Why is zero considered an important number?

[physio]

My question may come as a surprise to many people but I have not understood the importance of zero and its significance. How can nothing be something? Why do we have zero as a number? If it's used for measuring then we can very well do without zero, for example if there is a fellow counting pebbles then after say 9 pebbles he can count 11,12 etc. upto infinite. Also I don't understand the reason for negative numbers. How can something be less than nothing?

 

[Charmar]

If someone has 146 dollars in their bank account and they spend 200 by way of a check how much do they have? -54 dollars (Ignoring bouncing fees) They need to deposit 54 to make the account even.

http://en.wikipedia.org/wiki/Negative_number Many other examples and reasons available here.

 

[SteveL27]

My question may come as a surprise to many people but I have not understood the importance of zero and its significance. How can nothing be something? Why do we have zero as a number? If it's used for measuring then we can very well do without zero, for example if there is a fellow counting pebbles then after say 9 pebbles he can count 11,12 etc. upto infinite. Also I don't understand the reason for negative numbers. How can something be less than nothing?

There are really two different discoveries here. One is the discovery of zero as a numeral; and the other is the discovery of zero as a number.

The numeral zero made possible the invention of positional notation. The number 12304 means 10,000 + 2000 + 300 + 4. This notational trick gave us the ability to write down and manipulate really large numbers. Try multiplying VIII by XL and you can see how positional notation is vastly superior to Roman numerals.

There's a lot of power in a good notation!

The other aspect of zero is as a number.This is a lot more subtle. It involves accepting that if I haven't got any tomatoes, that I have zero tomatoes. That a number can be used to indicate the absence, as well as the presence, of something. Or even a deficit. If I owe you two tomatoes, I have negative 2 tomatoes.

The discovery that zero is a number, and that negative numbers can be used according to rules and give answers that make sense, are both great leaps of abstraction for humanity. And abstraction is something we're really good at.

Can you relate to negative numbers in terms of quantities of things owed? And to the number zero in terms of describing the quantity of things you don't have? Someone could have two cats and zero dogs, right?

 

[chiro]

Hey physio.

You can think of zero as the introduction (or rather the realization) of a point of reference.

The number zero provides a way to relate different things in both of the polarities (i.e. positive and negative).

You can interpret these in so many ways (absence of something vs presence of something, negative units being distance to the left, positive units being distance to the right and so on), but the above idea is the same for all of them.

When you have a way to relate anything in the above way, you can then look at doing more general ways of relating things through algebra and this is what is done in areas like geometry with angles and distance.

Every application has its own interpretation and particular thing that it focuses on, but mathematics justs considers how they relate at the abstract level and doesn't provide a specific interpretation.

 

[Studiot]

The importance of zero it that it allows us to solve many equations that cannot be solved it didn't exist.

The simplest is

zero is the only solution (value of x) that satisfies the equation a + x= a, where a is any other number.

The second unique property of zero allows us to solve many more equations. Since any number multiplied by zero equals zero, but the no product of two non-zero numbers gives this result, we can say

if ab = 0 then either a or b = 0, that is both a and b cannot be nonzero.

 

[Antiphon]

If it's used for measuring then we can very well do without zero, ...

It becomes highly relevant if you count the number of times your hide has been branded with a glowing hot iron. Zero is your friend.

 

[HallsofIvy]

It becomes highly relevant if you count the number of times your hide has been branded with a glowing hot iron. Zero is your friend.

Not for everyone!

 

[phinds]

My question may come as a surprise to many people but I have not understood the importance of zero and its significance. How can nothing be something? Why do we have zero as a number? If it's used for measuring then we can very well do without zero, for example if there is a fellow counting pebbles then after say 9 pebbles he can count 11,12 etc. upto infinite. Also I don't understand the reason for negative numbers. How can something be less than nothing?

Have you ever tried to do arithmetic with Roman numerals, or any other non-positional notation? Have fun with that.

Positional notation's need for zero is just one of MANY good reasons why zero is useful.

 

[physio]

Thanks all for the answers! What I actually want to ask is that why do we require zero at all? I mean can't we have in our number system 9 distinct numbers (1 to 9) and then have a combination of these with weighted positions like 1298 where 1 has the highest weight and then 2 and so on..We can very well do without zero...

 

[Charmar]

I mean can't we have in our number system 9 distinct numbers (1 to 9) and then have a combination of these with weighted positions like 1298 where 1 has the highest weight and then 2 and so on

I may be reading what you are asking differently than how you are intending it.

We could use a base 9 number system, I don't see what, if any advantage it could have over our current use of base 10.

http://en.wikipedia.org/wiki/Radix

What I actually want to ask is that why do we require zero at all?

We can very well do without zero...

If you don't want to use 0 how do you answer what is 42 - 42 =? or e^(i*∏) +1 =?

Lets say you have 75 red stones and 1 blue stone in a bag, and first try you pick out the blue stone, what are your chances of picking out another blue stone from the bag?

How would your system enumerate these answers?

My hero, zero.
www.youtube.com/watch?v=3huvvxUHDmM

This helped me when I was younger, while also being entertaining.

 

[physio]

At last I found the answer to my question! Now its crystal clear! Zero obviously does exist deduced from the following argument...

"Say you have 6 apples. Now, I'm taking 2 of them from you. How many apples do you have left? That's right, 4.
Now suppose I'm taking 1 more. You are now left with 3.
You might notice that while I'm taking apples from you, you are left with what you call a certain "number" of apples.
Now say I'll take your last 3 apples. How many are you left with?
Hum. None? So what is none? Is it also a number?
And what if I want to take even more apples now? Why can't I? It works in the bank, doesn't it? When you owe someone apples, how many apples do you have? Still none? So when I give you an apple, you'll have 1? But you owe some apples to people, so you'll still have none... so now something's wrong, are there different kinds of "none"? Or do you accept the concept of a negative number of apples?


To ignore the existence of "0" would be to ignore the existence of negative numbers, and eventually you'll start seing things like "No apples + 1 apple= No apples".
So is 1 actually none? So are there no positive numbers either?"

Thanks to everyone who contributed...! A weight of my mind!