Data Representation

Why Do Computers Use Binary?

Imagine you’re building a machine that only understands on and off—like a really simple light switch. A computer is essentially a massive collection of these switches, called transistors, which can either be on (1) or off (0).

Since computers operate using electrical signals, having just two states—on (1) and off (0)—makes things much simpler and more reliable. This is why everything inside a computer is stored as binary (base-2) instead of decimal (base-10), which we humans use.

Everything from images to sound and text is stored in binary. Even a YouTube video is just a massive collection of 1s and 0s! 

Since computers only understand 0s and 1s, we need a way to represent numbers using just those two symbols.

How Does Binary Work?

Think of decimal numbers:

• The number 325 means:

  • 3 hundreds, 2 tens, and 5 ones → (3 × 100) + (2 × 10) + (5 × 1).

Binary works the same way, but instead of 10s, it uses powers of 2:

• The binary number 101 means:

  • (1 × 4) + (0 × 2) + (1 × 1) = 5 in decimal.

To convert binary back to decimal, just add up the place values where there’s a 1.

💡 Example: Convert 10011101 to decimal 1️⃣ Look at the positions of the 1s:

• 10011101

• (128 + 16 + 8 + 4 + 1) = 157 ✅

So 10011101 = 157 in decimal.

✅ Binary is the language of computers—only 1s and 0s.

✅ Binary place values double each time (1, 2, 4, 8, 16…).

✅ To convert decimal to binary, subtract the biggest power of 2 first.

✅ To convert binary to decimal, add up the 1s' place values.

💡 Practice tip: Use the divide-by-2 method or a place value table to help with conversions.

Why Do We Need Floating Point Numbers?

Binary numbers are great for whole numbers, but what about decimals like 3.75 or 0.125?

Computers can’t store decimal points directly, so they use floating point representation, which works like scientific notation.

📏 The Two Parts of a Floating Point Number

A floating point number has two key parts:

1️⃣ Mantissa – The main number (stores the precision).

2️⃣ Exponent – Tells us where to move the decimal point.

Example:

In scientific notation, we write 0.58125 × 10⁴.

• Mantissa = 0.58125

• Exponent = 4

💡 In binary, the same concept applies, but with powers of 2 instead of 10.

🔢 How Floating Point Works in Binary

A floating point number in binary might look like:

👉 1.101 × 2³

• 1.101 (Mantissa) → The actual value.

• 2³ (Exponent) → Moves the decimal 3 places to the right.

So:

1.101 × 2³ = 1101.0 in binary = 13 in decimal.

📌 Key Takeaways

✅ Floating point allows computers to store decimal numbers.

✅ Uses a mantissa (value) and an exponent (scale).

ASCII characters fall into two categories: 

✔ Printable Characters → Letters (A-Z, a-z), numbers (0-9), symbols (@, #, &).

✔ Control Characters → Non-visible instructions (Escape, Delete, Enter, Tab).

💡 Example: Pressing the Enter key sends the ASCII code 13 (Carriage Return) to the computer.

✅ ASCII stores text by assigning each character a unique number.

✅ 8-bit Extended ASCII stores up to 256 characters.

✅ Control characters like Enter, Delete, and Tab don’t print but send instructions.

✅ Unicode was introduced to store more languages and symbols.

✅ To calculate storage, multiply characters by 8 bits per character.

Computers store images in different ways, depending on what kind of image it is. 

The two main types are:

• Bitmap Graphics (Pixel-based)

• Vector Graphics (Formula-based)

Each type has its strengths and weaknesses!

💡 The more pixels an image has, the higher the resolution and detail.

✅ Bitmap images = made of pixels, detailed but can become pixelated.

✅ Vector images = made of math formulas, can be resized without losing quality.

✅ Bitmaps are best for photos, vectors are best for logos, icons, and illustrations.

✅ File size: Bitmaps are bigger, vectors are smaller.