CS 101 Aug 31 Interpreting binary decimal Decimal

CS 101 – Aug. 31 • Interpreting binary decimal √ • Decimal binary • Shorthand ways of expressing binary – Octal – “Hex” • Negative numbers

Number of Possible Values • 1 bit: 0 and 1 • 2 bits: 00, 01, 10, 11 • 3 bits: 000, 001, …? • For a given number of bits, how many patterns?

decimal binary • You already know binary decimal • But given a decimal number like 45, what is binary form? • Binary numbers are longer than decimal. – (Ex. 5 digit number may require 15 bits) • My technique is the “binary store”

Binary store • At the Binary Store, price is a power of 2: $1, $2, $4, $8, $16… • • If you had $45, what could you buy? 45 = 32 + 8 + 4 + 1 Write powers of 2: 25 + 23 + 22 + 20 Write binary number: 101101

Another example • Let’s convert 61 to binary: • Go to binary store with $61… 61 = 32 + 16 + 8 + 4 + 1 61 = 25 + 24 + 23 + 22 + 20 Finally: 111101

Octal • Octal means base 8: each digit is a power of 8. • Because 8 = 23, each octal digit corresponds to 3 bits • 4618 = 100 110 0012 • 73258 = ?

Hexadecimal (“hex”) • Base 16: each digit is a power of 16 • Since 16 = 24, each hex digit corresponds to 4 bits. • Hex also means we have 16 different digits. • a = 10, b = 11, c = 12, d = 13, e = 14, f = 15

Hex examples • 96416 = 1001 0110 01002 • d 12316 = 1101 0010 00112 • Now let’s go the other way: • 1110002 = ______16 • 100111112 = _______16

Decimal octal, hex • Often the best way to come up with octal or hex is to go thru binary first. • Ex. What is 71 in octal? – Binary store gives: 71 = 64 + 2 + 1 – Binary number is 1000111 – Grouped as octal: 001 000 1112 = 1078 • We can check answer.

Why couldn’t the computer scientist tell the difference between Halloween (Oct 31) and Christmas (Dec 25) ?

Negatives? • We can used a “signed” representation. • We want half the rep’ns to be negative. • Ex. 5 bits 32 numbers. – 16 numbers are negative – Thus, range is – 16 to +15. • For n bits: range is – 2 n– 1 to 2 n – 1

How to represent negatives • In ordinary (unsigned) binary, this was impossible! • In signed: 3 steps to represent –x: 1. Find rep’n of +x. 2. Invert the bits. 3. Add 1. • Try some examples.