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April 21, 2010

The Binary Side Of -1 + 1 = 0

Filed under: Main — Tags: — admin @ 12:01 am

Welcome to a nerdy blog post on how computers deal with negative numbers. I don’t mean numbers with a bad attitude, but rather numbers like -1, -4, and on down to -∞.

Computers count in binary, which is represented to us humans as 1s and 0s. So to a computer, the binary value 1101 means thirteen and not one thousand, one hundred, one.

In decimal, where most humans count, each digit represents a power of 10:

101 means 1 hundreds, 0 tens, and 1 ones.

In binary, where most computers count, each digit represents a power of 2:

101 means 1 fours, 0 twos, and 1 one. That translates into the value 5; binary 101 is decimal 5.

In most cases, even programmers don’t need to worry about instantly recognizing binary numbers. The computer easily translates them into decimal when necessary. After all, binary numbers can have dozens of digits and get long and confusing quickly:

The value 1,053,276,123 is represented in binary as 1111110110001111011011111011011.

One thing programmers do have to deal with, however, are negative numbers.

If you’ve been programming, then you know that there are such things as signed and unsigned integers.

An integer is simply a whole number, one that can easily be represented in binary. A signed integer is one that can be positive or negative in value. An unsigned integer is always positive.

Figuring out unsigned integers is easy. Here is an animation of a four-digit binary number counter, which displays values from 0000 through 1111, or zero through fifteen decimal:

So how do you make a negative binary number?

You’ll be careful to note that there is no – sign available above, which is how negative values are represented in decimal.

Well, actually, there is a – sign. It’s called a sign bit. When you specify a signed value in programming, you’re telling the computer that the far left bit determines whether the value is positive or negative.

When the bit is on, or 1, the value is negative. When the bit is zero, the value is positive.

The following animation shows the signed values 0000 through 1111, which represents values from 0 through 7 and then -8 through -1:

Once that far left bit hits 1, the values become negative. Now it may seem like the value 1001 should be -1. After all, binary 0001 is one and with the first bit set, 1001 seems fair game for -1. Not so, however.

Binary values wrap from the highest 0111 to the lowest 1000. That should make sense because 1001 is one greater than 1000.

And if it doesn’t make sense, just trust me: It works.

Or just keep staring at the animation above and you’ll get it, or you’ll start seeing things.

The size of the bits being handled define the range. For four bits (above), the ranges are from 0 to 15 unsigned and from -8 to 7 signed. Here are the ranges for other, more common integer values used in programming:

Type Bits Unsigned Range Signed Range
char 8 0 to 255 -128 to 127
short int 16 0 to 65,535 -32,768 to 32,767
int / long int 32 0 to 4,294,967,295 -2,147,483,648 to 2,147,483,647

Does this information help you in real life? Of course not! But it’s yet another curious thing about computers, and it also explains why you see values such as 127 or 255 or 65535 again and again in technology.

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