|
|
|
|
|
|
by throwawaymath
2882 days ago
|
|
|
> Like, I didn't understand how division works -- if someone were to ask me what (3/4) / (5/6) even means conceptually I would not have been able to provide a coherent, accurate explanation. "Uh... it's like taking 5/6 of 3/4... wait no that's multiplication... you need to flip the second fraction over... for some reason..." In case you (or others reading this) still struggle to formalize division, a very nice way to conceptualize it is as the inverse of multiplication. This neatly sidesteps the problem of trying to figure out a clean analogue for what it means to to multiply a fraction of something by another fraction of something, since the intuitive group-adding idea of multiplication sort of breaks down with ratios. Addition is a straightforward operation, but subtraction is trickier. For all real x there exists an additive inverse -x satisfying x + (-x) = 0. So to subtract 3 from 4 we instead take the sum 4 + (-3) = 1. Likewise to multiply 3 by 4 we add four groups of 3: 3 + 3 + 3 + 3 = 12. We accomplish division by using a multiplicative inverse: for all real x there exists a 1/x such that x(1/x) = 1. So (3/4) / (5/6) is equal to (3 * 1/4) / (5 * 1/6). In other words, take the multiplicative inverse of 4 and 6 and multiply them by 3 and 5 respectively. Then multiply the first product by the inverse of the second product. This is the axiomatic basis of division as "repeated subtraction": subtraction is the sum of a number and another number's additive inverse, and multiplication is repeated addition. Then division is the product of a number and another number's multiplicative inverse. From this perspective you need not even understand division computationally if all you'll ever deal with are fractions and not decimals. |
|
|