Binary

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Addition:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 + carry 1

Subtraction:
0 - 0 = 0
0 - 1 = 1 (borrow 1)
1 - 0 = 1
1 - 1 = 0

Multiplication:
x * 2^n = Shift all bits n places to the left.

Division:
x / 2^n = Shift all bits n plaves to the right.

Not:
Replace 1's with 0's and vice versa.

Or:
0 or 0 = 0
0 or 1 = 1
1 or 0 = 1
1 or 1 = 1

And:
0 and 0 = 0
0 and 1 = 0
1 and 0 = 0
1 and 1 = 1

XOR:
0 xor 0 = 0
0 xor 1 = 1
1 xor 0 = 1
1 xor 1 = 0