I Is the equation 0!=1 based on flawed reasoning?

[Pete Mcg]

Yes .

How does 10% of almost nothing sound?

 

[Pete Mcg]

1 = 1 is not a definition. In general for all kinds of definitions, including words found in a dictionary, you can't define something in terms of itself.
The equation 1 = 1 is an example of the reflexive property of the equality relation. This property says that any number is equal to itself. Other relations, such as < or >, do not have this property. For example, $5 \nless 5$ and $2 \ngtr 2$.

Thank you for that. As stated previously in my original post, I'm not a Mathematician and am learning, every little bit counts and I appreciate your and everybody's input . (Learned today that you when you multiply a negative number by a negative number the result is a positive number - they didn't teach this in Maths @ school I went to so it took a bit of getting my head round it - at first I went HUH?? but it's starting to makes sense.) This a new world to me, good fun -talk about fun - this Godel incompleteness thing has got me intrigued!
As in learning any new 'language' one needs to understand the rules and conventions (another thing learned today was BOMDAS / PEMDAS)Even the supposedly simplest thing like 'number' has a myriad of different 'meanings' - natural numbers, rational numbers, irrational etc etc etc ...Anyway, enough of my rambling. Cheers.

 

[fresh_42]

they didn't teach this in Maths @ school I went to so it took a bit of getting my head round it - at first I went HUH??

A product is an area. An area is oriented. It makes a difference whether you circle it clockwise or counter-clockwise, one is noted as a positive number, the other one by a negative number. Which is which is up to you. The lines are oriented, too, up and down, left and right. Now $(+1)\cdot (+1)$ has the same orientation as $(-1)\cdot (-1),$ and $(+1)\cdot (-1)$ is of opposite orientation.

 

[jbriggs444]

Another motivation is the distributive law: $a \times (b+c) = a \times b + a \times c$

Say you have $-2 \times (1-1)$ then clearly that should evaluate as $-2 \times 0 = 0$. If we apply the distributive law, it should be equal to $( -2 \times 1 ) + ( -2 \times -1 )$. But if a negative times a negative is a negative, that formula evaluates as $-2 + -2$ for a result of $-4$ which is wrong.

So if a negative times a negative yields a negative, the distributive law breaks.

 

[Mark44]

(Learned today that you when you multiply a negative number by a negative number the result is a positive number - they didn't teach this in Maths @ school I went to so it took a bit of getting my head round it

Sorry to hear that it wasn't taught at your school. My first exposure to signed-number arithmetic was in ninth grade, back when I was 14.

 

[pbuk]

Another motivation is the distributive law: $a \times (b+c) = a \times b + a \times c$

So if a negative times a negative yields a negative, the distributive law breaks.

If a negative times a negative yields a negative there are no multiplicative inverses for negative numbers so we haven't even got a group.