Like if you square 1 or get the root of 1 it is still 1. however that doesn’t apply to 1.01 or 0.99, why is that?
That’s not different rules. If I square 1, I get 11, whereas if I square 2, I get 22. Same rules in both cases, and likewise for roots. Now, if you’re asking why 1 times any number is that number, that’s because 1 is the multiplicative identity. Basically, whenever you have any mathematical system which can be described as multiplication, there will be exactly one element called the identity, such that identityx = xidentity = x, for all x. In the integers (and in the rational numbers, and the reals, and the complex numbers), we happen to call the identity “one”. That’s how it’s defined (when it’s defined at all).
0 and 1 are the two solutions to x[sup]2[/sup] - x = 0. I don’t know if there’s anything deep to it. You may as well ask why [symbol]i[/symbol] and -[symbol]i[/symbol] are the only numbers whose additive inverses and multiplicative inverses are the same. It’s cause they’re the two solutions to 1/x = -x.
It is not special. The definition of “square root of X” is “a number which multiplied by itself yields X” and that is true for 1 as it is for .99 and for 1.01. Nothing special there. Either that or every single number is “special”. The sqrt of 4 is 2 which cannot be said of 5 so you can say 4 is “special”. All numbers are “special” then.
but ‘logically’ if you square or take the root of any number other than 1 you get a different number than the original. 1 is the only number that doesn’t apply to and i didn’t see why exactly.
There is no deep reason why. 1 is a fixed point of the function sqrt(x). -40 is a fixed point of the function 9x/5 + 32. -2 is a fixed point of the function x[sup]2[/sup] + 3x. Are -40 and -2 special? If not, why is 1 special?
Think about it concretely.
Take a handful of coins. Arrange them in a square such that each side is two coins wide/tall. You have two, squared. There is a total of four coins. Make a square that has sides three wide/tall. Three, squared. Total of nine. In the same vein, a “square” one wide, one tall will only be the one coin.
Make a square with a total of sixteen coins. It’ll come out four by four, with four being the square root of sixteen. Work backwards and you’ll see that making a “square” with a total of one coin will come out one by one.
As Chronos said, 1 is special, because it is the multiplicative identity for the real numbers. A few other special things happen for 1 because of that:
1/1 = 1
1^1 = 1
x^0 = 1 for all x except 0 (0^0 is undefined)
1 is the only positive integer with is neither prime nor composite
Similarly, 0 is sepecial because it’s the additive identity. But the existence of 0 and 1 (the two identities) is part of the rules of the real number system.
Where in the definition of squaring does it say “you get a different number”? There’s nothing logical about this statement.
If you take any number greater than 1 its sqrt is smaller than itself. If you take any number smaller than 1 its sqrt is greater than itself. 1 is the limit between these two situations and it makes sense to me.
There is a page on the WWW with a list of numbers starting with 1 and showing why they are all special but now I cannot find it. I saw it only days ago with the numbers in rainbow colors and every number had a description of the property which made it special. Can somebody find it?
Only for the past… say hundred years. Back when primes were first being talked about, 1 was considered a prime. Later on, for reasons I’ve never been clear on, it moved into the rather odd position it holds now.
It’s not just the square root function for which this happens, but also the cube root, fourth root, fifth root, sixth root, and so on. One and zero are the only non-negative real numbers which, when multiplied by themselves any number of times, always yield themselves again. Therefore, any root you take of zero or one always yields itself.
Here’s a way to look at it visually. Graph your favorite continuous function, like the square-root function for example. Then superimpose a graph of the function y = x. If your chosen function crosses the y = x line anywhere, then there exists a value of x that yields itself when plugged into your function. There is also a value y that yields itself when plugged into the inverse of your function.
Hope this helps.
There are probably several good reasons to have 1 not be a prime; one is that if 1 is prime, then you don’t get unique factorization.
If 1 is not prime, then any positive integer can be factored into primes in exactly one way: 12 = 223, 50 = 255, etc. However, if 1 is prime, then 12 = 1223 = 1122*3, etc.
The fundamental theorem of arithmetic states that every integer can be uniquely decomposed as a product of powers of primes. If 1 is prime, you either lose uniqueness, or you have a significantly more complicated statement.
If no one else covers units and zero divisors, I will, but I’m kinda pressed for time right now.
Ok. Unique factorization sounds like a very good reason to make 1 a special case in primality. It would need to be a special case in any statement of the rather obvious unique factorization property anyway.
The Unique Factorisation argument as presented above is incorrect. Since I cannot remember the precise names (one was something like Cauché), interested readers should investigate the history of Fermat’s Last Theorem where the aforementioned was working on a proof using Unique Factorisation but forgot to consider complex numbers. A German (Kummel?) disabused him.
if you square or take the root of zero, you get zero back.
This “unique factorization argument” is called the Fundamental Theorem of Algebra, and is stated essentially correctly by ultrafilter and borschevsky. Every positive integer greater than one may be expressed as the product of positive powers of distinct primes. This factorization is unique up to the arrangement of the prime factors. By “product,” of course, I include the case of a prime being the product of exactly one prime–itself.
If 1 were considered to be a prime, uniqueness of factorization would be destroyed, as has been stated before by others.
I think you’re thinking of the Fundamental Theorem of Arithmetic. The Fundamental Theorem of Algebra is something different.
Just to muddy the waters, you can take a square root of 1 and get -1.