I bet you can't prove that 0 is the smallest natural number.

@Heartlander 0 is a number, otherwise you couldn’t perform arithmetics with it.

Or how else would you define addition for example?

@Luke73 You can't perform all arithmetic with it. You can't divide zero by zero because there are no dividable parts of zero. nor can you divide any number by zero. It's the average of +1 and -1, always equidistance from equal positive and negatives.

5+0 = 5, never more than 5 nor less than five, thus zero has no magnitude, no substance, no anything.

@Heartlander I never claimed that you can perform every arithmetic with it. I was asking if you could define addition without 0. And we’re talking about natural numbers. There are no negative numbers in them.

And 0 is not always equidistant from positives and negatives for example -1 and 3. The arithmetic medium is 1.

No numbers are "provable" they are abstractions related to countables. Open your hand and if you are like most you see five fingers. Cut off 5 fingers and how many fingers do you have left? Zero.

@Heartlander So I take you can’t define addition and subtraction.

And you contradicted yourself just now. By asking how many, you ask for a number that can quantify. And not just count.

As a side node, there are number sets that are uncountable, like the real numbers.

@Luke73 all numbers are either countable, measurable, relatable or comprehendible.

Draw a circle. Put 3 dots in it. Then erase the 3 dots. How many dots are left in the circle? Zero. That's the only definition of subtraction you will ever get. Do it backwards to define addition.

@Heartlander Can you define any of those words?

With your „definition“ I can’t calculate 1+1 for example… And also how do you define a circle? Where on the circle are the points? Does it matter?

@Luke73 Draw a circle put one dot in it. add another dot. How many dots in the circle? 2. Addition defined. Doesn't matter where IN the circle you put the dots.

Countable - what you can do with fingers and toes, and if necessary someone else's fingers and toes.
Measurable - what can be measured, like what's a half inch, or a mile and a half?
Relatable - like pi is the relation between the perimeter and diameter of a circle
Comprehendible - the square root of -2
Add "undefinable" - what you get when you divide by zero.

@Heartlander You still haven’t defined what a circle is.

So you can only add countable numbers?

So you can’t count to one trillion, as there are not as many toes and fingers for it, right?

That definition of measurement doesn’t make any sense in mathematics. How would you measure exactly one mile for example? You can’t. There are always tolerances.

The last two aren’t even definitions, they’re just examples.

And there are instances where you can define that. For example take x/sin(x) at x=0 you would get 0/0 but if you take the limit, you get as answer is 1.

@Luke73 To count to a trillion you would need lots of figures and toes.
measurements are easy, use a ruler or yardstick, practically all measurements are approximations to some degree of accuracy.
relations are just that. like half the group is male, the other half female. The relationship can be used in arithmetic. Trig is about relationships last I thought about it.

@Luke73 gotta run. Was fun.

@Heartlander It’s ok to say you don’t know what you’re talking about ;)

@Luke73 Good grief, are you still insisting that zero is when it isn't? How many zeroes would you need to add together to amount to one?

@Heartlander It's obvious that you have no idea about mathematics and are just making things up. Otherwise the word monoid or group would have been mentioned already in this context, or even neutral element.

@Luke73 If you cut off all 5 fingers you would have zero fingers left, not the smallest number of fingers left; group, monoid and neural element notwithstanding

@Heartlander You even prove my point, you don't care about definitions or anything else. For you only your own (illogical) imagination matters.

@Luke73 Natural, real, integers, etc. are all collections or subsets of one another. They transition naturally to one another when when we perform math operations. They are not limitations or boundaries. "the smallest number" implies that there is something there, somehow quantifiable or measurable. At zero there is no there there.

@Heartlander Yes, they're contain each other but they all have different properties. Natural numbers, integers and rational numbers are all countable. Where as the real numbers aren't countable. And there is a smallest number in a closed interval, even in the real numbers. A smallest element can exist when a total order exsists.

@Luke73 math operations can produce different results if you don't bridge the transitions carefully. As I recall, it caused Intel a heap of trouble 25 or 35 years ago and $$ when their math chip or integrated chip treated zero as a very small number rather than zero.

While this discussion may seem of inconsequence, not so for computer programmers who suddenly had to explain the mysterious penny that materialized or vanished 10 million calculations later :)

I'm old school, self thought 8080 BAL. And yes I had my own vocabulary, often with a 4-letter prefix, when mixing real and integers.

@Heartlander You know that computers can't do math, do you? They can approximate some times. But when you are as good with computers as you claim to be, you must have known that. Take an 8-bit unsigned integer for example and try to calculate 200 + 200, what's the result? Not 400.

@Luke73 It's been 30 years (40 years?) since I peeked inside, but processor instructions always included a byte add and maybe even a word add that is deadly accurate as long as long as the answer didn't exceed the byte of word length. Pretty much everything beyond that needed some programming instructions. But send 200+200 through a floating point math package or math chip and it may give you almost 400 and hide how far off it is. But adding 200+200 internally will give an accurate 400 if you are willing to mess with the carry bit.

@Heartlander Well if you have a carry bit, it's not 8-bit anymore, is it?

And floating point is even more inaccurate, for example 0.1 + 0.2 is not 0.3, on no hardware, no matter how old or modern.

Computers can't do math correctly.

@Luke73 Right, too many bits, thus participation by the programmer was needed to come up with the accurate answer.

@Heartlander So you agree with me that computers can't do math, right?

@Luke73 It can't do some math. But with some math it's deadly accurate. Overall, it only has be accurate enough to satisfy the need for accuracy, when needed.