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Q: What are the definitions of integers?

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It follows from the definitions of the two operations.

There are two conflicting definitions of a "natural number": these are "The set of positive integers", or "The set of non-negative integers".According to the first definition, the list of positive integers does not include 0. However, according to the second definition, this does include zero.

Actually it is. Well, it depends what definition of "whole number" you use. Some definitions include only positive integers, some use it as a synonym of "integers". Therefore, due to this ambiguity, the phrase "whole numbers" would better be avoided in math. Rather, use words or phrases like "integers", "positive integers", or "non-negative integers", to convey the exact meaning, without ambiguity.

real number that is not rational: any real number that cannot be expressed as the exact ratio of two integers, e.x. √2 and π

We looked on-line for definitions of "integer", and every one we found defined "integer" as meaning "whole number". So, yes.

It depends. Some authors consider "Whole Numbers" to be the positive integers, some consider them to be the non-negative integers, and some consider them to be all integers. For the first two definitions, numbers like -3 would not be considered "whole numbers". With the last definition, negative numbers like -3 would be considered a "whole number".

There is some disagreement. You can make a rectangle that is 4 x 8, but some definitions specify consecutive integers, which means 32 wouldn't qualify.

There is some disagreement. You can make a rectangle that is 6 x 8, but some definitions specify consecutive integers, which means 48 wouldn't qualify.

I do definitions.

definitions

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Negative integers, zero and the positive integers, together form the set of integers.

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