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science/math/Binary System.md

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+# Binary System
+The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation.
+
+Negative numbers are commonly represented in binary using two's complement.
+
+## Two's complement
+Two's complement of an integer number is achieved by:
+- Step 1: Start with the absolute value of the number.
+- Step 2: inverting (or flipping) all bits – changing every 0 to 1, and every 1 to 0;
+- Step 3: adding 1 to the entire inverted number, ignoring any overflow. Accounting for overflow will produce the wrong value for the result.
+
+For example, to calculate the decimal number **−6** in binary:
+- Step 1: _+6_ in decimal is _0110_ in binary; the leftmost significant bit (the first 0) is the sign (just _110_ in binary would be -2 in decimal).
+- Step 2: flip all bits in _0110_, giving _1001_.
+- Step 3: add the place value 1 to the flipped number _1001_, giving _1010_.
+
+To verify that _1010_ indeed has a value of _−6_, add the place values together, but _subtract_ the sign value from the final calculation. Because the most significant value is the sign value, it must be subtracted to produce the correct result: **1010** = **−**(**1**×23) + (**0**×22) + (**1**×21) + (**0**×20) = **1**×−8 + **0** + **1**×2 + **0** = −6.
+
+| Bits:                | 1               | 0          | 1          | 0          |
+| -------------------- | --------------- | ---------- | ---------- | ---------- |
+| Decimal bit value:   | **−**8          | 4          | 2          | 1          |
+| Binary calculation:  | **−**(**1**×23) | (**0**×22) | (**1**×21) | (**0**×20) |
+| Decimal calculation: | **−**(**1**×8)  | **0**      | **1**×2    | **0**      |