Base 10 Number System

Base 10 number system uses digits 0 to 9 to represent numbers and thus has a base of 10. It is commonly known as the decimal number system. The English name was derived from the Latin word ‘decimus’, which means ‘tenth.’ Mathematically, it is written as (2,145)₁₀.

This system represents numbers that we commonly use today for all our calculations and measurements. The base 10 number system helps to simplify calculations as it uses 10 unique digits to represent numbers or values.

Discovery

The discovery of this number system dates back to centuries. As early as around 300 BCE, bamboo slips discovered from China were among the first archeological evidences. Later on, in the 7th century CE, Indian mathematicians proposed the base-10 number system. It was also independently developed by Babylonians and Aztecs using multiples of 10.

Base 10 Numerals

We use the place value of each digit in a number to determine the value of a numeral in the decimal number system. Here, the first 4 successive places left of the decimal point are units, tens, hundreds, and thousands. Each place is 10 times larger than the place value to its right.

Determining the Value of a Digit

Let us consider the decimal number (2,145)₁₀. To identify whether this number is represented in the base-ten number system, we will multiply each digit based on its place value. Each digit is multiplied by 10, raised to the power of its position in the numeral, and counted from the rightmost digit. 

For example, the number 2145 is written as:

= (2 × 10³) + (1 × 10²) + (4 × 10¹) + (5 × 10⁰)

= 2000 + 100 + 40 + 5

= 2145

Suppose there are numbers further left in a base ten. In that case, the numeral is multiplied by a factor of 10, which continues forever.

Solved Examples

Write the base ten numeral for the given expanded form.
(4 × 10000) + (7 × 1000) + (2 × 100) + (9 × 10) + (1 × 1)

Solution:

The given expanded form is:
(4 × 10000) + (7 × 1000) + (2 × 100) + (9 × 10) + (1 × 1)
The above form can be written as:
= (4 × 10⁴) + (7 × 10³) + (2 × 10²) + (9 × 10¹) + (1 × 10⁰)
= 47,291
Thus, the base 10 numeral is (47,291)₁₀

Write the base ten numeral of the following:
7 hundreds + 2 tens + 1 ones +  0 tenths +  7 hundredths +  9 thousandths.

Solution:

The given expanded form is:
7 hundreds + 2 tens + 1 ones +  0 tenths +  7 hundredths +  9 thousandths.
The above form can be written as:
= (7 × 100) + (2 × 10) + (1 × 1) + (0 × 1/10) + (7 × 1/100) + (9 × 1/1000)
= (7 × 10²) + (2 × 10¹) + (1 × 10⁰) + (0 × 10^-1) + (7 × 10^-2) + (9 × 10^-3)
= 721.079
Thus, the base 10 numeral is (721.079)₁₀
base 10 number system

Verify if the given number 6721 is represented in the base-ten number system.

Solution:

The given number is 6721.
To verify whether the given number is represented in the base-ten number system, we will multiply each digit by 10, raise it to the power of its position in the numeral, and count from the rightmost digit.
6721
= (6 × 10³) + (7 × 10²) + (2 × 10¹) + (1 × 10⁰)
= 6000 + 700 + 20 + 1
= 6721Hence, the given number is represented in the base ten number system (6721)₁₀