Short answers, pulled from the story.
An integer is the number zero, a positive natural number such as 1, 2, or 3, or the negation of a positive natural number such as -1, -2, or -3. An integer is a real number that can be written without a fractional component, so 21, 4, 0, and -2048 are integers while 9.75, 5/4, and the square root of 2 are not.
The word integer comes from the Latin integer, meaning whole or literally untouched, from in meaning not plus tangere meaning to touch. The word entire derives from the same origin through the French word entier, which means both entire and integer.
The letter Z comes from the German word Zahlen, meaning numbers, and its choice has been attributed to David Hilbert. Its earliest known textbook use appears in Algèbre by the collective Nicolas Bourbaki, dated 1947, and by 1961 modern algebra texts generally used Z for the positive and negative integers.
Leonhard Euler defined integers to include both positive and negative numbers in his 1765 Elements of Algebra. Before that, the term applied only to positive integers and was synonymous with the natural numbers.
Whole numbers were synonymous with the integers until the early 1950s. In the late 1950s, as part of the New Math movement, American elementary school teachers began teaching that whole numbers meant the natural numbers and excluded negatives, while integer included the negative numbers, and the term whole numbers remains ambiguous today.
The integers are not a field because they lack multiplicative inverses, which means they are not closed under division. They form an integral domain, and the smallest field that contains them as a subring is the field of rational numbers.
Yes, the set of integers is countably infinite, meaning each integer can be paired with a unique natural number through a bijection. Its cardinality equals aleph-null, the same as the natural numbers.