Numbers In Base Two

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In today’s Mathematics class, We will be discussing Numbers in Base 2. We hope you enjoy the class!

Binary Numbers (Base 2 Numbers)

Understanding Binary Numbers:

Binary numbers are a bit different from the regular numbers we use every day (called decimal or Base 10 numbers). In binary, we only have two digits – 0 and 1. It’s like an on/off switch where 0 could mean off, and 1 means on. This system is fundamental in the world of computers and electronics.

Addition in Base Two (Binary Addition)

Basic Rules:

0 + 0 = 0 (Just like adding nothing to nothing)
0 + 1 = 1 (Like adding 1 to nothing)
1 + 0 = 1 (Same as the above, order doesn’t matter)
1 + 1 = 10 (This is interesting! Instead of ‘2’, we write ‘0’ and carry over 1 to the next column)

Worked Examples:

Example 1: 1110 + 1001

Start from the rightmost digits: 0 + 1 = 1
Next: 1 + 0 = 1
Then: 1 + 0 = 1
Finally, at the leftmost digits: 1 + 1 = 10 (write 0, carry 1)
Adding the carried 1: 1 + 0 = 1
Solution: 10111

Example 2: 1111 + 1101

Proceeding from right to left, and remembering to carry over 1s, we get 1 1100

Subtraction in Base Two (Binary Subtraction)

Basic Rules:

0 – 0 = 0 (Taking nothing from nothing leaves nothing)
1 – 0 = 1 (If you have 1 and take away nothing, you still have 1)
10 – 1 = 1 (10 in binary is like ‘2’ in decimal)
11 – 1 = 10 (11 in binary is like ‘3’ in decimal)

Worked Example: 1110 – 1001

Starting from the right: 0 – 1 (can’t do, so borrow from the left, 10 – 1 = 1)
Next: 0 – 0 (borrowed previously) = 0
Then: 1 – 0 = 1
Finally, leftmost: 1 – 1 = 0
Solution: 101

Addition and Subtraction with Bicimals (Binary Decimals)

Bicimals: Just like decimals, but in binary. We align the binary points (like decimal points) and perform the operations.

Addition Example: 1.1011 + 10.1001

Align the binary points and add each column, carrying over as needed.
Solution: 100.0010

Subtraction Example: 101.101 – 11.011

Align the binary points. Subtract and borrow as needed.
Solution: 10.01

Multiplication & Division in Base Two

These operations follow the same principles as in Base 10 but only using 0s and 1s.

Multiplication Example: 10001 x 11

Multiply each digit of the second number with the entire first number and add the results.
Solution: 110011

Evaluation Exercises
Give these a try:

Simplify: 101 + 101 + 111
Multiply: 10001 x 11

Remember, practice makes perfect in mathematics, especially when learning a new concept like binary numbers.

That’s a wrap for today’s class on Binary Numbers! We hope you found it engaging and insightful.

If you have any questions, don’t hesitate to drop them in the comment section. We’re here to help you understand and succeed. Keep practicing, and see you soon for more mathematical adventures!