Binary

• The minimum and maximum values for a given number of bits are 0 and 2^n - 1, for 2^n total values
• We can multiply binary numbers by breaking the multiplier down into numbers that are 2^n, performing multiple left shifts, and adding the results together
• When adding positive and negative binary numbers together, the final carry does not matter if it would normally cause an overflow - it can be disregarded
• Negative binary numbers can be represented using two's complement.
• A signed binary number uses the leftmost bit to determine if it is positive (0) or negative (1)
• We can perform binary subtraction by converting the number to be subtracted into being negative, and then performing addition
• To convert a positive binary number to a negative, start from the right and copy everything up to and including the first 1. Then, flip every remaining bit.
• We can use a floating point to represent fractions, where the decimals continue to decrease by half
• A floating point consists of the mantissa - the actual number, and the exponent - that determines where the floating point is
• A positive exponent means the final number will be bigger than the mantissa. A negative exponent means it will be smaller.
• We can backfill a number with 0s if it is positive or 1s if it is negative, because these are not significant
• An exponent tells us how many places to shift the mantissa
• Some numbers cannot be represented exactly in binary, only approximately represented, because continuing to half the size will never reach the accuracy some things need, eg 0.1
• The absolute error is the difference between the actual value and the expected value. The relative error is a percentage, which is the absolute error divided by the target value.
• Relative errors scale with magnitude, but absolute errors don't. An absolute error could be miniscule for one value but massive for another
• With fixed point, there is a trade off between range and precision depending on where the binary point is. Processing speeds are faster than floating point
• With floating point, the same number of bits can represent a large range or great precision depending on how they are distributed between the mantissa and exponent - just not at the same time. Processing speeds are slower than fixed point
• A normalised floating point number must always have the mantissa start with 01 if positive or 10 if negative
• To normalise a floating point number, move the point so that there is only one digit to the left (the bit that determines if it is positive or negative) - everything else should be 'decimal'. Then include the relevant exponent
• An underflow error occurs if a number is too small to be represented in the provided format. Tends to result in a stored value of 0
• An overflow error occurs if a number is too large to be represented in the provided format, such as the result of calculations.