Foundations of Binary Numbers

TL;DR

Binary numbers use only two digits, 0 and 1, and are the fundamental language of computers. Each position in a binary number represents a power of two, unlike our familiar base-10 system. Understanding binary is key to grasping how computers perform calculations and store data.

1. The Mental Model

Think of binary like a series of light switches: each switch is either ON (1) or OFF (0). When you have several switches in a row, they can represent a number, with the rightmost switch being the smallest value and each switch to its left being twice as valuable.

2. The Core Material

Binary is a base-2 number system, meaning it only uses two digits: 0 and 1. This is different from the decimal system (base-10) you're used to, which uses digits 0-9. Each position in a binary number represents a power of 2, starting from 2^0 (which is 1) on the far right, then 2^1 (2), 2^2 (4), 2^3 (8), and so on.

To convert a binary number to decimal, you multiply each binary digit by its corresponding power of 2 and then add the results.

Converting Binary to Decimal

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Let's look at the binary number 1011:

• Rightmost digit (1): This is in the 2^0 position. So, 1 * 2^0 = 1 * 1 = 1.
• Second digit from right (1): This is in the 2^1 position. So, 1 * 2^1 = 1 * 2 = 2.
• Third digit from right (0): This is in the 2^2 position. So, 0 * 2^2 = 0 * 4 = 0.
• Leftmost digit (1): This is in the 2^3 position. So, 1 * 2^3 = 1 * 8 = 8.

Adding these values: 1 + 2 + 0 + 8 = 11. So, binary 1011 is decimal 11.

Converting Decimal to Binary

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To convert a decimal number to binary, you repeatedly divide the decimal number by 2 and record the remainder. The binary number is formed by reading the remainders from bottom to top.

Let's convert decimal 13 to binary:

1. 13 / 2 = 6 remainder 1
2. 6 / 2 = 3 remainder 0
3. 3 / 2 = 1 remainder 1
4. 1 / 2 = 0 remainder 1

Reading the remainders from bottom to top gives you 1101. So, decimal 13 is binary 1101.

Here's how the conversion process works visually:

graph LR
    A["Start with Decimal Number"] --> B{"Is Number > 0?"};
    B -- Yes --> C["Divide Number by 2"];
    C --> D["Record Remainder (0 or 1)"];
    D --> E["Update Number to Quotient"];
    E --> B;
    B -- No --> F["Collect Remainders (Bottom to Top)"];
    F --> G["Binary Result"];

3. Worked Example

Let's convert the binary number 110.101 to its decimal equivalent. This introduces fractions, where positions to the right of the decimal point represent negative powers of 2 (1/2, 1/4, 1/8, etc.).

• Left of the decimal point (integer part):

  • 0 * 2^0 = 0 * 1 = 0
  • 1 * 2^1 = 1 * 2 = 2
  • 1 * 2^2 = 1 * 4 = 4
  • Sum of integer part: 0 + 2 + 4 = 6

• Right of the decimal point (fractional part):

  • 1 * 2^-1 = 1 * (1/2) = 0.5
  • 0 * 2^-2 = 0 * (1/4) = 0
  • 1 * 2^-3 = 1 * (1/8) = 0.125
  • Sum of fractional part: 0.5 + 0 + 0.125 = 0.625

Combining both parts: 6 + 0.625 = 6.625.

So, binary 110.101 is decimal 6.625.

4. Key Takeaways

• Binary is a base-2 system using only digits 0 and 1.
• Each position in a binary number represents a power of two (2^0, 2^1, 2^2, etc.).
• To convert binary to decimal, sum the products of each digit and its corresponding power of two.
• To convert decimal to binary, repeatedly divide by 2 and record remainders from bottom to top.
• Binary is the native language of computers and crucial for understanding digital logic.
• Fractional binary numbers use negative powers of two (2^-1, 2^-2) for digits after the decimal point.

Common mistakes to avoid:
- Confusing powers of two with powers of ten.
- Reading remainders for decimal-to-binary conversion from top to bottom instead of bottom to top.
- Forgetting that 2^0 equals 1, not 0.
- Miscalculating negative powers of two for fractional binary numbers.

5. Now Try It

Convert the decimal number 25 to binary, and then convert the binary number 11010 back to decimal.
What to do:
1. Take decimal 25 and perform the repeated division by 2, recording the remainders.
2. Take binary 11010 and calculate its decimal equivalent by summing the products of each digit and its power of 2.
What success looks like: You should be able to show that decimal 25 converts to binary 11001, and binary 11010 converts to decimal 26.