Number Systems and Conversions

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Questions and Answers

What is the base used in the binary number system?

Base 16

Base 8

Base 10

Base 2 (correct)

In the number 7896, what value does the digit 8 represent?

8,000

800 (correct)

How are positional numbers represented in any base?

By squaring the digit values

By adding the digits together

By multiplying the digit by the base raised to its position (correct)

By converting them to base 10 first

Which of the following correctly indicates a base 10 number?

252ten Signup and view all the answers

What does the notation 0011b represent?

3 in base 2 Signup and view all the answers

In the decimal number 1247, what is the position of the digit 2?

2 Signup and view all the answers

What defines a binary digit?

A binary digit is referred to as a bit Signup and view all the answers

What does the most significant bit (MSB) represent in two's complement representation?

The sign of the number Signup and view all the answers

What is the result of converting the positive number 4 to two's complement in a 4-bit system?

0100 Signup and view all the answers

Why do computer systems often use two's complement instead of signed magnitude?

It simplifies hardware implementation Signup and view all the answers

Which of the following correctly describes the process of converting a negative number to two's complement?

Flip the bits and then add 1 Signup and view all the answers

What is the two's complement representation of the binary number `0011`?

<code>1101</code> Signup and view all the answers

What is the binary representation of the decimal number 13?

1101 Signup and view all the answers

Which character is not used in the hexadecimal number system?

G Signup and view all the answers

What is the base of the octal number system?

8 Signup and view all the answers

How many bits does one hexadecimal digit represent?

4 bits Signup and view all the answers

What are the steps to convert a decimal number to binary?

Divide by 2 and track the remainders Signup and view all the answers

What is the octal representation of the decimal number 27?

33 Signup and view all the answers

Which of the following correctly pairs a binary and its decimal equivalent?

1010 - 10 Signup and view all the answers

In the context of the binary number system, what does the digit '0' signify?

The absence of value Signup and view all the answers

What does dividing a decimal number by 2 and noting the remainder accomplish?

It aids in obtaining the binary representation Signup and view all the answers

What is the smallest unit of information in a digital computer?

Bit Signup and view all the answers

How many bits are contained in a byte?

8 bits Signup and view all the answers

What is represented by the least significant bit (LSB) in a binary number?

The smallest value contributing to the overall number Signup and view all the answers

What does the term 'endianness' refer to?

The order in which bytes are stored in memory Signup and view all the answers

What is the main role of the word size in a computer architecture?

To represent the largest number that can be stored Signup and view all the answers

Which of the following correctly describes a word in computing?

A collection of multiple bits processed as one unit Signup and view all the answers

In contemporary computers, what is a common word size used?

32 bits Signup and view all the answers

What is the position value of the most significant bit (MSB) in a 32-bit binary number?

31 Signup and view all the answers

If a word contains 2 bytes, how many bits does it have?

16 bits Signup and view all the answers

Which of the following correctly states the role of the binary representation in positive integers?

It defines the maximum limit of values that can be represented Signup and view all the answers

What is one disadvantage of using a special sign bit in binary representation?

It introduces the possibility of both +0 and -0. Signup and view all the answers

What is an effect of overflow in binary addition?

It leads to an erroneous result that cannot be represented. Signup and view all the answers

In a 1-bit binary addition, what do the carry-out bits indicate?

They indicate the need for additional bits in further addition. Signup and view all the answers

What should be done first when using signed magnitude binary arithmetic to add two numbers?

Convert both numbers to binary and arrange them as a sum. Signup and view all the answers

Why are arithmetic circuits complicated by the calculation of sign?

They must handle both +0 and -0 representations. Signup and view all the answers

What does the term 'carry bit' refer to in binary addition?

It is the portion that overflows to the next higher bit. Signup and view all the answers

When adding the binary numbers representing 107 and 46, what happens to the carry from the seventh bit?

It is discarded due to overflow. Signup and view all the answers

Which of the following is true regarding signed magnitude binary arithmetic?

It requires separation of sign and magnitude bits during addition. Signup and view all the answers

What is the impact of using only 1-bit for binary addition?

It restricts the range of representable sums. Signup and view all the answers

How are the results of binary addition verified when signed magnitude is used?

By converting results back to decimal. Signup and view all the answers

Study Notes

Number Systems

Numbers can be represented in any base
Humans typically use base 10
Base 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Number analysis based on position and powers of 10
Example: 3245 = (3 x 10³) + (2 x 10²) + (4 x 10¹) + (5 x 10⁰)

Positional Number Systems

Value of a digit depends on its position within the number
Position values increase from right to left, starting from position 0
The value of a digit is the digit multiplied by the base raised to the power of its position.

Binary Numbers

Base 2 is commonly used in computers.
Binary digits are 0 and 1.
A single binary digit is called a bit (binary digit).
The binary system corresponds to high/low signals (1/0).
Example: binary numbers with two bits: 00, 01, 10, 11

Decimal to Binary Conversion

Step 1: Divide the decimal number by 2 and note the remainder.
Step 2: Divide the quotient from the previous step by 2 and note the remainder.
Step 3: Repeat steps 1 and 2 until the quotient is 0.
Step 4: Write the remainders in reverse order.

Hexadecimal Numbers

Base 16
Uses digits 0–9 and letters A–F to represent the numbers 0–15
One hexadecimal digit corresponds to four bits in binary

Octal Numbers

Base 8
Uses digits 0–7
Each octal digit corresponds to three bits in binary

Bits, Bytes, and Words

Bit: A single binary digit (0 or 1)
Byte: A collection of 8 bits
Word: Two or more adjacent bytes

Representing Positive Integers

Binary numbers use 32 bits (or more)
Least significant bit (LSB): Position 0
Most significant bit (MSB): Position 31

Endianness

Important for how bytes are stored in computer memory.
Big-endian: First byte is most significant.
Little-endian: First byte is least significant.

Representing Negative Integers

Signed magnitude

Signed-magnitude Representation

One bit for sign (+ or - )
Remaining bits for magnitude
Two zeros are possible (+0 and -0)

Arithmetic Operations (Part 1)

1-bit binary addition: 4 possible cases
Carry bit propagation
Overflow detection during addition (carry in vs. carry out)

Arithmetic Operations (Part 2)

Two's complement representation: Negation is bit inversion + 1
Overflow detection in 2's complement addition
Binary subtraction using 2's complement: Subtract by adding the 2's complement of the subtrahend

Floating-Point Numbers

Representing very large and small numbers
Uses scientific notation (sign, mantissa, exponent)
Single-precision and double-precision representations

Non-numerical Data (Characters)

ASCII and Unicode
Represent characters using binary patterns

Non-numerical Data (Bitmaps)

Representing graphical images using a grid of pixels
Each pixel is represented by a bit (or bits).

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Description

This quiz covers the fundamentals of number systems, including decimal and binary representations. Understand how positional systems work and learn the steps for converting decimal numbers to binary. Test your knowledge on the importance of different bases in mathematics and computing.