Question

Division is weird

Answer 1

Addition and subtractions are okay
multiplication is a multiple of additions, and multiple of additions can link to exponents
but division... it feels out of place
But division function pops out of no where, it's more abstract, like a compliment of multiplication and it changes the game by too much. I was thinking about this guy's question, "use the number 8,15,23 to form some number, using the operators : addition,subtraction, multiplication and division." I was thinking, well all the operations are fine except division. It jumps easily, because once division is allowed, even if only one* number is given you are allowed the space of all real numbers!

However if division is not allowed, you quickly become constricted to smaller number domains. I dont think you can hit irrationals and fractions, you quickly become restricted to integers.

You also cant escape the factor space of the number you are given

Answer 2

You can always change an division problem into an multiplication problem
\[\dfrac{a}{b}=x\iff a=bx\]

ofcourse with some caution to \(b\)

Answer 3

that guy's problem is not well defined, there are infinitely many solutions using addition/subtraction alone

Answer 4

right, i did notice that, the numbers he had you can end up with a 1

Answer 5

is there a way to see if a 1 is not posible with some given set of numbers, under addition and subtraction

Answer 6

like u can clearly see if its, 2 evens nothing can be done

Answer 7

\[8x+15y = c\]

for a given integer \(c\), this equation always has infinitely many solutions

Answer 8

infinitely many integer solutions right

Answer 9

yes

Answer 10

is it because the GCD between them is 1?

Answer 11

Exactly!
\[ax+by=c\]
has infinitely many integer solutions iff \(\gcd(a,b)\mid c\)

Answer 12

it kinda feels like vector spaces lol..

Answer 13

like if 2 numbers have different primes.. they form a number basis under linear combination

Answer 14

\[32x+24y=26\]

has no solution because :

\[32x+24y= 8(4x+3y) = 26\]

that says 26 is a multiple of 8, which is false. so no solutions.

Answer 15

yes we can think of two coprime integers as a basis i guess...

Answer 16

that gcd thing is good for testing existence of solutions
below link has a simple way to find all integer solutions of a given equation
http://math.stackexchange.com/questions/897356/how-to-find-natural-solutions-of-an-equation/897369#897369

Answer 17

heeey u got a gold medal lol :)

Answer 18

lol there is a minor mistake in that reply but nobody cared to point it out

Answer 19

ya u i see lol u ssaid (-t,t) at first

Answer 20

but u fixed it anyway :P

Answer 21

wow you're alert! i saw that silly mistake today oly when i pulled it up to review

Answer 22

hehe i try to write out each word of the solution in my head so i can keep along

Answer 23

but yeah it wont affect the final answer as (-t, t) is as much a null solution as (t, -t)

Answer 24

yeah since t can be negative

Answer 25

ill go see how to work out the diophantine from this knowledge