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Binary numbers are a base 2 numeral system.

A binary number is a number expressed in the base-2 numeral system, which uses only two symbols: 0 and 1. Each digit is referred to as a bit. 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.

Two of Leibniz's binary calculation examples

Conversion

Break down the binary value in chunks of 4, multiply each 1 by its equivalent value, either 8, 4, 2 or 1. Add the resulting numbers together to get the result. For example, the value 1100, or (8*1 + 4*1), is equal to C(decimal 12).

1101 0101(D5)
11010101Binary
1286432168421Values
128641641Result: 213

Binary Arithmetic

In the first column, divide the first number by 2 by removing the last bit, until 1 is reached. In the second column, multiply by 2 by adding an extra bit of 0. The answer is found by adding the numbers in the second column with odd numbers in the first column. A binary number ending with 1 is odd.

3519
100011 10011
10001 100110
1000 1001100
100 10011000
10 100110000
11001100000
1010011001

This example multiplies 35 by 19, to arrive at a result of 665. The result 1010011001 can be deconstucted as:

10 1001 1001 = 1 + 8 + 16 + 128 + 512 = 665

Binary numbers can be multiplied and divided by multiples of 2, by rotating one bit left to multiply by 2, or one bit right to divide by 2.

2200010110
44ROL00101100
11ROR00001011

AND, or "both", sets individual bits to 0. AND is useful for masking bits, for example, to mask the high order bits of a value AND with $0F: $36 AND $0F = $06. ORA(OR), or "either one or both", sets individual bits to 1. OR is useful for setting a particular bit, for example, $80 OR $08 = $88. EOR(XOR), or "one or the other but not both", inverts individual bits.

inputoutputANDORAEOR
01011
00000
11110
10011

To activate the 1st, 2nd and 4th bits:

00001011
unsigned char num = 0;
num |= (1 << 0);
num |= (1 << 1);
num |= (1 << 3);

To deactivate the 1st, 4th and 6th bits:

11010110
unsigned char num = 255;
num &= ~(1 << 0);
num &= ~(1 << 3);
num &= ~(1 << 5);

To read value of bit:

(num >> bit) & 1;

If Bit 7 is not set (as in the first example) the representation of signed and unsigned numbers is the same. However, when Bit 7 is set, the number is always negative. For this reason Bit 7 is sometimes called the sign bit.

BinaryUnsignedSigned
0010 00113535
1010 0011163-93
1111 1111255-1
1000 0000128-128

A fixed-point number is a number that has a fixed number of digits after the decimal point. If, for example, we use 8 bits to store a number with decimal points, we could decide to store it this way. The high and low nibbles have the same resolution.

Integer PartDecimal Points
000000000.0
001110103.a
010001014.5

To turn a regular integer into fixed point, shift left by the number of fractional bits(width << bits), and to turn a fixed point into integer, shift right by the number of fractional bits(width >> bits).

To multiply, you do the multiply, and then you shift right by the number of fractional bits.

(3.8 * 2.0) >> 8

To divide, you first shift the numerator left by the number of fractional bits, then you do the division.

(3.8 << 8) / 2.0

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