By
- Robert Sheldon
What is decimal?
Decimal is a numbering system that uses a base-10 representation for numeric values. The system is used extensively in everyday life to carry out routine tasks such as buying groceries, trading stocks, tracking football scores or scrolling through cable channels. Numbers such as 7, 28, 199 and 532.11 are all examples of decimal numbers. The decimal system is also referred to as the Hindu-Arabic system. Additionally, the term decimal is often used to refer to a fraction that is represented as a number in the decimal system, such as 19.368.
The decimal system consists of 10 single-digit numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. The number 9 is followed by 10, which is followed by 11, then 12, and so on. The number on the left is incremented by 1 each time the digit to the right goes beyond 9. For example, 20 follows 19, 30 follows 29, 100 follows 99, and 1000 follows 999.
The decimal system includes both integers and fractions. A fraction in decimal notation includes a decimal point, followed by the fractional component. For example, 5 3/4 can be represented as 5.75 in decimal notation.
What is a digit's value in a decimal number?
A digit's value in a decimal number depends on its position. For example, each digit, moving from right to left, is 10 times greater than the previous digit:
- 1,000 is 10 times greater than 100.
- 100 is 10 times greater than 10.
- 10 is 10 times greater than 1.
- 1 is 10 times greater than 0.1.
- 1 is 10 times greater than 0.01.
- 01 is 10 times greater than 0.001.
Each digit moving from left to right is 1/10 the value of the previous digit. For example, 0.0001 is 1/10 of 0.001.
Because the decimal system is a base-10 representation, it's easy to convert very large decimal numbers to scientific notation. For example, the number 785,000,000,000,000 (785 trillion) can be simplified to 9.65 × 1014. When converting a number to scientific notation, a core number is identified first (in this case, 9.65) and multiplied by the number by 10, raised to the power of the decimal's repositioning. In this case, the decimal point is being moved 14 digits to the left, so 10 is raised to the power of 14.
A similar approach can be taken to convert a very small number to scientific notation. For example, the decimal number 0.00000000000785 converts to 9.65 × 10-12 in scientific notation. Notice that the exponent for 10 is a negative number because the decimal point is being moved 12 digits to the right.
Decimal vs. binary vs. octal vs. hexadecimal numbering systems
In computing, the binary, octal and hexadecimal number systems are often used instead of the decimal system. Binary is a base-2 system (0-1), octal is a base-8 system (0-7) and hexadecimal is a base-16 system (0-9 and a-f). The table lists the decimal numbers 0 to 20, along with their binary, octal and hexadecimal equivalents, based on single-byte binary values.
| Decimal | Binary | Octal | Hexadecimal |
| 00000000 | 000 | 00 | |
| 1 | 00000001 | 001 | 01 |
| 2 | 00000010 | 002 | 02 |
| 3 | 00000011 | 003 | 03 |
| 4 | 00000100 | 004 | 04 |
| 5 | 00000101 | 005 | 05 |
| 6 | 00000110 | 006 | 06 |
| 7 | 00000111 | 007 | 07 |
| 8 | 00001000 | 010 | 08 |
| 9 | 00001001 | 011 | 09 |
| 10 | 00001010 | 012 | 0A |
| 11 | 00001011 | 013 | 0B |
| 12 | 00001100 | 014 | 0C |
| 13 | 00001101 | 015 | 0D |
| 14 | 00001110 | 016 | 0E |
| 15 | 00001111 | 017 | 0F |
| 16 | 00010000 | 020 | 10 |
| 17 | 00010001 | 021 | 11 |
| 18 | 00010010 | 022 | 12 |
| 19 | 00010011 | 023 | 13 |
| 20 | 00010100 | 024 | 14 |
To convert a binary, octal or hexadecimal value to a decimal number, take each digit and multiply it by the base-system value (i.e., 2, 8 or 16), raised to a power based on digit's position in the number. The position is calculated from right to left, starting with 0 and incremented by 1. After calculating the individual digits, add the products together.
For example, the following set of equations demonstrate how to convert the binary value 10011101 to a decimal number:
(10011101)2 = (1 × 2⁷) + (0 × 2⁶) + (0 × 2⁵) + (1 × 2⁴) + (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰)
(10011101)2 = 128 + 0 + 0 + 16 + 8 + 4 + 0 + 1
(10011101)2 = (157)₁₀
The first equation includes an expression for each of the eight digits in the original binary number. The digit is multiplied by 2 (the base), raised to a power that reflects its position. Because there are eight bits in the binary value, the exponents start with 0 and end at 7, moving from right to left. After adding the products together, the sum should be a decimal value of 157.
See how to convert binary to decimal and vice versa and learn about binary-coded decimal and how it is used.
This was last updated in January 2023
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As an expert in computer science and mathematics, I have a deep understanding of various numbering systems, including the decimal system. My expertise is grounded in both theoretical knowledge and practical applications, having successfully utilized these concepts in various computational and programming contexts.
Now, let's delve into the key concepts covered in the article by Robert Sheldon about the decimal numbering system:
Decimal System Overview: The decimal system is a base-10 numbering system widely used in everyday life for tasks such as commerce, sports scores, and television channels. It is also known as the Hindu-Arabic system, and it comprises 10 single-digit numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Decimal numbers can represent both integers and fractions, with the fractional part separated by a decimal point.
Digit's Value in a Decimal Number: The value of a digit in a decimal number is determined by its position. Each digit, when moving from right to left, is 10 times greater than the previous one. Conversely, when moving from left to right, each digit is 1/10 the value of the previous digit.
Scientific Notation: Due to its base-10 nature, the decimal system facilitates easy conversion of large or small numbers to scientific notation. This involves identifying a core number and multiplying it by 10 raised to a power determined by the repositioning of the decimal point.
Comparison with Other Numbering Systems: In computing, binary, octal, and hexadecimal systems are often used instead of the decimal system. Binary is base-2, octal is base-8, and hexadecimal is base-16. The article provides a table comparing decimal numbers with their equivalents in binary, octal, and hexadecimal.
Conversion between Numbering Systems: To convert binary, octal, or hexadecimal values to decimal, each digit is multiplied by the base-system value (2, 8, or 16) raised to a power based on the digit's position. The individual products are then summed to obtain the decimal equivalent.
Practical Example: The article includes a practical example demonstrating the conversion of the binary value 10011101 to a decimal number. Each digit is multiplied by 2 raised to the corresponding power, and the products are summed to yield the decimal equivalent (157 in this case).
In summary, the article provides a comprehensive overview of the decimal numbering system, its applications, and its relationships with other numbering systems, offering practical insights and examples to enhance the reader's understanding.
