Wikipedia:Reference desk/Archives/Mathematics/2023 September 11

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September 11

On Negative Numbers

Why is it that a positive number added to a negative number creates a sum that travels in the positive direction but a positive number multiplied by a negative number creates a product that travels in the negative direction? Negative numbers always trip me up because they don't feel logical. The rules for calculating how negative numbers work with other numbers doesn't feel like conclusions that can be reached with common sense. Does the fact that I struggle so much with negative numbers make me incorrigibly stupid? ~~Anon 50.237.188.108 (talk) 14:33, 11 September 2023 (UTC)

Sorry, I made a typo. I said that the rules "doesn't" feel logical. I meant they "don't" feel logical. 50.237.188.108 (talk) 14:35, 11 September 2023 (UTC)

Sorry, I made a mistake. I wrote that I "said" something when in reality i just wrote it. I didn't say it aloud and even if I did it wouldn't be relevant, I also forgot to sign this post. ~~Anon 50.237.188.108 (talk) 14:41, 11 September 2023 (UTC)

You also put a comma splice in your last sentence. Tut tut.  --Lambiam

To your last question in the first part, certainly not. To the first one, let me try to make a coherent explanation (maybe a picture would help, but oh well):

• For addition, think of walking forward a certain amount, and then walking backward some other amount (positive plus negative). You might end up in front of where you started (a positive distance), or you might end up behind where you started (a negative distance), depending on how far you walked each direction.
• For multiplication, think of making groups of things. If I have a debt of $5, that's a negative amount of money. If I owe that same amount to three different people (-$5 x 3), I'm a total of $15 in debt - still a negative amount of money.

Hopefully that might be of at least a little help. LittlePuppers (talk) 14:44, 11 September 2023 (UTC)

Thanks, LittlePuppers, that makes a lot of sense. How would you make a negative subtracted by a negative fit into that money analogy? ~~Anon 50.237.188.108 (talk) 14:54, 11 September 2023 (UTC)

The best I can think of there is if you have a debt, and then some of that debt is taken away. (I don't know if that's a great analogy, because for it to end up positive the person would have to take away more than the debt that you have - maybe you paid $8 on your $5 debt or something.) I'll think and see if I can come up with something better for negatives subtracted from negatives. LittlePuppers (talk) 14:59, 11 September 2023 (UTC)

Thank you, LittlePuppers! -- 50.237.188.108 (talk) 13:46, 12 September 2023 (UTC)

If it makes you feel any better, it took centuries for mathematicians to accept negative numbers and the arguments against them were the same as the arguments given in your question. The nonnegative integers are called "natural numbers", so does that make everything else "unnatural numbers"? In the end, including negative numbers as numbers was done because 1) it's useful to do so and 2) it can be done in a way that's logically consistent. What I mean by logically consistent here is that the laws of arithmetic, such as the distributive law, still hold when you include the negative numbers in your "number system", provided that you define the operations, addition, multiplication, etc. the right way. For example why is the product of two negatives numbers a positive number? The answer is we define the product that way so the distributive and other laws still hold. In any case, I'd encourage you to read our article on negative numbers to fill in some of the details. Also, mathematics is a discipline where you're required to prove what you say is true, and a healthy skepticism is required to be a good mathematician. So if you're not convinced of something then speak up.

For future reference, you can just edit your forum posts to correct spelling, grammatical, typographical, and other minor errors. I do so myself frequently, despite proofreading multiple times before publishing. The rule of thumb I use is that if it's a minor change that doesn't affect the meaning then just edit the post; but if you're correcting the substance in any way then leave the original and make the correction a new post. --RDBury (talk) 17:35, 11 September 2023 (UTC)

You can also correct an error in your own posting by crossing it out and inserting a correction, like this: "A negative number times a positive number is positive negative." Or just cross it out and make a separate post with the correction. --142.112.221.184 (talk) 07:43, 12 September 2023 (UTC)

Thank you, RDBury! -- 50.237.188.108 (talk) 13:48, 12 September 2023 (UTC)

• This lesson from 3blue1brown has some good information on a way to conceptualize addition and multiplication using group theory which is quite good. You can either watch the video, or you can read the two sections that describe them under the headers "additive group for numbers" and "multiplicative group for numbers". It's an interesting perspective, and may help you understand mathematical operations like this from a different perspective. --Jayron32 18:16, 11 September 2023 (UTC)

  Thank you, Jayron32! -- 50.237.188.108 (talk) 13:48, 12 September 2023 (UTC)