Decimal Number Conversions
A number can be represented in different number systems. Although the digits and the way a number is written may change, its actual value can remain the same. Decimal number conversion is the process of expressing a number from another number system using the decimal (base 10) system.
This note explains how to convert binary, octal, and hexadecimal numbers into decimal by using positional values.
1. Understanding Positional Values
The value of a digit in a number depends on its position and the base (radix) of the number system.
The positional value of each digit is determined by raising the base to the position number. Positions are counted from right to left, starting at zero.
For a number in base (b), the place values from right to left are:
[ b^0,\ b^1,\ b^2,\ b^3,\ldots ]
To convert a number to decimal:
- Identify the base of the number.
- Write the positional value for each digit, starting from (b^0) at the right.
- Multiply each digit by its positional value.
- Add all the products.
Key Idea: To convert a number from any base to decimal, multiply each digit by the base raised to its position, then add the results.
2. Binary to Decimal
The binary number system has base 2 and uses only two digits: 0 and 1. Each position represents a power of 2.
The positional values, from right to left, are (2^0, 2^1, 2^2, 2^3,\ldots), which are 1, 2, 4, 8, 16, and so on.
Example: Convert (10110_2) to decimal| Digit | 1 | 0 | 1 | 1 | 0 |
|---|---|---|---|---|---|
| Position | 4 | 3 | 2 | 1 | 0 |
| Positional value | 16 | 8 | 4 | 2 | 1 |
Multiply each digit by its positional value and add the products:
[ \begin{aligned} 10110_2 &= (1 \times 16)+(0 \times 8)+(1 \times 4)\ &\quad +(1 \times 2)+(0 \times 1)\ &=16+0+4+2+0\ &=22 \end{aligned} ]
Therefore,
[ \boxed{10110_2=22_{10}} ]
The zero digits contribute zero to the total, but their positions still matter because they determine the place values of the other digits.
3. Octal to Decimal
The octal number system has base 8 and uses the digits 0 to 7. Each position represents a power of 8.
The positional values, from right to left, are (8^0, 8^1, 8^2,\ldots), which are 1, 8, 64, 512, and so on.
Example: Convert (57_8) to decimal| Digit | 5 | 7 |
|---|---|---|
| Position | 1 | 0 |
| Positional value | 8 | 1 |
Multiply each digit by its positional value:
[ \begin{aligned} 57_8 &= (5 \times 8^1)+(7 \times 8^0)\ &=(5 \times 8)+(7 \times 1)\ &=40+7\ &=47 \end{aligned} ]
Therefore,
[ \boxed{57_8=47_{10}} ]
Remember that an octal number cannot contain the digits 8 or 9.
4. Hexadecimal to Decimal
The hexadecimal number system has base 16. It uses the digits 0 to 9 and the letters A to F to represent values from 10 to 15.
| Hexadecimal digit | Decimal value |
|---|---|
| A | 10 |
| B | 11 |
| C | 12 |
| D | 13 |
| E | 14 |
| F | 15 |
Each position represents a power of 16. From right to left, the positional values are (16^0, 16^1, 16^2,\ldots), which are 1, 16, 256, 4096, and so on.
Example: Convert (3A_{16}) to decimalFirst, replace the hexadecimal digit A with its decimal value, 10.
| Digit | 3 | A (10) |
|---|---|---|
| Position | 1 | 0 |
| Positional value | 16 | 1 |
Then multiply each digit by its positional value and add the results:
[ \begin{aligned} 3A_{16} &= (3 \times 16^1)+(10 \times 16^0)\ &=(3 \times 16)+(10 \times 1)\ &=48+10\ &=58 \end{aligned} ]
Therefore,
[ \boxed{3A_{16}=58_{10}} ]
When converting hexadecimal numbers, always replace letters A to F with their corresponding decimal values before calculating.
5. Comparing the Conversion Methods
The same positional-value method is used for all three conversions. The only difference is the base.
| Number system | Base | Digits used | Place values (from right) |
|---|---|---|---|
| Binary | 2 | 0, 1 | (2^0, 2^1, 2^2,\ldots) |
| Octal | 8 | 0–7 | (8^0, 8^1, 8^2,\ldots) |
| Decimal | 10 | 0–9 | (10^0, 10^1, 10^2,\ldots) |
| Hexadecimal | 16 | 0–9, A–F | (16^0, 16^1, 16^2,\ldots) |
For example, the rightmost digit in any whole-number representation has position 0. Its positional value is therefore (b^0=1), regardless of the base.
6. Quick Practice
Convert each number into decimal.
- (1101_2)
- (11001_2)
- (72_8)
- (346_8)
- (1F_{16})
- (2C_{16})
| Question | Working | Decimal answer |
|---|---|---|
| (1101_2) | (8+4+0+1) | 13 |
| (11001_2) | (16+8+0+0+1) | 25 |
| (72_8) | (7 \times 8+2) | 58 |
| (346_8) | (3 \times 64+4 \times 8+6) | 230 |
| (1F_{16}) | (1 \times 16+15) | 31 |
| (2C_{16}) | (2 \times 16+12) | 44 |
7. Important Examination Points
- Identify the correct base before calculating.
- Start assigning positions from zero at the rightmost digit.
- Multiply each digit by the corresponding power of the base.
- Add the products to obtain the decimal value.
- In hexadecimal, remember that A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15.
- Check that every digit is valid for the number system. For example, binary digits can only be 0 and 1.
Key Idea: The conversion changes the representation of a number, not its value. Use the base and digit positions to calculate that value accurately.
Summary
Decimal conversion from binary, octal, or hexadecimal is performed using positional values. Assign each digit a position starting from zero on the right, multiply it by the base raised to that position, and add the products. Use base 2 for binary, base 8 for octal, and base 16 for hexadecimal. For hexadecimal, convert letters A–F to their decimal values before performing the calculation.