Post 3086578 by JordiGH@mathstodon.xyz
(DIR) More posts by JordiGH@mathstodon.xyz
(DIR) Post #3086257 by JordiGH@mathstodon.xyz
2019-01-18T01:28:06Z
0 likes, 0 repeats
One thing that always seems kind of tricky to explain is, why is negative times negative positive?I know that sometimes we ask this question, and I can think of a number of answers, but most of the time we just say "because it is", and memorise the rule and use it and move on to other things.Maybe this is okay? I know people who do mathematics like to explain and understand, but what's the point, ultimately, to come up with some kind of story as to why \(-1\times-1=1\)?
(DIR) Post #3086258 by JordiGH@mathstodon.xyz
2019-01-18T01:31:02Z
0 likes, 0 repeats
By the way, the best "explanation" I can think of it, take, say, 10 positive and negative particles. This adds up to zero because all the positives pair up with the negatives and cancel.Now take away 7 negative particles.Now 7 positive particles have no pair so and 3 positive particles still are cancelled out by 3 negative particles.Thus,10 + (-10) = 00 - (-7) = +7
(DIR) Post #3086259 by clacke@libranet.de
2019-01-18T01:45:46Z
0 likes, 0 repeats
I don't see where your explanation connects with multiplication on that level. Subtracting a negative is easier to understand intuitively, but if multiplying with a negative is unintuitive, then it's not obvious that subtracting a negative is the same as adding a negative multiplied by -1.The way I learned it was just that multiplying with a negative multiplier flips the sign of the multiplicand, but that's a bit of a "just because".
(DIR) Post #3086577 by kwarrtz@mathstodon.xyz
2019-01-18T01:56:39Z
0 likes, 0 repeats
@JordiGH If you think of arithmetic as operations on the real line, where addition corresponds to translation and multiplication corresponds to scaling, then multiplication by a negative number is a scale and a flip, so two flips cancel out. This is still pretty abstract, but it is at least geometric/visual and (I think) fairly intuitive
(DIR) Post #3086578 by JordiGH@mathstodon.xyz
2019-01-18T02:00:00Z
1 likes, 0 repeats
@kwarrtz The number line explanation of negative times negative never made sense to me.But then again, that's never really stopped me.
(DIR) Post #3086605 by JordiGH@mathstodon.xyz
2019-01-18T01:59:06Z
0 likes, 0 repeats
@clacke Multiplication by a negative is repeated subtraction. If you understand why removing a negative results in positive, then repeatedly subtracting a negative should result in repeatedly adding a positive quantity.
(DIR) Post #3086606 by clacke@libranet.de
2019-01-18T02:04:14Z
0 likes, 0 repeats
@JordiGH Hmm yes, this could work. Good point.Maybe not for everyone, but nothing does, so it seems like a good one to have in the toolbelt when teaching.
(DIR) Post #3100409 by Breakfastisready@mathstodon.xyz
2019-01-18T10:00:38Z
0 likes, 0 repeats
@JordiGH Here's a more pretentious answer.Suppose \(\mathbb{Q}\) is the set of rational numbers and you want it to be a field. You know the inverses of all positive numbers. You want to find out what are the multiplicative inverses of negative numbers.We prove that for any \( 0 \neq x \in \mathbb{Q} \), the additive inverse of the multiplicative inverse of \( x \) is the multiplicative inverse of the additive inverse by \[ x(x^{-1} + ( - x^{-1})) = 1 + x(-x^{-1}) \]
(DIR) Post #3100410 by clacke@libranet.de
2019-01-18T13:10:29Z
0 likes, 0 repeats
@Breakfastisready I got the impression that @JordiGH 's audience would be closer to 12 years old than 25 years old. Probably not enough patience for a mathematical proof. 😀
(DIR) Post #3106212 by charles@bitsof.tech
2019-01-18T16:44:22.223965Z
0 likes, 0 repeats
@JordiGH because two "-" symbols can be combined to make a "+", of course