Comment on SBA #119 maths
Sadbutdru@sopuli.xyz 2 weeks agoDivision and subtraction are syntactic sugar for multiplication and addition.
Can you tell me a bit more about how you mean this? I searched a bit but only basic primary school level resources about the relationship between addition and subtraction came up.
Do you mean like subtraction is just adding a negative number, and division is just multiplication by the inverse of a number? In that case I don’t really see how it simplifies things much because negatives and inverses still need as much definition. Or are you talking about bit-wise operations like a computer would use to do these things?
But hard to write this with the limitations of text but essentially it can be written as multiplication of fractions.
2 ÷ 2 ÷ 2 ÷ 2
2 x ½ x ½ x ½
Personally, I think the second form is easier to visualize and reason about, it can also can be simplified to use an exponent.
FishFace@piefed.social 2 weeks ago
When you set up arithmetic from absolute first principles you typically say that there is an operation called addition, an operation called multiplication, that they are related by the distributive law, and then you assert that there are identities for these operations, which do nothing: adding zero does nothing, and multiplying by one does nothing, so zero and one are the additive and multiplicative identities.
Next you add that each element has an inverse under these operations. The additive inverse of a number is its negative, so the inverse of 2 is -2. The multiplicative inverse of a number is its reciprocal. The defining feature is that the inverses cancel the operation leaving just the identity, so 2 + -2 = 0, and 3 × 1/3 = 1. The exception is that 0 does not have a multiplicative inverse.
With this, the operation of subtraction is defined in terms of addition and additive inverse, and division is defined in terms of multiplication and multiplicative inverse.
Saying that one is “syntactic sugar” for the other is standard in higher mathematics, but of course you could go through the same procedure starting with subtraction and division, and defining inverses in terms of those operations - the result is no different. The reason starting with inverses is preferred though is because there are lots of structures which have an operation and an identity for it, in which you can ask the question “are there inverses” because inverses are defined purely in terms of an operation and its identity. You can’t ask the question “can you divide in this structure” if the only way to even have a concept of division is to start with one.
This topic is the beginnings of abstract algebra which starts with group theory and builds from there.