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Bit Representation

Module by: Don Johnson

Summary: How computers represent numbers in bits.

Computers express numbers in a fixed-size collection of bits, commonly known as the computer's word length. Today, word-lengths are either 32 or 64 bits, corresponding to a power-of-two number of bytes (8-bit "chunks"). This design choice restricts the largest integer (in magnitude) that can be represented on a computer.

Exercise 1

For both 32-bit and 64-bit integer representations, what is the largest number that can be represented? Don't forget that the sign bit must also be included.

Solution 1

For b b-bit signed integers, the largest number is 2b-1-1 2 b 1 1 . For b=32 b 32 , we have 2,147,483,647, and for b=64 b 64 , we have 9,223,372,036,854,775,807 or about 9.2×1018 9.218 .

When it comes to expressing fractions, positional notation is easily extended to negative exponents using the decimal point convention: Wherever the decimal point occurs in a string of digits, we know that the first digit to the left corresponds to the zero exponent, and the one to the right is -1 -1 . In this way, we have 2.5=2×100+5×10-1 2.5 20 5-1 . Computers don't use this convention to represent numbers with fractional parts; instead, they use a generalization of scientific notation. Here, a number (not necessarily an integer) x is expressed as mbe m b e , with the mantissa, m, lying within a proscribed range (scientific notation requires 1≤|m|<10 1 m 10 ), b is the base, and e is the integer exponent. Computers use a convention known as floating point, where base 2 is used and the mantissa must lie between one-half and one. Thus, two and one-half in the computer's number representation scheme becomes five-eighths times four: .101×22 .101 2 2 . The mantissa is stored in most of the bits in a computer's word, and the exponent stored in the remaining bits. Both the exponent and mantissa have sign bits associated with them. Thirty-two-bit floating point numbers, known as single-precision floating point, have eight bits representing the exponent (including the sign bit) and the remaining twenty-four representing the mantissa (again, also including a sign bit).

Exercise 2

What are the largest and smallest numbers that can be represented in 32-bit floating point? 64-bit floating point that has sixteen bits allocated to the exponent? Note that both exponent and mantissa require a sign bit.

Solution 2

In floating point, the number of bits in the exponent determines the largest and smallest representable numbers. For 32-bit floating point, the largest (smallest) numbers are 2±127=1.7×1038×5.9×10-39 2 ± 127 1.738 5.9-39 . For 64-bit floating point, the largest number is about 109863 10 9863 .

So long as the integers aren't too large, they can be represented exactly in a computer using the binary positional notation. Electronic circuits that make up the physical computer can add and subtract integers without error. However, this statement isn't quite true; when does addition cause problems? Floating point representation handles numbers with fractional parts, but only some with no error. Similar to the integer case, the number could be too big or too small to be so represented. More fundamentally, many numbers cannot be accurately represented no matter how many bits are used for the exponent and mantissa. For example, you know that the fraction 1/3 13 has an infinite decimal (base 10) expansion; no matter how many decimal digits you have in your calculator, you will never represent 1/3 13 with complete accuracy. It also has an infinite binary expansion as well.

Exercise 3

Can 1/3 13 be expressed in a finite number of digits in any base? If so, what bases do the job?

Solution 3

If the base is a multiple of three, then 1/3 13 will have a finite expansion. For example, in base 3: 1/3=0.13 13 0.1 , and in base 6: 1/3=0.26 13 0.2 .

Clearly, many numbers have infinite expansions, and this situation applies to binary expansions as well. Consequently, number representation and arithmetic performed by a computer cannot be infinitely accurate in most cases. The errors incurred in most calculations will be small, but this fundamental source of error can cause trouble at times.

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