
ReplyDeleteFor the concept of it as a "number", if you're doing just about anything in abstract algebra, you have the concept of the "additive identity" -- some element e in your set such that, if you add* it to any element, you just get that element back. Regular arithmetic with real numbers is the obvious example; 0 plus any other number is just that number. Another example is modular arithmetic: a number is equivalent to r (mod n) if it's equal to q*n + r, where r is between 0 and n-1 (in other words, if you divide by n, you get remainder r). The additive identity is still 0: if a number is r (mod n), adding 0 (mod n) is equivalent to adding some multiple of n: The remainder will still be r, so 0 + r = r (mod n). If you're actually trying to do something with these constructs, it's important that 0 be an actual element of your set, not just an abstraction.
I hope those help; coming up with good, easy-to-understand examples is not my strongest suit.
*"adding" in theoretical algebra can be defined in some very strange ways, but that's another story.
One example of why the difference matters: We can state a series of rules about how integers work that don't work unless 0 is even. So, for example, if you add or subtract two integers, you can state whether the result will be even or odd depending on what the two inputs are. If you add or subtract two even numbers or two odd numbers, the result is always even; if you add or subtract one even and one odd number, the answer is always odd:
ReplyDeleteE + E = E: 2 + 4 = 6
E – E = E: 4 – 2 = 2
O + O = E: 5 + 3 = 8
O – O = E: 5 – 3 = 2
E + O = O: 4 + 5 = 9
E – O = O: 6 – 3 = 3
O – E = O: 7 – 2 = 5
Plug in any even and odd numbers you want and this works. But this only works if 0 is even, since an even number ± 0 is even and an odd number ± 0 is odd. So if 0 is not even, these rules break unless you fold in all sorts of exceptions.
Likewise with multiplication: the product of two integers is even unless they are both odd. So:
E × E = E: 6 × 4 = 24
O × E = E: 5 × 2 = 10
O × O = O: 3 × 7 = 21
This pattern works for any integers you plug in. But it doesn't work unless 0 is even, since any number multiplied by 0 is 0. Otherwise, 4 × 0 = 0 would break the rule.
Another, somewhat minor example: The number line alternates between even and odd numbers. E.g., 1, 2, 3, 4, 5 = O, E, O, E, O. This is equally true for negative numbers: –5, –4, –3, –2, –1 = O, E, O, E, O. But the number line spans 0, so if it isn't even, you have a disruption in the pattern: –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5 = O, E, O, E, O, E, O, E, O, E, O.
(I just realized this is basically the same thing sconstant said, but hopefully having it spelled out in this much detail will help.)
You're begging the question.
ReplyDeleteIt is not the case that we have "no number" of apples. There is a supply of apples in the universe, and we have have a number of them, and we can count the number of apples that we have. We can count them easily, in this case, because that number is zero.
I think this is the part where I just mention, as a formality, that I love this blog. I'll toy with these in the morning. You all are the best.
ReplyDeleteIt's the number that says "You owe me 1 apple."
ReplyDeleteSo isn't 0 the number that says "I may not have any apples, but at least I don't owe you any apples"?
ReplyDelete