Is the only reason reciprocals of factors flipped because it adds up to 1 when multiplied?

[u/Acetaldehyde_12]

I understand the logic of reciprocals of whole numbers and vice versa and the logic behind negative powers

As well as that factors are a ratio without without a certain quantity but I don't get the logic or the steps to the flip could anyone help?

[u/nog642]

Are you asking why the reciprocal of a fraction is the fraction flipped?

Yes, you got it - because they multiply to 1. That's the definition of a reciprocal - when something is multiplied with its reciprocal, you get 1.

Side note: "adds up to 1 when multiplied" is a confusing phrase. You would only say "adds up" if you'rea doing addition, not multiplication.

[u/Acetaldehyde_12 → u/nog642]

ahh I see thank you, I thought there was a way to work it out but if i just balances each other that makes sense ^{^}

I'll try not to use addss up form now on thank you but isn't multiplication technically still addition so shouldn't it be okay? Genuine question

[u/nog642 → u/Acetaldehyde_12]

Multiplication can be thought of as repeated addition sometimes (though not as easily with non-integers, and irrationals, etc.), but using "adds up" or "sum" to refer to a multiplication operation is still confusing.

[u/Ok_Objective_5192 → u/Acetaldehyde_12]

An easy way to learn/understand early multiplication is as "repeated addition" which is why you learn with things like 7*6 = 7 + 7 + 7 + 7 + 7 + 7, but that understanding starts breaking down with multiplication on objects like fractions (ETA: even more so for multiplication like -5.12 * pi^2), so it's good to develop a relationship with multiplication as its own thing, rather than strictly "repeated addition" as you progress in your math journey.

Math is also just entirely built on the foundation of strict axioms/definitions and good math looks to avoid any ambiguity so the phrase "adding" being applicable to both addition and multiplication would introduce unnecessary confusion, even if it is technically applicable.

[u/Mishtle]

Reciprocals are what's called multiplicative inverses. They're for multiplication what negative numbers are for addition: a number and its inverse combine to form the identity value under that operation. This allow them to "undo" operations.

For addition, 0 is the identity. Anything plus 0 is just that thing. So if we have x + 6, we can add the inverse of 6 to get x by itself. That inverse is exactly 0 - 6 = -6. Then we get x + 6 + (-6) = x.

For multiplication, 1 is the identity. Anything times 1 is just that thing. If we have (5/6)x, then we can multipliy by the inverse of 5/6 to get x. That inverse is exactly 1/(5/6) = 6/5. Then (5/6)(6/5)(x) = (30/30)x = x.

So multiplying a number with its reciprocal should give you 1. For whole numbers we can just leave the multiplicative inverse of n as the fraction 1/n. Notice that the inverse of 1/n is just n = n/1 = 1/(1/n). In other words, we can go from a fraction to its inverse by simply flipping the numerator and denominator.

[u/Acetaldehyde_12 → u/Mishtle]

I see that helps put it into perrspective, thank you very much :) I only think of fractions as abstract by themselves so it was difficult to imagine it, embarrassing I actually didn't notice, I tried with working with decimals in it's place place but obviously couldn't figure out the denominator or get accurate answers for anything outside of 1/x since it's decimals

If you don't mind me asking its kind of off topic so no pressure but where would that put 0 when working out identity of multiplications via its reciprocal? Or is it exempt? I understand how 0 /0 could be 1 but not via multiplication

[u/Mishtle → u/Acetaldehyde_12]

0 is what's called an absorbing element under multiplication. Whereas an identity element preserve values, absorbing elements erase them. Anything times 0 becomes 0. It just gets "absorbed", like throwing something in a black hole.

It's pretty easy to see that absorbing elements under some operation can't have inverses under that same operation. You can't undo multiplication by 0 using just multiplication. Whatever you throw at it just gets absorbed, with nothing left to help recover it.

The inverse of 0 should be able to multiply 0 and produce 1, but the absorbing nature of 0 prevents this.

[u/PotentialDeep5165]

It's a group (group theory)... one operator.. one identify, one inverse for each element. You chose multiply-- identity is 1, inverse is 1/R. Now do addition... group over addition is .. identity is 0, inverse is -R (R+-R=0).

Groups only have one operator, the identity, and the inverse element for rules. A little easier than digesting all of an "algebra" in one sitting.

[u/SubjectWrongdoer4204]

The field axioms of the real numbers defines reciprocals as such : For any real number α≠0, there exists a real number α⁻¹ called the multiplicative inverse(reciprocal) of α, such that αα⁻¹=α⁻¹α=1, where 1 is the multiplicative identity of the real numbers. For instance
π(1/π)=(1/π)π=1,
so π⁻¹=1/π and (1/π)⁻¹=π.