Natural Number vs. Whole Number

Attribute	Natural Number	Whole Number
Definition	Positive integers starting from 1	Non-negative integers starting from 0
Inclusion of Zero	No	Yes
Symbol	N	W
Cardinality	Countable infinity	Countable infinity

Definition

Natural numbers are a set of positive integers starting from 1 and continuing indefinitely. They are often denoted by the symbol N. Whole numbers, on the other hand, include all the natural numbers along with zero. They are denoted by the symbol W. In essence, natural numbers are a subset of whole numbers.

Representation

Natural numbers are typically represented as N = {1, 2, 3, 4, ...}. Whole numbers, on the other hand, are represented as W = {0, 1, 2, 3, ...}. This means that while natural numbers start from 1, whole numbers start from 0 and include all the positive integers thereafter.

Properties

One key property of natural numbers is that they are closed under addition and multiplication. This means that when you add or multiply two natural numbers, the result will always be a natural number. Whole numbers also share this property, as they include all the natural numbers along with zero.

Another property of natural numbers is that they are not closed under subtraction. For example, subtracting 5 from 3 would result in a negative number, which is not a natural number. Whole numbers, however, are closed under subtraction since they include zero.

Use in Mathematics

Natural numbers are often used in counting and ordering objects. They are fundamental in arithmetic operations such as addition, subtraction, multiplication, and division. Whole numbers, on the other hand, are used in a similar manner but also include zero, making them useful in situations where zero is a relevant quantity.

Applications

Natural numbers are commonly used in everyday life for tasks such as counting money, measuring distances, and keeping track of quantities. Whole numbers, with the inclusion of zero, are used in various scientific and mathematical calculations where zero plays a significant role, such as in temperature scales and coordinate systems.

Conclusion

In conclusion, natural numbers and whole numbers are closely related sets of integers with some key differences. While natural numbers start from 1 and continue indefinitely, whole numbers include zero along with all the natural numbers. Both sets have unique properties and applications in mathematics and everyday life, making them essential concepts to understand.