is negative numbers exist?

Question

-5=5
-5²=5²
25=25
√25=√25
5=5
if they not exist why we using them?
if they exist how can it possible?
anyone can solve this dilemma?

Answer 1

You're proof is perfectly valid, but is not sound. That is, from your assumption that -5 = 5, you have correctly proven that 5 = 5, but your assumption that -5 = 5 is false, so the proof is erroneous. If 5 = 5 is indeed true (which it is), it is not due to your argument.



You could try running the proof backwards:



5 = 5

√25 = √25

25 = 25

(-5)² = 5²

-5 = 5



where you assume that 5 = 5 (which is true), and attempt to prove that -5 = 5, except now the final step is not logically valid. The problem is that squaring is not one-to-one, and cannot be undone in this fashion. Yes, some operations cannot be undone in this fashion. This is not just an issue with negative numbers. For example, check out this argument:



0 = 0

0 * 1 = 0 * 2

1 = 2



The problem is the step of cancelling 0 from both sides. Multiplying by 0 is not one-to-one, so it also cannot be undone.



So, yeah, don't be alarmed, the integers are not broken. Not every operation is as nice as addition, multiplication, taking negatives, etc. Sometimes you have to watch the flow of logic, lest you make some bad arguments.



In terms of the question, do negative numbers exist? The answer I would have to give is no. Positive numbers exist, and arguably, so does 0. Positive numbers, according to Bertrand Russell, and I tend to agree with him, are properties of collections of objects. If I pick a dozen roses, that bunch shares a property with a hand of 12 uno cards, and the collection of distinct hours that have elapsed since I woke up this morning, specifically the property of "twelveness". The digits "12" are simply symbols we invented to denote these properties.



So, what about negative numbers? Well, they're certainly not properties of collections, like positive numbers. There's no such thing as -1 roses, or uno cards, or hours. Negative numbers exist more as an extension of the positive numbers (as do rational, real, and complex numbers). Negative numbers, like the other extensions, have nice properties. It turns out that the integers, positive, negative, and otherwise, form a nice algebraic structure known as an integral domain (in fact, even better, a unique factorisation domain). These structures are concepts that you don't study until 2nd-3rd year university of a maths degree, so I won't go into detail. Essentially, they have all the nice properties that we expect from the integers, such as the ability to add and multiply numbers in any order, the ability to undo addition and multiplication.



For example, without the negative numbers, we would be unable to calculate 1 - 2, so we would have to be careful with what numbers we subtract. It would also mean, by extension, that 1 - 2 + 2 would also be meaningless. We get such nice results, plus the ability to model certain situations like credit/debt, that we decided to keep the negative numbers, despite their not having any solid meaning in the real world.



Anyway, I hope that helps.

Answer 2

Negative numbers exist in the same sense that numbers exists, meaning that no, they don't. It's all theory.



By suggesting that -5=5, he is suggesting that it might be possible, and his other steps follow leading for him to conclude that -5 and 5 are the same. But in "reality" there are four different outcomes;

5=5

5=-5

-5=5

-5=-5

This is because the square root of 25 is both 5 and -5.



Weird eh?

Answer 3

In reality, they of course don't. You will never have an apple that costs -$3. But in Mathematics, the negative numbers is one of the composition of the number line or a x-y coordinate graph wherein the basis of the solutions can be evidently acquired. Also, you can do your algebra using the negative numbers; you can compute for your sister's age, or even know your father's company sales. In Physics, the negative sign represents the direction of the magnitude. If I say, the velocity of the car is -3m/s, meaning, the car is moving backwards.



You know, you don't underestimate these negative numbers, they are one of the foundations of the improvement of science and mathematics today.



And of course, in real life, a person always have a positive and negative aspect in him/her. Same applies with numbers. :)

Answer 4

It is true that negative numbers physically do not exists.Negative and positive are interpreted as directions.So on a Number line there is a Zero.All numbers right to it are positive and to the left are negative.The concept helps us to understand the properties of the solutions of say polynomials.So when we take a sqrt of a positive number we get two values eg sqrt()=+3 and -3.In money transactions +ve means profit negative means loss.Filling means positive rate and emptying means negative rate.In electricity we have +ve potential and -ve potential.There is a positive charge and negative charge which really exists.Convex Lens is negative curvature and concave lens is positive curvature etc.In set theory -ve is the additive inverse of a number.and so on.....

Ivan

Answer 5

Other than your first step ... - 5 = 5 <=== wrong



The last step is where your problem is.



Square root of 25 is

plus OR minus 5

Answer 6

They're imaginary. You can only use them if you believe in them. If you can't see them when you imagine them, then you'll fail Maths.