Existence of negative numbers

In summary, the existence of counting/positive numbers implies the existence of negative numbers through a bijection that maps the lowest counting number to the highest and vice versa. This mirroring or reversal of order is possible for any set with a strict order and the negative numbers can be seen as a dual space of the positive numbers. This bijection is a permutation and can be applied to integers and sets more generally. However, the usual construction of the integers is through the Grothendieck construction, which involves defining addition and multiplication. Permutations also have a broader use in group theory.
  • #1
JanEnClaesen
59
4
If counting/positive numbers exist, do they imply the existence of negative numbers?

I'd say yes, because there's always a bijection that maps the lowest counting number of the set to the highest, then the second lowest to the second highest, etc. This reversal of order/mirroring is possible for any set with a strict order. The negative numbers are then some sort of dual space of the positive numbers. This bijection is a mirror permutation, can the idea of permutation groups be applied more generally to integers/sets?

I apologize for the borderline vague statements.
 
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  • #2
JanEnClaesen said:
If counting/positive numbers exist, do they imply the existence of negative numbers?

What do you mean with "exists"? Do you somehow want to construct the negative numbers from the positive counting numbers?
 
  • #3
They certainly exist in any mathematical sense and, yes, they can be constructed from the positive counting numbers- so the negative integers exist in then same sense that the positive integers do.
 
  • #4
"exists" as in constructing: mapping the highest to the lowest, second highest to the second lowest, etc.
This construction is a permutation, do permutations have a more general use for ordered sets?
 
  • #5
JanEnClaesen said:
"exists" as in constructing: mapping the highest to the lowest, second highest to the second lowest, etc.
This construction is a permutation, do permutations have a more general use for ordered sets?

That's not the usual construction of the integers. The usual construction is called the Grothendieck construction. It's not that your proposal of constructing the negatives is wrong, but there are annoying details such as defining addition and multiplication and checking the properties.

And yes, permutations are useful outside constructing the integers too. For example, in group theory.
 

1. What are negative numbers?

Negative numbers are numbers that are less than zero. They are represented by a minus sign (-) in front of the number, or by being placed to the left of the number on a number line.

2. How can negative numbers exist if you can't physically have -2 of something?

Negative numbers were originally created to help solve mathematical equations and problems that involved calculating with numbers that were less than zero. They are a concept and do not always represent physical objects. For example, if you owe someone $2, you could represent that as -2 on a number line.

3. Can you add or subtract negative numbers?

Yes, you can add and subtract negative numbers just like any other numbers. When adding a negative number, it is equivalent to subtracting a positive number. For example, -3 + (-2) is the same as -3 - 2, which equals -5. The same concept applies for subtraction.

4. Why do we use negative numbers in math?

Negative numbers are used in math to represent values that are less than zero. They are essential in solving equations and problems that involve values that are below zero. They also help us understand and represent real-world situations, such as debt or temperature below zero.

5. Can you multiply or divide negative numbers?

Yes, you can multiply and divide negative numbers just like any other numbers. However, there are some rules to keep in mind. When multiplying or dividing two negative numbers, the result will always be a positive number. When multiplying or dividing a positive and a negative number, the result will always be a negative number.

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