How Computers Add - A Logical Approach
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How Computers Add: A Logical Approach
Introduction
In our last discussion, we explored number systems and counting in binary (see "It's a Binary World - How Computers Count"). To recap, computers operate using binary, composed of 0s and 1s. For example, the binary number 1010 is equivalent to 10 in decimal. These binary numbers can also be represented in octal (8) or hexadecimal (16) systems, where 1010 becomes 12 in octal or A in hex.
Understanding Bits and Bytes
The standard PC code uses 8-bit bytes, building upon the hex system. Processors, along with their operating systems, have evolved from 8-bit to 16-bit, 32-bit, and now 64-bit architectures. This progression means a computer can process 1, 2, 4, or 8 bytes simultaneously. Don't worry if this sounds complex; it's not essential for understanding how computers perform addition.
The Mathematics of Addition
Let's delve into the mathematics. It might seem daunting, but by thinking logically, like a computer, you’ll find it straightforward. We'll introduce Boolean Algebra, a simple yet powerful tool that explains how computers function.
Boolean Algebra Basics
Boolean Algebra is named after George Boole, a 19th-century English mathematician who devised this logic system for digital computers. Instead of using standard operations like addition or subtraction, Boolean Algebra employs logical operations such as AND, OR, and XOR (exclusive OR).
- x OR y = z: z is true if either x or y (or both) are true.
- x AND y = z: z is true only if both x and y are true.
- x XOR y = z: z is true if either x or y is true, but not both.
That's the math you need to understand how computers add numbers.
Digital Logic and Gates
In computing, we use electronic circuits known as gates to apply this logic to binary numbers. These gates?"AND, OR, and XOR?"are built using transistors and handle operations on binary numbers stored in memory.
Adding in Binary
In decimal addition, adding 9 + 3 gives 12 by carrying over units. Similarly, in binary, when you add 1 + 1, it results in 10, indicating a carry to the next bit position. This represents a decimal value of 2.
To construct an adder, we replicate this binary addition process with logic circuits. An adder uses three inputs (two bits and a carry in) and two outputs (a result and a carry out). We use XOR, AND, and OR gates to create a 1-bit adder.
Building Adders and Logic Blocks
By combining individual adders, we form a logic block known as an ADDER, which can perform additions at various bit positions. Computers integrate these blocks to execute complex calculations at incredible speeds.
Truth Table Example
Consider a truth table for an adder:
- No Carry In:
- A B C -> O Carry
- 0 0 0 -> 0 0
- 1 0 0 -> 1 0
- 0 1 0 -> 1 0
- 1 1 0 -> 0 1
- With Carry In:
- A B C -> O Carry
- 0 0 1 -> 1 0
- 1 0 1 -> 0 1
- 0 1 1 -> 0 1
- 1 1 1 -> 1 1
Adding Numbers
To add 2 + 3 in decimal (010 + 011 in binary), we need three ADDER blocks:
1. Least Significant Bit:
- Input A: 0, Input B: 1, Carry: 0
- Result: 1, Carry: 0
2. Next Bit:
- Inputs A: 1, B: 1, Carry: 0
- Result: 0, Carry: 1
3. Most Significant Bit:
- Inputs A: 0, B: 0, Carry: 1
- Result: 1, Carry: 0
Thus, the result is 101 in binary, which is 5 in decimal.
Conclusion
While this process might seem laborious, computers leverage hundreds of adders to perform millions of calculations per second efficiently. Next time, we'll explore how computers handle more complex operations, and you'll find it's simpler than it seems!
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