CS132 - Numerical Representations and Fundamental Knowledge

What is a bus?

A shared communication pathway that transfers data, addresses, or control signals between components.

What is a bit?

A single binary digit: 0 or 1.

What is a byte?

Usually 8 bits.

What is a word?

The number of bits a processor can process as a unit, e.g. 16, 32 or 64 bits.

What is the most significant bit (MSB)?

The bit with the highest place value, usually the leftmost bit.

What is the least significant bit (LSB)?

The bit with the lowest place value, usually the rightmost bit.

What does active high mean?

A signal/component is active when its value is 1.

What does active low mean?

A signal/component is active when its value is 0. A bar over a signal name often means active low.

What base is decimal?

Base 10, using digits 0 to 9.

What base is binary?

Base 2, using digits 0 and 1.

What base is octal?

Base 8, using digits 0 to 7.

What base is hexadecimal?

Base 16, using digits 0 to 9 and A to F.

How do you convert decimal to another base?

Repeatedly divide by the target base, record remainders, then read the remainders from bottom to top.

How do you convert binary to octal?

Group binary bits in 3s from the right, because 2^3 = 8.

How do you convert binary to hexadecimal?

Group binary bits in 4s from the right, because 2^4 = 16.

How do you convert any base to decimal?

Multiply each digit by base^position and add the results.

Why is binary addition similar to decimal addition?

You add from the least significant bit and carry to the next column when the result is too large for one digit.

How can subtraction be treated as addition?

Use A - B = A + (-B), where -B is represented using two's complement.

What is signed magnitude?

A signed representation where the first bit is the sign bit and the remaining bits give the magnitude.

In signed magnitude, what do sign bits 0 and 1 mean?

0 means positive; 1 means negative.

Give one advantage of signed magnitude.

The sign is easy to inspect and sign/magnitude can be handled separately, useful for signed magnitude multiplication.

Give two disadvantages of signed magnitude.

It has +0 and -0, and addition/subtraction circuits are more complicated.

What is two's complement?

A binary representation for signed integers where negative x is found by inverting the bits of x and adding 1.

How do you find the two's complement negative of a binary number?

Invert all bits, then add 1.

How do you subtract using two's complement?

Convert the second number to two's complement, add it to the first number, and ignore any final carry beyond the fixed width.

Why is two's complement useful for subtraction?

The same binary adder circuit can perform addition and subtraction.

What is overflow?

A result is too large or too small to fit in the fixed number of bits available.

When can signed overflow occur?

When adding two positive numbers gives a negative result, or adding two negative numbers gives a positive result in fixed-width signed arithmetic.

What is biased form?

A way to store signed exponents by adding a fixed bias: stored exponent = real exponent + bias.

Why is biased exponent form useful?

It represents positive and negative exponents without using two's complement.

What is fixed point representation?

A representation where the binary point is assumed to be in a fixed position.

Give two advantages of fixed point.

It is simple, fast and predictable.

Give two disadvantages of fixed point.

It is less flexible and has a limited range/precision layout fixed in advance.

What is IEEE 754 floating point used for?

Representing very large and very small fractional numbers.

What are the three parts of an IEEE 754 floating point number?

Sign bit, biased exponent, and mantissa/significand.

What does the sign bit do in floating point?

It shows whether the number is positive or negative.

What does the biased exponent do in floating point?

It stores the scale/power of two, using a bias to allow positive and negative exponents.

What does the mantissa/significand do in floating point?

It stores the precision/detail of the number.

Fixed point vs floating point: when is fixed point preferable?

When hardware should be simple, fast and predictable, and the range of values is known.

Fixed point vs floating point: when is floating point preferable?

When a much wider range of very small and very large values is needed.

How do you multiply unsigned binary numbers?

Use long multiplication with binary rules: 0 times anything is 0; 1 times 1 is 1; then add shifted partial products.

How do you multiply signed magnitude numbers?

Ignore sign bits, multiply magnitudes as unsigned numbers, determine the sign separately, then attach the correct sign bit.

Why is two's complement multiplication more complex than signed magnitude multiplication?

The sign is built into the bit pattern, so a special algorithm such as Booth's algorithm may be needed.

Basic Booth's algorithm idea?

Look at pairs of multiplier bits: 10 subtract, 01 add, 00 or 11 do nothing, then arithmetic shift right and repeat.

How do you multiply two IEEE 754 floating point numbers?

XOR sign bits, add exponents and subtract the bias once, multiply mantissas, normalise, then round if needed.

Why is floating point multiplication simpler than floating point addition?

The exponents do not need to be aligned before multiplying mantissas.

What is a possible advantage of ternary/base-3 over binary?

It can represent more values per digit, so fewer digits may be needed for the same range.

What is a disadvantage of ternary/base-3 hardware?

Voltage ranges are closer together, making noise/error margins worse and hardware design more difficult.