ITSC-2181 Test 3 Number Systems

Converting to Decimal

Converting to Decimal

To convert a number from any base to decimal using transposed tables, you can follow these adjusted steps:

1. Create a Digit Row: Write down the digits of the number in a single row from left to right, matching their original order.

2. List Powers of the Base: Below the digits, list the corresponding powers of the base in a new row, starting with (0) for the rightmost digit and increasing as you move left.

3. Calculate Products: Directly below each power of the base, calculate the product of the digit and its corresponding power of the base.

4. Sum the Products: Add all the products listed in the bottom row to obtain the decimal value of the number.

Understanding Hexadecimal

The hexadecimal system uses sixteen symbols: 0-9 to represent values zero to nine, and A-F (or a-f) to represent values ten to fifteen. Here's a quick reference:

0 in Hexadecimal = 0 in Decimal

9 in Hexadecimal = 9 in Decimal

A in Hexadecimal = 10 in Decimal

F in Hexadecimal = 15 in Decimal

Why Hexadecimal is Important

• Compactness: Hexadecimal allows for a more compact representation of binary numbers. This compactness is particularly beneficial in computing for displaying memory addresses, representing color values in digital graphics, and encoding data where binary numbers can become lengthy and harder to interpret.

• Ease of Conversion: Converting between binary and hexadecimal (and vice versa) is straightforward compared to binary-to-decimal conversions. This ease of conversion makes hexadecimal a preferred choice for many applications in computer science and engineering.

• Alignment with Byte Boundaries: Since one hexadecimal digit represents four binary digits, hexadecimal aligns perfectly with byte boundaries (1 byte = 8 bits), simplifying the representation and manipulation of data at the hardware level.

Converting Binary to Hexadecimal

To convert a binary number to hexadecimal, follow these steps:

1. Group Binary Digits: Starting from the right, divide the binary number into groups of four digits. Add leading zeros if necessary to make the last group complete.

2. Convert Each Group: Convert each group of four binary digits to its hexadecimal equivalent using the reference table mentioned above.

3. Concatenate: Join the hexadecimal digits together in the same order to get the final hexadecimal number.

One of the earliest and simplest methods for representing negative numbers in binary is…

…the sign-and-magnitude representation

Understanding Sign-and-Magnitude Representation

Sign-and-magnitude representation uses the leftmost bit of a binary number to indicate the sign, with the remainder of the bits representing the magnitude (absolute value) of the number. In this system:

• The sign bit is 0 for positive numbers and 1 for negative numbers.

• The remaining bits indicate the magnitude of the number in binary.

For example, in an 8-bit sign-and-magnitude representation:

• The positive number +5 would be 00000101.

• The negative number -5 would be 10000101.

The first bit is the sign bit, and the next seven bits represent the magnitude of the number.

Sign-and-magnitude is one of several methods for representing negative numbers in binary, alongside…

…Two's Complement and One's Complement.

The One's Complement system is a method for representing signed numbers in binary form,…

offering an approach to encode negative numbers distinct from the more intuitive sign-and-magnitude representation.

Understanding One's Complement

In One's Complement, negative numbers are represented by inverting all the bits of their positive counterparts. To find the One's Complement of a binary number, you simply change all 1s to 0s and all 0s to 1s. This operation effectively creates a "mirror image" of the number across the midpoint of the possible value range in a given bit width.

Features of One's Complement

• Negative Zero: Like sign-and-magnitude, One's Complement has both a positive and a negative zero. For example, in an 8-bit system, 00000000 represents +0, and 11111111 represents -0.

• Bitwise Inversion: To negate a number (i.e., change its sign), all bits are inverted. This simple operation allows for the representation of negative numbers.

From One's Complement to Decimal

To convert a One's Complement number to decimal:

1. Determine if the number is negative (if the leftmost bit is 1).

2. If negative, invert the bits to find the magnitude.

3. Convert the binary magnitude to decimal.

4. If the original number was negative, negate the decimal result.

From Decimal to One's Complement

To convert a decimal number to One's Complement:

1. Convert the absolute value of the number to binary.

2. If the number is negative, invert the bits.

Basic Rules of Binary Addition

Binary addition follows three primary rules, similar to decimal addition but simplified due to only having two digits (0 and 1):

1. 0 + 0 = 0: Adding two zeros results in zero, with no carry to the next higher bit.

2. 1 + 0 = 0 + 1 = 1: Adding a zero and a one, in any order, results in one, without generating a carry.

3. 1 + 1 = 10: Adding two ones results in zero, with a carry of one to the next higher bit.

These rules are applied bit by bit, starting from the least significant bit (rightmost bit) and moving towards the most significant bit (leftmost bit), similar to how you would add numbers in decimal form.

Two's Complement representation is the most widely used method for encoding signed integers in computer systems. This system allows for…

…efficient arithmetic operations, including addition, subtraction, and comparison of both positive and negative numbers.

Fundamentals of Two's Complement

Two's Complement operates by allocating the highest-order bit as the sign bit: 0 for positive numbers and 1 for negative numbers. The remaining bits represent the magnitude of the number.

From Decimal to Two's Complement

To convert a decimal number to Two's Complement, start by determining whether the number is positive or negative.

• For positive numbers and zero, simply convert the decimal number to its binary equivalent and ensure it fits within the chosen bit width (e.g., 8 bits).

• If the number is negative:

  • Convert the absolute value of the decimal number to binary first.

  • Then, invert all the bits of this binary representation, which is equivalent to applying the One's Complement.

  • Finally, add 1 to the inverted binary number to obtain the Two's Complement representation.

This process guarantees that the highest-order bit (leftmost bit) acts as the sign bit, where 0 indicates a positive number and 1 indicates a negative number.