Decimal Number System (Base 10)

A decimal number system is the number system that we use daily, based on the 10 digits. In mathematics, a number system is considered the notation of numbers by using digits or symbols. The number system consists of four main types, namely the binary number system, decimal number system, octal number system, and hexadecimal number system. The decimal number system is also known as the Hindu-Arabic or Arabic number system, since in ancient civilizations, it was difficult to multiply and divide large numbers by using hands. In this article, let us learn more about the decimal number system and the conversion from the decimal number system to other systems in detail.

Outlines

Decimal Number System

• What is meant by decimals?
• What is the Decimal Number System?
• Place Value Chart of the Decimal Number System
• What is the origin of the base-10 number system?
• Why Decimal System Called Base 10

Expanded Form of the Decimal Number System

Conversion of the Decimal Number System

• Decimal to Binary Conversion (Base 10 to Base 2)
• Decimal to Octal Conversion (Base 10 to Base 8)
• Decimal to Hexadecimal Conversion (Base 10 to Base 16)
• Practice questions

Conversion from other bases to the decimal number system

• Binary to Decimal Conversion (Base 2 to Base 10)
• Octal to Decimal Conversion (Base 8 to Base 10)
• Hexadecimal to Decimal Conversion (Base 16 to Base 10)
• Key Points to Remember
• Practice Questions

Applications of the Decimal Number System in Real Life

Conclusion

Frequently Asked Questions (FAQs)

Decimal Number System

• What is meant by decimal?

Decimals are numbers that consist of two parts, namely, a whole/integral number part and a fractional/decimal part separated by a decimal point. 

For example, 12.5 is a decimal number.

• What is the Decimal Number System?

The decimal system, in mathematics, is a positional numeral system that employs 10 as its base and requires 10 different numerals, namely the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It also requires a dot (decimal point) to represent decimal fractions. In this scheme, the numerals used in denoting a number take different place values depending upon position.

In a base-10 system the number 543.21 represents the sum (5 × 10²) + (4 × 10¹) + (3 × 10⁰) + (2 × 10⁻¹) + (1 × 10⁻²).

Each digit in the decimal system has a position, and every digit is ten times more significant than the previous digit. Suppose 25 is a decimal number, then 2 is ten times more than 5. Some examples of decimal numbers are:-

(12)₁₀, (345)₁₀, (119)₁₀, (200)₁₀, (313.9)₁₀

• Place Value Chart of the Decimal Number System

Most Significant Digit			Decimal Point			Least Significant Digit
10²	10¹	10⁰		10⁻¹	10⁻²	10⁻³
100	10	1	.	0.1	0.01	0.001

• What is the origin of the base-10 number system?

By the 7th century, Indian mathematicians had perfected a decimal (or base ten) positional system, which could represent any number with only ten unique symbols. Over the next few centuries, Arab merchants, scholars, and conquerors began to spread it into Europe.

The base-10 number system is called the Hindu-Arabic number system or Arabic number system. This number system, with its associated arithmetic algorithms, has furnished the basis for the development of Western commerce and science since its introduction to the West in the 12th century CE.

• Why is the decimal system called Base 10

The decimal system is called base 10 because it is built on the number 10. It is called Base 10, why:

First, the decimal system uses ten digits only: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. After 9, counting starts again from 0, and a new place value is added.

Second, each place value is a power of 10.

For example: ones (10⁰), tens (10¹), hundreds (10²), thousands (10³), and so on. This shows that the whole system depends on 10.

Third, humans naturally count using ten fingers, which made base 10 the easiest and most practical system to use in daily life.

Because of these reasons, the decimal number system is known as the Base 10 Number System.

Expanded Form of Decimal Numbers

The expanded form of a decimal number means writing the number as the sum of the place values of its digits. In the base 10 (decimal) number system, each digit is multiplied by its corresponding power of 10. This method helps students clearly understand the value of each digit according to its position, both before and after the decimal point.

In expanded form, whole number parts use positive powers of 10, while decimal parts use negative powers of 10.

Examples of Expanded Form in Base 10

• (352)₁₀ = 3 × 10² + 5 × 10¹ + 2 × 10⁰
• (47.6)₁₀ = 4 × 10¹ + 7 × 10⁰ + 6 × 10⁻¹
• (5.203)₁₀ = 5 × 10⁰ + 2 × 10⁻¹ + 0 × 10⁻² + 3 × 10⁻³
• (908.45)₁₀ = 9 × 10² + 0 × 10¹ + 8 × 10⁰ + 4 × 10⁻¹ + 5 × 10⁻²

Conversion of the Decimal Number System

Conversion from the decimal number system (base 10) to other number systems is done by using division by the base of the required number system. The remainders obtained at each step are written in reverse order to get the final answer.

• Decimal to Binary Conversion (Base 10 to Base 2)

To convert a decimal number into binary, divide the number repeatedly by 2 and note the remainders.

Example 1: Convert (25)₁₀ into binary.

Solution: We solve step by step

25 ÷ 2 = 12 remainder 1 (MCB)

12 ÷ 2 = 6 remainder 0

6 ÷ 2 = 3 remainder 0

3 ÷ 2 = 1 remainder 1

1 ÷ 2 = 0 remainder 1 (LCB)

Reading the remainders from bottom to top:

(25)₁₀ = (11001)₂

Example 2: Convert (13)₁₀ to binary:

Solution: We solve step by step

13 ÷ 2 = 6 remainder 1 (MCB)

6 ÷ 2 = 3 remainder 0

3 ÷ 2 = 1 remainder 1

1 ÷ 2 = 0 remainder 1 (LCB)

Reading the remainders from bottom to top:

(13)₁₀ = (1101)₂

Example 3: Convert 18₁₀ to binary

Solution: We solve step by step

18 ÷ 2 = 9 remainder 0 (MCB)

9 ÷ 2 = 4 remainder 1

4 ÷ 2 = 2 remainder 0

2 ÷ 2 = 1 remainder 0

1 ÷ 2 = 0 remainder 1 (LCB)

(18)₁₀ = (10010)₂

• Decimal to Octal Conversion (Base 10 to Base 8)

To convert a decimal number into octal, divide the number repeatedly by 8.

Example 1: Convert (83)₁₀ into octal.

Solution: We solve step by step

83 ÷ 8 = 10 remainder 3 (MCB)

10 ÷ 8 = 1 remainder 2

1 ÷ 8 = 0 remainder 1 (LCB)

Reading the remainders from bottom to top:

(83)₁₀ = (123)₈

Example 2: Convert (45)₁₀ to octal

Solution: We solve step by step

45 ÷ 8 = 5 remainder 5 (MCB)

5 ÷ 8 = 0 remainder 5 (LCB)

Reading the remainders from bottom to top:

 (45)₁₀ = (55)₈

Example 3: Convert (100)₁₀ to octal.

Solution: We solve step by step

100 ÷ 8 = 12 remainder 4 (MCB)

12 ÷ 8 = 1 remainder 4

1 ÷ 8 = 0 remainder 1 (LCB)

Reading the remainders from bottom to top:

 (100)₁₀ = (144)₈

• Decimal to Hexadecimal Conversion (Base 10 to Base 16)

To convert a decimal number into hexadecimal, divide the number repeatedly by 16.

Letters are replaced as

A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

Example 1: Convert (254)₁₀ into hexadecimal.

Solution: We solve step by step

254 ÷ 16 = 15 remainder 14 (E)  (MCB)

15 ÷ 16 = 0 remainder 15 (F) (LCB)

Reading the remainders from bottom to top:

(254)₁₀ = (FE)₁₆

Example 2: Convert (31)₁₀ to hexadecimal

Solution: We solve step by step

31 ÷ 16 = 1 remainder 15 (F) (MCB)

1 ÷ 16 = 0 remainder 1 (LCB)

Reading the remainders from bottom to top:

(31)₁₀ = (1F)₁₆ 

Example 3: Convert (78)₁₀ to hexadecimal

Solution: We solve step by step

78 ÷ 16 = 4 remainder 14 (E)  (MCB)

4 ÷ 16 = 0 remainder 4 (LCB)

Reading the remainders from bottom to top:

(78)₁₀ =( 4E)₁₆

• Practice questions

Convert the following decimal numbers into the given base:

• Convert (18)₁₀ to binary
• Convert (45)₁₀ to binary
• Convert (64)₁₀ to octal
• Convert ( 83)₁₀ to octal
• Convert (100)₁₀ to hexadecimal
• Convert (254)₁₀ to hexadecimal

Conversion from other bases to the decimal number system

To convert a number from any base to decimal (base 10), we multiply each digit of the number by the power of its base according to its position and then add all the results.

• Binary to Decimal Conversion (Base 2 to Base 10)

Rule: Each digit is multiplied by powers of 2, starting from the rightmost digit with power 2⁰.

Example 1: Convert (1011)₂ to decimal.

Solution: Given (1011)₂, a binary number.

We need to multiply each binary digit by the decreasing power of 2. That is,

= 1×2³ + 0×2² + 1×2¹ + 1×2⁰

= 8 + 0 + 2 + 1

= (11)₁₀

Example 2: Convert (11001)₂ to decimal.

Solution: Given (11001)₂, a binary number.

We need to multiply each binary digit by the decreasing power of 2. That is,

= 1×2⁴ + 1×2³ + 0×2² + 0×2¹ + 1×2⁰

= 16 + 8 + 0 + 0 + 1

= (25)₁₀

Example 3: Convert (10010)₂ to decimal.

Solution: Given (10010)₂, a binary number.

We need to multiply each binary digit by the decreasing power of 2. That is,

= 1×2⁴ + 0×2³ + 0×2² + 1×2¹ + 0×2⁰

= 16 + 0 + 0 + 2 + 0

= (18)₁₀

• Octal to Decimal Conversion (Base 8 to Base 10)

Rule: Each digit is multiplied by powers of 8, starting from 8⁰ on the right.

Example 1: Convert (125)₈ to decimal.

Solution: Given (125)₈  is an octal number

Here, we have to multiply each octal digit by the decreasing power of 8, such as

= 1×8² + 2×8¹ + 5×8⁰

= 64 + 16 + 5

= (85)₁₀

Example 2: Convert (47)₈ to decimal.

Solution: Given (47)₈  is an octal number

Here, we have to multiply each octal digit by the decreasing power of 8, such as

= 4×8¹ + 7×8⁰

= 32 + 7

= (39)₁₀

Example 3: Convert (216)₈ to decimal.

Solution: Given (216)₈  is an octal number

Here, we have to multiply each octal digit by the decreasing power of 8, such as

= 2×8² + 1×8¹ + 6×8⁰

= 128 + 8 + 6

= (142)₁₀

• Hexadecimal to Decimal Conversion (Base 16 to Base 10)

Rule: Multiply each digit by powers of 16.

Letters are replaced as

A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

Example 1: Convert (2A)₁₆ to decimal.

Solution: Given (2A)₁₆

A = 10

Multiply each digit by a decreasing power of 16 to obtain an equivalent decimal number.

= 2×16¹ + 10×16⁰

= 32 + 10

= (42)₁₀

Example 2: Convert (4E)₁₆ to decimal.

Solution: Given (4E)₁₆

E = 14

Multiply each digit by a decreasing power of 16 to obtain an equivalent decimal number.

= 4×16¹ + 14×16⁰

= 64 + 14

= 78₁₀

Example 3: Convert (8F)₁₆ to decimal.

Solution: Given (8F)₁₆

F = 15

Multiply each digit by a decreasing power of 16 to obtain an equivalent decimal number.

= 8×16¹ + 15×16⁰

= 128 + 15

= 143₁₀

Key Points to Remember

• The rightmost digit always starts with a power of 0
• Base decides the power (2, 8, or 16)
• Add all multiplied values to get the decimal number
• Hexadecimal letters must be converted to numbers first

Practice Questions

Convert the following numbers into decimal (base 10):

• (10101)₂, (1110)₂, (100011)₂
• (27)₈, (145)₈, (306)₈
• (1A)₁₆, (2F)₁₆, (4C)₁₆, (8E)₁₆

Applications of the Decimal Number System in Real Life

The decimal number system is the most commonly used in daily life. The decimal number system is essential for everyday life because it is simple, accurate, and universally understood. It helps people perform calculations easily in education, business, science, and daily activities.

In money and banking, the decimal number system is used to write prices, salaries, profits, losses, and savings. Currency values such as rupees, dollars, and cents are expressed in decimal form. Calculations related to shopping, discounts, bills, and bank transactions are all performed using decimal numbers.

In education and examinations, marks, percentages, grades, and averages are calculated using the decimal system. Students’ results, GPA, and performance records are all expressed in decimal form, making comparison and evaluation easy.

In measurements, the decimal number system is used to measure length, weight, time, temperature, and volume. Units like meters, kilograms, liters, and degrees are written in decimal form. Scientific measurements also use decimals for accuracy and precision.

In science and engineering, decimal numbers are used for calculations, formulas, experiments, and data analysis. Measurements and results are often written in decimals to show exact values, especially in physics, chemistry, and biology.

In medicine and health, decimal numbers are used to measure body temperature, weight, height, medicine dosage, and medical test results. Accurate decimal values are important for proper diagnosis and treatment.

In business and commerce, decimal numbers are used in accounting, taxation, budgeting, and financial records. Profit margins, interest rates, and tax calculations depend on decimal values for correctness.

Conclusion:

In my view, the simplicity and clarity of the decimal number system make it ideal for everyday calculations. It is used in education, business, science, medicine, and daily life. All basic arithmetic operations and measurements are easily performed using this system. Understanding the decimal number system helps students learn other number systems such as binary, octal, and hexadecimal. Therefore, the decimal number system forms the foundation of mathematics and is essential for further learning in computers and digital technology.

Frequently Asked Questions (FAQs)

Q1. What is the decimal number system?

The decimal number system is a base-10 number system that uses digits from 0 to 9 to represent numbers.

Q2. Why is the decimal number system called base 10?

It is called base 10 because it uses ten digits, and place values are based on powers of 10.

Q3. What are place values in the decimal system?

Place values include ones, tens, hundreds, thousands, tenths, hundredths, and so on, depending on the position of the digit.

Q4. Why is the decimal number system important in daily life?

It is important because it is used in money, measurements, time, education, and all everyday calculations.

Q5. How is the decimal system related to other number systems?

The decimal system is used as a reference to convert numbers from binary, octal, and hexadecimal into understandable values.

Q6. Is the decimal number system used in computers?

Computers mainly use binary, but decimal numbers are used by humans to understand computer data.

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