TM112 Binary Data Representation

• TM112: Introduction to Computing and Information Technology 2 is the course the following information pertains too.
• This will cover block 1 (part 1) on the topic of binary data representation and computation

Introduction

• This section will provide a basic understanding of how computers represent and process data.
• Studying binary representations helps compare them and reason about their efficiency.
• Certain representations can lead to errors with far-reaching consequences.

Representing Integers and Text in Binary

• Symbols need to be distinguishable in any representation.
• Changes in electrical voltages or friction in mechanical systems can cause random fluctuations, or noise, which may distort how symbols are perceived.
• In a binary system, there are only two symbols, making them easier to distinguish.
• For example, Morse code specifies a dash should be three times as long as a dot.

Binary Representation Systems In Computers

• Early computers like the ENIAC tried to represent data using the base-10 system where 0 volts represented 0, 1 volt represented 1, etc with 9 volts representing the digit 9.
• Advantage of representing data in binary is that only two ranges of voltage need to be detected.
• A lot of circuitry distinguishes different voltages, which took up space and generated heat.

Binary to Decimal Conversion

• Binary numbers can be converted to decimal notation by adding the values based on place value.
• To convert 1001 to decimal notation, use a table.
• 1001 in binary = 8+1=9 in decimal.

Decimal to Binary Conversion

• The decimal number 25 can be converted to binary by dividing by 2 and noting the remainders
• The binary representation is read upwards; 25 in decimal = 11001 in binary

Representing Integers in Binary

• Unsigned integers are those greater than or equal to zero and are sufficient for non-negative values.
• The Decimal values that can be represented in binary depend on the # of bits available. With 3, 23 = 8 values. With 3 bits, encode unsigned integers from 0 to 7.
• With 3 bits, the largest decimal value is 7. Since there are 8 available codes, one code is used to represent 0.
• With n bits, 2n unsigned integers are representable, with the largest integer being 2n – 1.

Adding Unsigned Integers in Binary Notation

• Adding two positive integers is straightforward wherein the the values are added and the sign is automatically positive.
• If the values are represented as two unsigned integers in binary notation, their binary values can be added.
• Numbers are written one under the other, so that each column has the same place value
• The digits in each column are added, column by column from the right, carrying digits as necessary.
• To add the two unsigned integers 110 and 101 using the working below, the 'carry' digit must be indicated:
• 1 1 0
• 1 0 1 +
• 1 0 1 1

Sign-Magnitude Representation

• Signed magnitude is an intuitive method for representing unsigned numbers.
• The most significant bit (MSB) of a binary number is the "sign" of the number;
• An MSB of "1" denotes a negative number
• An MSB of "0" denotes a positive number
• In an N bit word signed magnitude system
• 1 bit signifies the number sign (MSB)
• N-1 bits are for the number magnitude
• 2N-1-1 signifies the largest integer
• -(2N-1-1) signifies the smallest integer
• Example: In an 8 bit word, the signed magnitude system gives the following decimal representations for 00000001 and 10000001:
• 00000001: MSB is 0, so it's positive. (0000001)2 = 110, so the number is +1
• 10000001: MSB is 1, so it's negative. (0000001)2 = 110, so the number is -1

Representing Text in Binary

• Most text encoding systems come from ASCII (American Standard Code for Information Interchange), developed in 1963.
• The original ASCII system encoded upper/lower-case letters, numbers, punctuation, other symbols, and control codes (carriage return, backspace, tab) in 7 bits.
• The encoding system became an 8-bit system to expand and include more symbols as computers became more common.
• When binary numbers were assigned to each character in the original ACSII system, the sequences of the alphabet/numerals aided a computer processor for easy common operations.
• To maintain backward compatibility, the original 127 ASCII codes were maintained in UTF-8 (Unicode Transformation Format-8) where the leftmost bit is padded out with a 0.
• Since 2007, UTF-8 uses up to 6 bytes to encode characters, which are in use across the world.

Floating-Point Numbers and Scientific Notation

• The decimal number 2343.56 can also be written as 23.4356 × 102, 0.234356 × 104, or 234356.0 × 10-2.
• The decimal point can 'float' to any position as long as the power of 10 is appropriate. Scientific notation is a special case of floating-point notation where there is a single non-zero digit between 1 and 9 (inclusive) to the left of the decimal point.
• Floating-point numbers in scientific notation - number 2343.56 = 2.34356 × 103.
• Number -0.000654 = -6.54 × 10-4 notation:
• Negative exponent (-4) indicates the decimal point moves 4 places left to return to the original.
• Scientific notation has three distinct parts:
• Sign
• Exponent (power of 10)
• Mantissa (decimal number part)

Representing Logic Operations and Logic Circuits

• Binary encodings are used to represent numerical and textual data.

• Operations, arithmetical operations (addition), & comparison operations (less than/equals), can be encoded as logic operations which act on the binary data representations.

• Encoding integers happens via binary representations, while the addition operator is encoded as logical operations, forming a 'truth table'.

• A truth table lists the possible input value combinations of input values and the resulting output value.

• We see how basic operations lead to logic circuits that perform complex operations.

• Simple operations are then used to build a logic circuit and add two binary numbers.

The NOT operation

• ‘Flipping’ a single bit or “a”: If x is 1, the result is 0, & if x is 0, the result is 1.
• The Not Operation is expressed as: x̅ or x’
• NOT operator: If x = 0, x̅ (x') = 1
• NOT operator: If x = 1, x̅ (x') = 0
• Electrical components represent fundamental operations or "logic gates", combined to create more complex operations.

The AND operation

• AND logical operations involve two inputs.
• A truth table for two binary inputs, x and y, has four rows:
• Where x = 0 and y = 0, xy = 0
• Where x = 0 and y = 1, xy = 0
• Where x = 1 and y = 0, xy = 0
• Where x = 1 and y = 1, xy = 1

The OR operation

• The Truth table for OR operations (designated by the symbol +):
• Where x = 0 and y = 0, x + y = 0
• Where x = 0 and y = 1, x + y = 1
• Where x = 1 and y = 0, x + y = 1
• Where x = 1 and y = 1, x + y = 1

Building Logic Circuits

• To build a logic circuit with two inputs, A and B, to test if B > A, start with a truth table:
• If B > A, the result is 1 (True), otherwise the result is 0 (False).
• To translate into a logic expression, follow this algorithm
• Identify the row where the outcome (B > A) is 1.
• If input A is 1, write A; otherwise write NOT A in the logic expression.
• If input B is 1, write B; otherwise write NOT B in the logic expression.
• Join with AND.
• The final equation is the sum of all the deduced logic expressions.
• The Resulting logic expression NOT A AND B tells tells us that the logic circuit is equivalent to each inputs combination with in the truth table using 2 logic gates"

What is Inside a Logic Gate

• A Logic gate comprises fundamental components of the on/off switch actions.
• Early computers use vacuum tubes or valves
• Modern computers incorporate transistors formed of layers of semiconducting material such as silicon.