Binary Number Systems
Why This Matters
This lesson introduces the binary number system, the fundamental language of computers. We will explore how binary numbers represent data, perform conversions between binary, denary (decimal), and hexadecimal, and understand the significance of bits and bytes.
Key Words to Know
Introduction to Number Systems
Computers operate using electrical signals that are either on or off, which can be represented by two states: 0 and 1. This forms the basis of the binary number system (base-2). In contrast, humans typically use the denary (decimal) number system (base-10), which employs ten digits (0-9). Understanding how these systems work and how to convert between them is crucial for comprehending computer operations. Each digit in a number system holds a positional value, which is a power of its base. For example, in denary, the number 123 means (1 * 10^2) + (2 * 10^1) + (3 * 10^0). Similarly, in binary, the number 101 means (1 * 2^2) + (0 * 2^1) + (1 * 2^0). The concept of positional value is fundamental to all number systems.
Binary to Denary Conversion
To convert a binary number to its denary equivalent, we sum the positional values of each '1' bit. Each position in a binary number represents a power of 2, starting from 2^0 (1) on the rightmost side and increasing by a power of 2 for each position to the left.
Steps:
- Write down the binary number.
- Above each digit, write its corresponding power of 2, starting from 2^0 on the right.
- For every position where there is a '1' in the binary number, add its corresponding power of 2.
- Ignore positions with a '0'.
Example: Convert 1101_2 to denary.
1 1 0 1
2^3 2^2 2^1 2^0
8 4 2 1
1101_2 = (1 * 8) + (1 * 4) + (0 * 2) + (1 * 1) = 8 + 4 + 0 + 1 = 13_10
This method allows for straightforward conversion of any binary number into its human-readable denary form.
Denary to Binary Conversion
Converting a denary number to binary can be done using the repeated division by 2 method or by finding the largest power of 2. The division method is generally more systematic.
Steps (Repeated Division by 2):
- Divide the denary number by 2.
- Note down the remainder (which will be either 0 or 1).
- Take the quotient and divide it by 2 again.
- Repeat until the quotient becomes 0.
- Read the remainders from bottom to top to get the binary number.
Example: Convert 13_10 to binary.
13 / 2 = 6 remainder 1
6 / 2 = 3 remainder 0
3 / 2 = 1 remainder 1
1 / 2 = 0 remainder 1
Reading remainders from bottom to top: 1101_2.
Alternatively, you can find the largest power of 2 less than or equal to the denary number, place a '1' at that position, subtract it, and repeat for the remainder.
Hexadecimal Number System
The hexadecimal (base-16) number system uses 16 unique symbols: 0-9 and A-F (where A=10, B=11, C=12, D=13, E=14, F=1...
Bits, Bytes, and Data Storage
The smallest unit of data in a computer is a bit, which can hold a value of either 0 or 1. A byte is a group of ...
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Exam Tips
- 1.Practice conversions regularly: Be proficient in converting between binary, denary, and hexadecimal in both directions. Use a systematic approach like repeated division for denary to binary.
- 2.Memorise powers of 2: Knowing powers of 2 up to 2^8 or 2^10 will significantly speed up binary-to-denary conversions and help with understanding data sizes.
- 3.Understand positional values: Always remember that each digit's position in a number system determines its value (e.g., 2^0, 2^1, 2^2, etc., for binary).
- 4.Show your working: Even for simple conversions, clearly show the steps you take. This allows for partial credit if your final answer is incorrect but your method is sound.
- 5.Be precise with notation: Use subscripts (e.g., 101_2, 13_10, A5_16) to clearly indicate the base of the number you are working with.