Integers are defined as whole numbers.
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They are the integers.
Yes, 513 is a whole number. Whole numbers are non-negative integers that do not have any fractional or decimal parts. They include all positive integers starting from zero. Therefore, 513 fits the criteria of being a whole number.
Natural numbers are sometimes defined to include zero, sometimes not. Equivalent terms therefore may include: positive integers, non-negative integers, whole numbers, positive whole numbers ("whole number" is ambiguous; some take it to include negative numbers, some not).
Integers on the number line are the whole numbers.
No. Every rational number is not a whole number but every whole number is a rational number. Rational numbers include integers, natural or counting numbers, repeating and terminating decimals and fractions, and whole numbers.