Today, we are going to take a closer look at the elements that make up the set of **natural numbers**. If you remember from a previous blog, this set represents all positive integers within the set of real numbers.

All real numbers (which include natural numbers) are made up of one or more **digits**. The digits we use in our modern numbering system are *0,1,2,3,4,5,6,7,8,9*. Note that there are ten numbers that are made up of one digit (0-9). There are ninety two-digit numbers (10-99), and there are one thousand three-digit numbers (100-999). All real numbers can be expressed in this decimal notation, even fractions. This will be looked into in a future blog.

The **position** of the digits within a numeral is as important as the digits themselves. The numeral “476” is not the same as the numeral “674” even though both numerals use the same three digits: 4, 7, and 6. In order to understand why this is the case, we need to understand the **decimal number system**. I’ll show the structure of this system by illustrating the number “six hundred and seventy-four” in three different ways.

In this system, each position represents a **multiplier** used to determine the magnitude of the digit in its slot (**A**). The rightmost slot in any number has a multiplier of 1. The slot to its left represents a multiplier of 10. The third slot from the right represents a multiplier of 100. Theoretically, this can continue indefinitely. Very large (or very small) multipliers can also be expressed using scientific notation.Each slot represents a different multiplier that is ten times more than the slot to the right. Since all the multipliers in this system are multiples of ten, the decimal notation system is also referred to as a **base-10 system**. The **arithmetic expression** (B) is a way of expressing the relationship between the multipliers and its digits. The **number** (**C**) is a shorthand way of expressing this arithmetic expression. There are other numerical systems that use a multiplier other than ten. We’ll explore these systems and their practical applications in a future blog.

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