GCSE
Computer Science
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Introduction to GCSE Computer Science -
1.1 Systems Architecture -
1.2 Memory and Storage -
1.3 Computer Networks, Connections and Protocols -
1.4 Network Security -
1.5 Systems Software -
1.6 Ethical, Legal, Cultural and Environmental Impacts of Digital Technology -
2.1 Algorithms -
2.2 Programming Fundamentals -
2.3 Producing Robust Programs -
2.4 Boolean Logic -
2.5 Programming Languages and Integrated Development Environments
1. Computer Systems
In this lesson, we will explore how computers use the binary system to represent numbers. We will cover denary-to-binary conversion, binary addition, overflow errors, hexadecimal, and binary shifts.
Converting Denary Whole Numbers to Binary Numbers
The binary system uses only two digits, 0 and 1, to represent the on-and-off states that computers use to process data. Converting denary (decimal) whole numbers to binary involves dividing the number by 2 and recording the remainder. This process continues until the quotient becomes 0. This method can be visualised using a table, as shown below:
Draw out a table like the one below:
| 128 (2⁷) | 64 (2⁶) | 32 (2⁵) | 16 (2⁴) | 8 (2³) | 4 (2²) | 2 (2¹) | 1 (2⁰) |
|---|---|---|---|---|---|---|---|
If we wanted to convert a denary number to binary using a table, start at the left most column (128 or 2⁷) and compare the number in the column heading with the one we wish to convert.
- If the column header is greater than the number we wish to convert, it is not needed and we place 0 in the box below.
- If the column header is less than the number we wish to convert, it is needed and we place in a 1 in the box, then subtract the column header from our denary number.
Repeat this process across the columns.
Analogy
Think of converting denary to binary like giving out change using coins. You always start with the biggest coin (128p), and if it fits, you use it. If not, you move to the next smallest coin (64p), and so on, until you’ve built the exact amount.
Example
If we want to convert the number 13 to binary, we would follow these steps:
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 |
Start with 128:
- 128 > 13, so write '0'.
- 64 > 13, so write '0'.
- 32 > 13, so write '0'.
- 16 > 13, so write '0'.
- 8 < 13, so write '1'. Subtract 8 from 13 to get 5 (13 - 8 = 5).
- 4 < 5, so write '1'. Subtract 4 from 5 to get 1 (5 - 4 = 1).
- 2 > 1, so write '0'.
- 1 = 1, so write '1'.
This gives us the binary number 00001101 for the denary number 13.
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Create a free accountConverting Denary Whole Numbers to Binary Numbers
- Denary (decimal) numbers can be converted to binary by using powers of 2 from left to right.
- Place 1 under powers of 2 that fit into the number and 0 elsewhere.
Converting Binary Numbers to Denary Whole Numbers
- Each binary digit is multiplied by the corresponding power of 2.
- The results are added together to get the denary value.
Binary Addition and Overflow Errors
- Binary addition follows these rules:
- 0+0=0
- 1+0=1
- 1+1=10
- 1+1+1=11
- Overflow errors occur when the result exceeds the number of bits available.
Converting Denary Whole Numbers to 2-Digit Hexadecimal Numbers
- Convert the denary number to 8-bit binary.
- Split the binary into two 4-bit nibbles and convert each nibble into a hexadecimal digit.
Converting Binary Integers to Hexadecimal Equivalents
- Group the binary number into two 4-bit nibbles.
- Convert each nibble into its hexadecimal equivalent.
Binary Shifts
- A left shift multiplies a binary number by 2 for each shift.
- A right shift divides a binary number by 2 for each shift.