# Place Value

The convention in mathematics is to work in a base-ten system. This system has unique symbols (numerals) to represent the numbers 0-9. There is no unique symbol to represent 10, or any larger number. Nor is there a unique symbol to represent any of the infinite numbers between 0 and 1, for example. The *decimal number system* uses the numerals 0-9, arranging them in strings where the position of the numeral within the string determines its value.

#### Contents

## Decimal Number System

In the decimal number system, the value of a digit depends on its position in the number. Numbers get larger moving from right to left.

In the number 863, the **place value** of 3 is *ones*, the place value of 6 is *tens*, and the place value of 8 is *hundreds*.

In the number 863.951, the **place value** of 9 is *tenths* \(\left(\dfrac{1}{10}\right) \), the place value of 5 is *hundredths* \(\left(\dfrac{1}{100}\right) \), and the place value of 1 is *thousandths* \(\left(\dfrac{1}{1000}\right)\). These continue on in a similar manner.

A common mistake is to say that the place value of 9 is *oneth*. This term does not exist.

In the number 12.345, what is the place value of the digit 4?

The digit 4 is in the hundredths place. \(_\square\)

Place value is not to be confused with *face value*. The face value of a number is the value of the numeral itself.

In the number 12.345, what is the face value of the digit in the thousandths place?

The digit 5 is in the thousandths place, and the face value of 5 is 5. \(_\square\)

## Uses

The decimal number system conveys information that can be discerned with a quick glance. A measurement of 3.001 mg is much more precise than a measurement of 3 mg. Similarly, a person receiving a bank statement that says $XXX,XXX,XXX.XX may not be worried about the number in the tens place, while a person whose account reads $XX.XX is certainly going to be interested in the digit in the tens place.

Knowing the names of the place values is not required for solving an arthimetic problem, but it can be useful for teaching and discussion. You don't need to understand the parts of speech to write a good sentence, but if you are editing a friend's essay, knowing terms like *noun* and *verb* can help you give specific, useful feedback. Similarly, if your younger brother is learning to subtract and you're checking his homework, it's useful to be able to say "look and the hundredths place" to help him find an error.