How Do We Explain the Multiplication of Two Negative Numbers?

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In summary: In general, a polynomial is irreducible if it can't be factored into two polynomials of lower degree). Gradually, the elements of the quotient ring are represented as polynomials of lower degree than X^2 + 1, so, the quotient ring can be identified with the set of polynomials of degree 1.That's one of the constructions that you asked for, but i'm not sure if this is what you wanted.In summary, complex numbers are defined as numbers with both real and imaginary parts, and their existence can be proved through various mathematical constructions. The fundamental theorem of algebra guarantees the existence of complex numbers, and their usefulness in various applications has solidified their accepted definition in the mathematical world.
  • #1
jobyts
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-ve number_of_times ?

Let's say I'm trying to teach a kindergartener about addition/subtraction/multiplication of positive/negative numbers.

Let' say she has 5 candies. I can ask her - how many candies you 'have'? The answer is +5.
Or, if she has no candies and 'needs' 5 candies, that could be represented as -5.

If one kid needs 5 candy and another kid needs 3 candy, totally they need 8 candy,
hence (-5) + (-3) = (-8) => -ve number + -ve number = -ve number

If one kid needs 5 candy and there are 3 such kids, in total, they need 15 candies, hence
(-5) * (+3) = (-15) => -ve number * positive number of times = -ve number.

The question I have is how do I say the candy example for multiplication of 2 -ve numbers.

How's multiplication of 2 negative numbers, makes into a positive number?

Shouldn't '-ve number_of_times' be considered as an 'undefined' operation, like divide_by_zero? Did we just define a meaningless thing, and to make things work, we further kept on defining more fancy things like complex numbers.., real and imaginary part...
 
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  • #2


if you insist on that example, if each kid need 5 candy and we have 3 such confused kids, then they actually have 15 candies. Here a "confused kid" is a kid who thinks he need but actually have a given number of candies, and vice verse!
 
  • #3


Shouldn't '-ve number_of_times' be considered as an 'undefined' operation, like divide_by_zero?

a negative number times a negative number is positive because: given a, -a is defined to be the unique number such that a + (-a) = 0

Given a positive number a, -a is a number. Hence there exists -(-a) such that -(-a) + (-a)=0

But a+(-a) = 0. As -(-a) is the unique number with this property, we deduce -(-a) = a

And this is why two negatives gives a positive.

The problem with all your examples that you're trying is that the number of kids is a natural number, not an integer, so you're struggling to find an example of a negative number of kids. In fact, there are very few basic examples where you multiply two things together and neither of them is naturally 'counted'

EDIT: Here are a couple of good ones:
http://mathforum.org/dr.math/faq/faq.negxneg.html
 
  • #4


Wow! That has to be the worth method ever. Damn.

First, get the child curious. That gives motivation.

Second, get a new lesson.
 
  • #6


Thanks for all the answers.

Another question: Is there a proof that a number should have an imaginary part, or it is a thin airconcept?
 
  • #7


Every COMPLEX number has both a real and imaginary part. There is no proof, that is the definition of a complex number.
 
  • #8


matticus said:
Every COMPLEX number has both a real and imaginary part. There is no proof, that is the definition of a complex number.

Is there a proof for complex number to exist? Can I just make a statement the imaginary part of a complex number can have a real part and imaginary' part and goes on infinitely recursively. What makes the maths world to accept the definition of complex numbers to be the current one, and reject all others? Is it the applications of it makes a definition standard?
 
  • #9


I'm not sure what "proof of existence" do you seek here, would you agree that the real numbers exist?
I'll restrict myself to the mathematics though, leaving existence and usefulness in waves, quantum mechanics, ... to physicists!

Given any field R, there exist a field C which contains R as a sub-field, this "larger" field is unique (up to k-isomorphism), and every non-constant polynomial over C (and consequently over R) has a root that belong to C.
Or, in other words, there's a unique(up to k-isomorphism) algebraically closed field extension for every field.

So that when we consider the real numbers, we know for sure (without construction) that a larger field must exists which is algebraically closed and is unique. (But thanks to the fundamental theorem of algebra, we know that the structure of the complex numbers meets these criteria)

Moreover, we are not restricted to single construction, as long as we have a field which has the reals R in it (as a subfield), and contains all roots of R.

Here's another construction of C which doesn't require the usual "adjoining i". Consider the quotient of the set of all polynomials in X over the real numbers by the ideal of the irreducible polynomial [itex](X^2 + 1)[/itex].
 
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Related to How Do We Explain the Multiplication of Two Negative Numbers?

1. What is a negative number?

A negative number is a number that is less than zero. It is represented by a minus sign (-) before the number. For example, -5 is a negative number.

2. How do you perform operations with negative numbers?

To add or subtract negative numbers, you can simply follow the same rules as with positive numbers. When multiplying or dividing negative numbers, there are specific rules to follow depending on the number of negative factors. For example, when multiplying two negative numbers, the product will be positive.

3. Can you have a negative number of times?

Yes, you can have a negative number of times. This is often used in mathematical equations, such as when finding the slope of a line or calculating velocity. A negative number of times indicates a decrease or opposite direction from the original value.

4. What is the difference between a negative number and a positive number?

The main difference between a negative number and a positive number is their position on the number line. Positive numbers are greater than zero and increase in value as you move to the right, while negative numbers are less than zero and decrease in value as you move to the left.

5. Can negative numbers be used in real-life situations?

Yes, negative numbers are used in many real-life situations, such as in finances (e.g. debt), weather (e.g. temperature), and sports (e.g. losing points). They are also used in science and engineering to represent values such as direction, force, and temperature.

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