Data Representation and Compression, Page 6 (2024)

• EK 2.1.1C At a higher level, bits are grouped to represent abstractions, including but not limited to numbers, characters, and color.
• EK 2.1.1D Number bases, including binary, decimal, and hexadecimal, are used to represent and investigate digital data.
• EK 2.1.1F Hexadecimal (base 16) is used to represent digital data because hexadecimal representation uses fewer digits than binary.

Place value notation uses only one digit in each place. We need to represent 0 through 15, and we have digits for 0-9, so we choose A-F for the remaining values:

1. Watch this Hexadecimal and Binary Timer Snap! program run. The top row counts in binary, the middle in decimal, the bottom in hex. Write a description of the hex counter's behavior.
   

Base 16 uses powers of sixteen instead of powers of two or ten. Place values in hexadecimal represent the units place (16⁰ = 1), the sixteens place (16¹ = 16), the two hundred fifty-sixes place (16² = 256), the four thousand ninety-sixes place (16³ = 4096), etc. So, for example:

3B₁₆ = 3 × 16¹ + 11 × 16⁰ = 48 + 11 = 59₁₀

To translate from hexadecimal (for example, 7B3₁₆) to base 10, first, write the digits on paper. Then write out the hexadecimal place values: start at the right with 1, then write 16 in the next place to the left, then write 256 (which is 16²), and so on. Each new place will be worth 16 times the one to its right.

7	B	3
256	16	1

So, 7B3₁₆ = (7 × 256) + (11 × 16) + (3 × 1) = 1792 + 176 + 3 = 1971₁₀.

FF₁₆
= (15 × 16¹ + 15 × 16⁰)₁₀
= (15 × 16 + 15)₁₀
= 255₁₀

2. Translate these hexadecimal numerals into base 10 notation:
   1. AF₁₆
   2. 5D₁₆
   3. 18₁₆
   4. 3E8₁₆

   A numeral such as 11 means eleven in the usual decimal notation, or three if it is a binary numeral, or seventeen in hexadecimal notation. The only way you know that the "18" in part (c) above doesn't mean eighteen is that subscript "16" means hexadecimal.

To translate from base 10 (for example, 299₁₀) to base 16, first write out the hexadecimal place values by multiplying by 16 moving left from the units place until you get to a value larger than your number (4096 for this example). Then think, "My number is smaller than 4096, so I leave that place blank. But I can subtract two hundred fifty-six once, so I write a 1 there, and there's 43 left. Now, I can subtract two sixteens, and there's 11 left. And 11 is B in hex."

4096	256	16	1
	1	2	B

Now, read the number off: 12B₁₆ = 299₁₀.

3. Translate these decimal numerals to hexadecimal notation:
   1. 59
   2. 144
   3. 229
   4. 316

4. This algorithm runs into a problem after base 36. What's the problem and what's the solution?
5. Convert 299₁₀ to base 7.