BINARY ARITHMETIC

Binary Addition

• 0 + 0 = 0 - When both bits are 0, the result is 0;
• 0 + 1 = 1 - When one bit is 0 and the other is 1, the result is 1;
• 1 + 0 = 1 - Similar to the previous case, the result is 1;
• 1 + 1 = 10 - When both bits are 1, the result is 10 in binary, which equals 2 in decimal;

Let’s illustrate binary addition with an example:

    0110
+   0100
---------
    1010

Starting from the rightmost bits:

• Adding the bits in the 1’s place: 0 + 0 = 0;
• Moving to the next bit: 1 + 0 = 1;
• Then: 1 + 1 = 10 (write down 0 and carry over 1);
• Finally: 0 + 0 (plus the carried-over 1) = 1;

So, the result is 1010 in binary.

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Decimal to Binary Conversion

Let’s convert 37 to Binary:

Division	Result	Remainder
37/2	18	1 (18*2 = 36) (36 + 1 = 37)
18/2	9	0
9/2	4	1
4/2	2	0
2/2	1	0
1/2	0	1

• I have the binary number 100101 when read from bottom to top, and then from left to right.

When I sum up these values:

> (1x2^0) + (0x2^1) + (1x2^2) + (0x2^3) + (0x2^4) + (1x2^5)
> 1 + 0 + 4 + 0 + 0 + 32 = 37