Introduction to Hexadecimal and Data Conversion

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Questions and Answers

What are computer systems fundamentally based on?

Binary sequences (correct)

Octal numbering

Hexadecimal numbers

Decimal numbers

Why is the binary system preferred over other bases for electronic implementation?

It's reliable in noisy environments (correct)

It's easier to convert to decimal

It requires more storage space

It can represent more values

What does a bit vector represent?

Any kind of data type once encoded properly (correct)

Float numbers exclusively

Just program instructions

Only integer values

How is the numeric value of a binary number calculated?

By multiplying each bit by 2 raised to the bit position Signup and view all the answers

What is the correct prefix for a binary number in written form?

0b Signup and view all the answers

What role do bits play in computer systems?

They are the foundational components of data representation Signup and view all the answers

What type of signals do binary systems represent?

Electrical signals in terms of high and low voltage Signup and view all the answers

What is the significance of the rightmost digit in a decimal number?

It represents the smallest place value Signup and view all the answers

What is the decimal equivalent of the hexadecimal number 1D8A?

7562 Signup and view all the answers

Which of the following correctly describes the process of converting binary to hexadecimal?

Convert every 4 binary digits to 1 hex digit Signup and view all the answers

Which logic gate outputs a high signal only if at least one input signal is high?

OR gate Signup and view all the answers

What is the output of an AND gate given the inputs 1 and 0?

0 Signup and view all the answers

In Boolean algebra, what is the output of the XOR gate when both inputs are 0?

0 Signup and view all the answers

If the binary number 10010 is converted to hexadecimal, what is the result?

12 Signup and view all the answers

What is the correct hexadecimal representation of the decimal number 255?

FF Signup and view all the answers

Which operation does NOT belong to the common logical operations represented in Boolean algebra?

MULTIPLY Signup and view all the answers

What is the correct way to denote the hexadecimal number FA1D37B?

0xFA1D37B Signup and view all the answers

How many bits does one hexadecimal digit represent?

4 bits Signup and view all the answers

Which of the following is the decimal value for the hexadecimal number 1D8A?

7562 Signup and view all the answers

In binary, what does the number 13 convert to?

1101 Signup and view all the answers

What is the range of values representable in a single byte in hexadecimal?

00 to FF Signup and view all the answers

What is the first step in converting a decimal number to hexadecimal?

Divide the number by 16 Signup and view all the answers

What is the hexadecimal equivalent of the binary number 1011 0001 0010 1111?

B12F Signup and view all the answers

What is the quotient when converting the decimal number 7562 to hexadecimal using the division method?

472 Signup and view all the answers

What condition leads to a negative overflow when adding two w-bit signed numbers?

TAddw(u,v) < –2w–1 Signup and view all the answers

What is the maximum possible value of an unsigned multiplication result for w-bit numbers?

22w – 1 Signup and view all the answers

What happens to the high order bits during the multiplication of signed and unsigned bit vectors?

They are ignored. Signup and view all the answers

Which operation is equivalent to multiplying a number by a power of 2?

Left shift Signup and view all the answers

What is the smallest possible value for a two's complement product of w-bit numbers?

–22w–2 + 2w–1 Signup and view all the answers

What is the result of converting a w-bit signed integer X to a (w+k)-bit integer X' using sign extension?

The sign bit is repeated k times on the left. Signup and view all the answers

What happens to the value when a 32-bit integer is truncated to a 16-bit short?

High-order bits are truncated, altering the value. Signup and view all the answers

What is the negation of a bit-vector X represented by the equation -X = ~X + 1?

It involves inverting all bits and adding one. Signup and view all the answers

In the addition of signed numbers, why is the true sum often disregarded?

Only w bits are needed for calculation. Signup and view all the answers

When performing modular addition on unsigned numbers, how are overflow bits treated?

They are ignored completely. Signup and view all the answers

What is the result of the operation 01101 + 11011 in modular arithmetic?

11000 Signup and view all the answers

What is the implication of performing operations on signed integers regarding the sign bit?

The sign bit is used for overflow detection. Signup and view all the answers

What does the operation ~X + X yield for any signed integer X?

It results in 0. Signup and view all the answers

Study Notes

Intro to Hexadecimal

Base 16
Requires 16 symbols, 0..9 + A,B,C,D,E,F
Shorthand notation for binary
Written in C as 0xFA1D37B or 0xfa1d37b

Advantage of Hexadecimal

Easy to convert between hexadecimal and binary since one hex digit represents four bits
Example: 1011000100101111 = 1011 0001 0010 1111 = B 1 2 F = 0xB12F

Bytes

One byte equals 8 bits
Range of values in binary: 000000002 to 111111112
Range of values in decimal: 010 to 25510
Range of values in hexadecimal: 0016 to FF16

Data Sizes

Kilo, Mega, Giga prefixes can be confusing because they are used in both binary and decimal systems
This course uses the binary sense of Kilo, Mega, Giga etc.

Conversions

Every four binary digits convert to one hex digit and vice versa.
Example: 11001101 = 1100 1101 = CD

Boolean Algebra Truth Tables

AND:
- 0 & 0 = 0
- 0 & 1 = 0
- 1 & 0 = 0
- 1 & 1 = 1
OR:
- 0 | 0 = 0
- 0 | 1 = 1
- 1 | 0 = 1
- 1 | 1 = 1
NOT (Complement):
- ~0 = 1
- ~1 = 0
XOR:
- 0 ^ 0 = 0
- 0 ^ 1 = 1
- 1 ^ 0 = 1
- 1 ^ 1 = 0

Logic Circuits

Logic gates perform specific logic operations
They are the building blocks of digital circuits
OR gate has two or more inputs and one output
The output is high if any or all inputs are high

Bits

Computers are binary machines
Bits are represented as 0 and 1
Everything on a computer is encoded as a set of bits, including programs and data

Why Bits?

Electronic implementation is easier using bi-stable elements
Bits are more reliable to transmit on noisy and inaccurate wires

Bit Vector

A bit vector is a sequence of bits, but its meaning depends on the type of information it represents and the encoding used.
Without context, a bit vector has no inherent meaning.

Converting Binary to Decimal

Sum the value of each binary digit times its power of 2.
Example: 11001 = 1 * 24 + 1 * 23 + 0 * 22 + 0 * 21 + 1 * 20 = 25

Converting Decimal to Binary

Divide the number by 2
The remainder of each division becomes the binary digit
Repeat until the quotient is 0
Example: 13 = 1101
- 13 / 2 = 6 (remainder 1)
- 6 / 2 = 3 (remainder 0)
- 3 / 2 = 1 (remainder 1)
- 1 / 2 = 0 (remainder 1)

Converting Decimal to Hexadecimal

Divide the number by 16
The remainder of each division becomes the hex digit
Repeat until the quotient is 0
Example: 7562 = 1D8A
- 7562 / 16 = 472 (remainder 10 = A)
- 472 / 16 = 29 (remainder 8)
- 29 / 16 = 1 (remainder 13 = D)
- 1 / 16 = 0 (remainder 1)

Converting Hexadecimal to Decimal

Multiply each hexadecimal digit by its corresponding power of 16.
Sum the results.
Example: 1D8A = (1 × 16³) + (13 × 16²) + (8 × 16¹) + (10 × 16⁰) = 7562

Negation

To compute -X for a bit-vector X, compute the complement and add 1: -X = ~X + 1.
Example: X = 011001 = 25, ~X = 100110 = -26, ~X+1 = 100111 = -25

Addition for Unsigned Numbers

Standard addition function ignores carry bits and implements modular arithmetic.
Formula: UAdd(u , v) = (u + v) mod 2w

Addition of Signed Numbers

True sum requires w+1 bits
Addition ignores the carry bit
If TAddw(u, v) >= 2w–1, then sum becomes negative (positive overflow)
If TAddw(u, v) < –2w–1, then sum becomes positive (negative overflow)

Multiplication

Task: Computing Exact Product of w-bit numbers x, y (either signed or unsigned)
Ranges of results:
- Unsigned multiplication requires up to 2w bits to store result: 0 ≤ x * y ≤ (2w – 1)2 = 22w – 2w+1 + 1
- Two's complement smallest possible value requires up to 2w-1 bits: x * y ≥ (–2w–1)*(2w–1–1)= –22w–2 + 2w–1
- Two's complement largest possible value requires up to 2w bits (in one case): x * y ≤ (–2w–1)2 = 22w–2
Maintaining exact results would need to keep expanding the word size with each product computed.

Multiplication by Power of 2 (Left Shift)

Multiplication by a power of two is equivalent to the left shift operation.
Example: u * 2k is the same as u << k

Sign Extension

Task: Given a w-bit signed integer X, convert it to a (w+k)-bit integer X' with the same value.
Solution: Make k copies of the sign bit.
Example: X = xw-1 xw-2...x1x0, X' = xw-1...xw-1xw-1 xw-2...x1x0
C automatically performs sign extension when converting from a smaller data type to a larger one.

Truncating

Example: From int to short (from 32-bit to 16-bit)
High-order bits are truncated, altering the value
This non-intuitive behavior can lead to bugs in code.

Example:

int i = 1572539;
short si = (short) i;
printf(" i = %i\n si = %i\n\n ", i, si );

This prints:
```
i = 1572539
si = -325 
```

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Description

This quiz covers the basics of hexadecimal notation and its advantages over binary representation. It includes concepts like byte size, data conversions, and Boolean algebra truth tables. Perfect for students learning about number systems in programming.