Other Numbering Systems

What we have studied so far only applies to positive integers. However, real-world numbers are not always positive integers. Negative numbers and numbers with decimals are also used.

Floating-Point Numbers

So far, the only numbers we've dealt with are integers - numbers with no decimal point. Computers have a general problem with numbers with decimal points, because computers can only store fixed-size, finite values. Decimal numbers can be any length, including infinite length (think of a repeating decimal, like the result of 1 / 3).

The way a computer handles decimals is by storing them at a fixed precision (number of significant bits). A computer stores decimal numbers in two parts - the exponent and the mantissa. The mantissa contains the actual digits that will be used, and the exponent is what magnitude the number is. For example, 12345.2 can be represented as 1.23452 * 10^4. The mantissa is 1.23452 and the exponent is 4 with a base of 10. Computers, however, use a base of 2. All numbers are stored as X.XXXXX * 2^XXXX. The number 1, for example, is stored as 1.00000 * 2^0. Separating the mantissa and the exponent into two different values is called a floating point representation, because the position of the significant digits with respect to the decimal point can vary based on the exponent.

Now, the mantissa and the exponent are only so long, which leads to some interesting problems. For example, when a computer stores an integer, if you add 1 to it, the resulting number is one larger. This does not necessarily happen with floating point numbers. If the number is sufficiently big, adding 1 to it might not even register in the mantissa (remember, both parts are only so long). This affects several things, especially order of operations. If you add 1.0 to a given floating point number, it might not even affect the number if it is large enough. For example, on x86 platforms, a four-byte floating-point number, although it can represent very large numbers, cannot have 1.0 added to it past 16777216.0, because it is no longer significant. The number no longer changes when 1.0 is added to it. So, if there is a multiplication followed by an addition it may give a different result than if the addition is performed first.

You should note that it takes most computers a lot longer to do floating-point arithmetic than it does integer arithmetic. So, for programs that really need speed, integers are mostly used.

Negative Numbers

How would you think that negative numbers on a computer might be represented? One thought might be to use the first digit of a number as the sign, so 00000000000000000000000000000001 would represent the number 1, and 10000000000000000000000000000001 would represent -1. This makes a lot of sense, and in fact some old processors work this way. However, it has some problems. First of all, it takes a lot more circuitry to add and subtract signed numbers represented this way. Even more problematic, this representation has a problem with the number 0. In this system, you could have both a negative and a positive 0. This leads to a lot of questions, like "should negative zero be equal to positive zero?", and "What should the sign of zero be in various circumstances?".

These problems were overcome by using a representation of negative numbers called two's complement representation. To get the negative representation of a number in two's complement form, you must perform the following steps:

1. Perform a NOT operation on the number

2. Add one to the resulting number

So, to get the negative of 00000000000000000000000000000001, you would first do a NOT operation, which gives 1111111111111111111111111111110, and then add one, giving 11111111111111111111111111111111. To get negative two, first take 00000000000000000000000000000010. The NOT of that number is 11111111111111111111111111111101. Adding one gives 11111111111111111111111111111110. With this representation, you can add numbers just as if they were positive, and come out with the right answers. For example, if you add one plus negative one in binary, you will notice that all of the numbers flip to zero. Also, the first digit still carries the sign bit, making it simple to determine whether or not the number is positive or negative. Negative numbers will always have a 1 in the leftmost bit. This also changes which numbers are valid for a given number of bits. With signed numbers, the possible magnitude of the values is split to allow for both positive and negative numbers. For example, a byte can normally have values up to 255. A signed byte, however, can store values from -128 to 127.

One thing to note about the two's complement representation of signed numbers is that, unlike unsigned quantities, if you increase the number of bits, you can't just add zeroes to the left of the number. For example, let's say we are dealing with four-bit quantities and we had the number -3, 1101. If we were to extend this into an eight-bit register, we could not represent it as 00001101 as this would represent 13, not -3. When you increase the size of a signed quantity in two's complement representation, you have to perform sign extension. Sign extension means that you have to pad the left-hand side of the quantity with whatever digit is in the sign digit when you add bits. So, if we extend a negative number by 4 digits, we should fill the new digits with a 1. If we extend a positive number by 4 digits, we should fill the new digits with a 0. So, the extension of -3 from four to eight bits will yield 11111101.

The x86 processor has different forms of several instructions depending on whether they expect the quantities they operate on to be signed or unsigned. These are listed in Appendix B. For example, the x86 processor has both a sign-preserving shift-right, sarl, and a shift-right which does not preserve the sign bit, shrl.