Multiplicative and additive identities as successors

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In summary, the conversation discusses the properties of the ring of integers Z, including its total ordering and discreteness, as well as the significance of the multiplicative identity 1 being the successor of the additive identity 0 in Z. It is noted that this has significance in terms of being the smallest countable unit, and it is suggested that if a ring has the same properties as Z, it must also be Z.
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Fact: The ring of integers Z is totally ordered: for any distinct elements a and b in Z, either a>b or a<b.

Fact: The ring of integers is discrete, in the sense that for any element a in Z, there exists an element b in Z such that there is no element c in Z with a<c<b, and the same argument holds with the greater than signs flipped. In other words, successors and predecessors exist in Z (but not in R, for example).

Observation: In Z, the multiplicative identity 1 is the successor of the additive identity 0.

Questions: Is this fact a coincidence? Does this have any significance? Must the multiplicative identity always succeed the additive identity in rings that have the same properties as Z, assuming there are any other?
 
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It has a significance. One is the smallest countable unit and as such the multiplicative unit as well as the smallest entity to count, hence the successor of zero.

If a ring has the same properties as ##\mathbb{Z}## then it is ##\mathbb{Z}##.
 

FAQ: Multiplicative and additive identities as successors

What is the difference between multiplicative and additive identities?

Multiplicative identity refers to the number 1, which when multiplied by any number, results in that number. Additive identity refers to the number 0, which when added to any number, results in that number. Essentially, the multiplicative identity is used for multiplication and the additive identity is used for addition.

Can you give an example of a multiplicative identity?

One example of a multiplicative identity is 1. For any number n, 1 multiplied by n is equal to n.

How are multiplicative and additive identities related to successors?

In mathematics, successors refer to the number that comes after a given number. In the case of multiplicative and additive identities, the successor of 1 is 2, and the successor of 0 is 1. This is because when we add 1 to 0, we get the multiplicative identity 1, and when we multiply 1 by 2, we get the additive identity 0.

Why are multiplicative and additive identities important in mathematics?

Multiplicative and additive identities are important because they serve as the starting point for multiplication and addition. They are also used in various mathematical operations and proofs.

Can you explain how multiplicative and additive identities are used in real-life situations?

In real-life situations, the multiplicative and additive identities are used in various ways, such as calculating discounts or interest rates. For example, if you have a 20% discount on an item, you are essentially multiplying the original price by 0.8, which is the additive identity of 1. Similarly, when calculating interest on a loan, the additive identity is used to represent the initial amount borrowed, and the multiplicative identity is used to calculate the interest rate on top of the initial amount.

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