What is the Base-10 Number System?

a hand working a colorfiul abacus
A base-10 abacus can be a great math teaching tool.

 Getty Images / ManoAfrica

If you've ever counted from 0 to 9, then you've used base-10 without even knowing what it is. Simply put, base-10 is the way we assign place value to numerals. It is sometimes called the decimal system because a digit's value in a number is determined by where it lies in relation to the decimal point. 

The Powers of 10

In base-10, each digit in a position of a number can have an integer value ranging from 0 to 9 (10 possibilities). The places or positions of the numbers are based on powers of 10. Each number is 10 times the value to the right of it, hence the term base-10. Exceeding the number 9 in a position starts counting in the next highest position.

Numbers greater than 1 appear to the left of a decimal point and have the following place values

  • Ones
  • Tens
  • Hundreds
  • Thousands
  • Ten thousands
  • Hundred thousands, and so on

Values that are a fraction of or less than 1 in value appear to the right of the decimal point:

  • Tenths
  • Hundredths
  • Thousandths
  • Ten thousandths
  • Hundred thousandths, and so on

Every real number may be expressed in base-10. Every rational number that has a denominator with only 2 and/or 5 as the prime factors may be written as a decimal fraction. Such a fraction has a finite decimal expansion. Irrational numbers may be expressed as unique decimal numbers in which the sequence neither recurs nor ends, such as pi. Leading zeros do not affect a number, although trailing zeros may be significant in measurements.

Using Base-10

Let's look at an example of a large number and use base-10 to determine each digit's place value. For instance, using the whole number 987,654.125, the position of each digit is as follows:

  • 9 has a place value of 900,000
  • 8 has a value of 80,000
  • 7 has a value of 7,000
  • 6 has a value of 600
  • 5 has a value of 50
  • 4 has a value of 4
  • 1 has a value of 1/10th
  • 2 has a value of 2/100th
  • 5 has a value of 5/1000th

Origin of Base-10

Base-10 is used in most modern civilizations and was the most common system for ancient civilizations, most likely because humans have 10 fingers. Egyptian hieroglyphs dating back to 3000 B.C. show evidence of a decimal system. This system was handed over to Greece, although the Greeks and Romans commonly used base-5 as well. Decimal fractions first came into use in China in the 1st century B.C.

Some other civilizations used different number bases. For example, the Mayans used base-20, possibly from counting both fingers and toes. The Yuki language of California uses base-8 (octal), by counting the spaces between fingers rather than the digits.

Other Numeral Systems

Basic computing is based on a binary or base-2 number system where there are only two digits: 0 and 1. Programmers and mathematicians also use the base-16 or hexadecimal system, which as you can probably guess has 16 distinct numeral symbols. Computers also use base-10 to perform arithmetic. This is important because it allows exact computation, which is not possible using binary fractional representations.