Understanding Binary Numbers: Conversion and Addition

Study Notes

Understanding Binary Numbers

Binary numbers form the foundational language of computers and digital systems. They're composed of just two digits: 0 and 1. In this guide, we'll explore the basics of binary, including converting binary to decimal, the structure of the binary numerical system, and binary addition.

Binary Numerical System

Binary numbers are based on the base-2 system, where each digit has a value of either 0 or 1. In contrast, our familiar decimal system is based on the base-10 system, using digits 0-9.

To count in binary, we start at 0 and increment by 1 for each position. For example:

0 (zero)
1 (one)
10 (two)
11 (three)
100 (four)
101 (five)
110 (six)
111 (seven)

Each new position to the left doubles the value of the previous position. In binary, the positions are called bits (short for binary digits).

Binary Number System

Converting Binary to Decimal

To convert binary to decimal, you multiply each bit by its position's value and then add up the results. For example, to convert the binary number 1101 to decimal:

Multiply the rightmost bit (1) by 1 (since it's in the 1's place): 1 * 1 = 1
Multiply the next bit (1) by 2 (since it's in the 2's place): 1 * 2 = 2
Multiply the next bit (0) by 4 (since it's in the 4's place): 0 * 4 = 0
Multiply the leftmost bit (1) by 8 (since it's in the 8's place): 1 * 8 = 8

Now, add up the results: 1 + 2 + 0 + 8 = 11. So, 1101 in decimal is 11.

Binary Addition

Binary addition follows a simple set of rules. To add two binary numbers, you can follow these steps:

Start from the rightmost bits and work your way to the left.
For each pair of bits, add the two numbers and carry over any excess value (using the same rules as decimal addition).

For example, let's add 1011 (9) and 1101 (11) in binary:

1 + 1 = 0 (with a carry of 1)
0 + 1 + 1 = 2
1 + 0 + 0 = 1
1 + 1 + 1 = 3

Now, add the carry: 0 + 1 = 1

So, the binary addition of 1011 and 1101 is 10110. This is equivalent to the decimal number 20 in decimal.

Binary numbers are a fundamental part of computer science and digital systems. With an understanding of binary, you can delve deeper into computer systems, programming, and other related fields.

Description

Explore the basics of binary numbers, including converting binary to decimal, the structure of the binary numerical system, and binary addition. Learn how to convert binary to decimal by multiplying each bit by its position's value and adding up the results. Discover how to add two binary numbers following simple rules.