Positive Binary Numbers

Overview

Computers store and process data using binary (base-2) representation, as they operate with electronic switches that can be either on (1) or off (0). Binary numbers, composed of just 0s and 1s, allow computers to efficiently represent, store, and manipulate data of various kinds, from numbers to text, images, and sound.

Binary Representation in Computing

Binary System: A number system that uses only two symbols, 0 and 1. Each digit in binary is called a "bit."
Switches in Computers: Computers use tiny switches that can be in two states: on (1) and off (0). Binary numbers align naturally with this on-off mechanism, making them ideal for storing data.
Data Interpretation: The same binary sequence can represent different types of data depending on context:
- Numeric: Represents positive or negative integers.
- Text: Character encoding schemes (e.g., ASCII) use binary values to represent letters and symbols.
- Images and Sound: Media data is stored as binary values in compressed or raw formats.

Binary to Decimal Conversion

Binary Place Values: Each position in a binary number represents a power of 2, starting from the right ( $2^0$ , $2^1$ , $2^2$ , etc.).

Conversion Process:

To convert a binary number to decimal, sum the products of each bit (0 or 1) and its place value.

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Example: Convert 1011 to decimal.

Starting from the right, the place values are $2^3$ , $2^2$ , $2^1$ , and $2^0$

Calculation:

1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11

Therefore, 1011 in binary is 11 in decimal.

Decimal to Binary Conversion

Repeated Division by 2:

To convert a decimal (denary) number to binary, repeatedly divide the number by 2, keeping track of the remainders.
Write down the remainders in reverse order to obtain the binary representation.

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Example: Convert 13 to binary.

$13 \div 2 = 6$ remainder 1
$6 \div 2 = 3$ remainder 0
$3 \div 2 = 1$ remainder 1
$1 \div 2 = 0$ remainder 1 Writing the remainders from bottom to top, 13 in decimal is 1101 in binary.

Examples

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Example 1: Binary to Decimal Convert the binary number 11010 to decimal.

1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 16 + 8 + 0 + 2 + 0 = 26

So, 11010 in binary is 26 in decimal.

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Example 2: Decimal to Binary Convert the decimal number 19 to binary.

$19 \div 2 = 9$ remainder 1
$9 \div 2 = 4$ remainder 1
$4 \div 2 = 2$ remainder 0
$2 \div 2 = 1$ remainder 0
$1 \div 2 = 0$ remainder 1 Writing the remainders from bottom to top, 19 in decimal is 10011 in binary.

Note Summary

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Common Mistakes

Misinterpreting Place Values: Not properly aligning binary digits with their respective place values (e.g., confusing the leftmost bit with $2^1$ instead of $2^3$ ).
Incorrect Remainder Order in Decimal to Binary Conversion: Forgetting to reverse the remainders after dividing.
Confusing Binary Representations: Not recognising that different data types can share the same binary structure, but are interpreted differently (e.g., 01000001 in binary represents 65 as a number but could represent the character A in ASCII).

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Key Takeaways

Binary Basics: Binary numbers use only 0 and 1, representing data through the on-off states of computer switches.
Conversions:
Binary to Decimal: Sum the products of each bit and its place value.
Decimal to Binary: Use repeated division by 2 and reverse the order of remainders.
Application: Understanding binary representation and conversion is essential for interpreting data in computer systems.

Positive Binary Numbers Simplified Revision Notes for A-Level OCR Computer Science

Positive Binary Numbers

Overview

Binary Representation in Computing

Binary to Decimal Conversion

Conversion Process:

Decimal to Binary Conversion

Examples

Note Summary

Common Mistakes

Key Takeaways

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