Decimal to binary number conversion

What is the binary form of -10? How it is ca开发者_StackOverflow中文版lculated?

To convert -10 (decimal) to binary:

Repeatedly divide the absolute value (|-10| = 10) of the number by 2 until you get 0 in the quotient:

(10 / 2 = 5 R 0)
(5  / 2 = 2 R 1)
(2  / 2 = 1 R 0)
(1  / 2 = 0 R 1) // zero is the value in the quotient so we stop dividing

Place the remainders in order to obtain the binary equivalent:

1010

For an 8-bit cell the answer is 0000 1010, 16-bit cell 0000 0000 0000 1010, and so on.

Take the one's complement by inverting the bits (we will assume an 8-bit cell holds the final value):

0000 1010
1111 0101 // bits are inverted

Now take the 2's complement by adding 1:

 1111 0101
+        1
----------
 1111 0110 // final answer

What happens with a 4-bit cell?

The one's complement would be:

1010
0101 // inverted bits

Taking the 2's complement produces:

 0101
+   1
 ----
 0110 // final answer for a 4-bit cell   

Since the number should be negative and the result does not indicate that (the number begins with 0 when it should begin with a 1) an overflow condition would occur.

Take the binary form of 10, invert all the bits, and add one.

10       0000 1010
invert   1111 0101
add 1    1111 0110

Follow this link. You can find the binary form of a negative number say -n by finding out the two's complement of n.

Your question has no "right" answer. First, you have to define the representation you want to use. Do you want to use two's complement, ones' complement, sign-magnitude, or something else? Then you have to define how many bits to use.

Let's say we are working with 5-bits wide representations.

+10: 0 1 0 1 0

Let's look at sign-magnitude system. In this system, the most significant bit (bit 5) is 1 if a number is negative, and 0 otherwise. The rest of the bits represent the magnitude (absolute value) of the number.

So we get:

-10: 1 1 0 1 0 (sign-magnitude, 5 bits)

Let's look at ones' complement now. Here, a negative number is represented by just changing 1s to 0s and vice-versa (hence the name ones' complement—you complement a number with respect to a long sequence of 1s).

So we get:

-10: 1 0 1 0 1 (ones' complement, 5 bits)

Finally, let's look at two's complement system. In this, we take a number in its ones' complement system, and then add 1 to it.

So we get:

-10: 1 0 1 0 1
             1
     ---------
     1 0 1 1 0 (two's complement, 5 bits)
     ---------

Thus, the binary representation of a negative number depends upon the system we use, and the number of bits available to us.

Also, you might have noticed the position of the apostrophe in ones' complement and two's complement. Why is is not one's complement or twos' complement? Then answer, from Knuth:

A two's complement number is complemented with respect to a single power of 2, while a ones' complement number is complemented with respect to a long sequence of 1s. Indeed, there is also a "twos' complement notation," which has radix 3 and complementation with respect to (2...22)₃