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.

• Bit: The smallest unit in computers. It is either a 1 or a 0.
• Nibble: Half a byte, or 4 bits.
• Byte: 8 bits together form one byte, a number from 0 to 255. Bits in a byte are numbered starting from the right at 0.
• Short: Two bytes put together are 16 bits, forming a number from 0 to 65535. The low byte is the rightmost eight bits.
• Big Endian: Stores data big-end first. When looking at multiple bytes, the first byte (lowest address) is the biggest.
• Little Endian: Stores data little-end first. When looking at multiple bytes, the first byte is smallest.

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).

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.

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.

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.

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

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

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

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.

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.

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

Binary Stack Encoding

A stack of zeros and ones can be encoded in a single number by keeping with bit-shifting and incrementing.

• Pushing a 0 onto the stack is equivalent to doubling the number.
• Pushing a 1 is equivalent to doubling and adding 1.
• Popping is equivalent to dividing by 2, where the remainder is the number.

int
push(int stack, int bit) {
	return (stack << 1) + bit;
}

int
pop(int stack, int *acc) {
	*acc = stack & 0x1;
	return stack >> 1;
}

int
dup(int stack) {
	int a, res = pop(stack, &a);
	return push(push(res, a), a);
}

int
swap(int stack) {
	int a, b, res = pop(pop(stack, &a), &b);
	return push(push(res, a), b);
}

int
echo(int stack) {
	int bit;
	printf(" .. ");
	while(stack > 1)
		stack = pop(stack, &bit), printf("%d ", bit);
	printf("< \n");
	return stack;
}

The above defines the primitives of a small stack machine that utilizes a single number as memory, this strategy is at the core of the Fractran stack machine.

echo(dup(swap(push(push(push(push(1, 0), 1), 0), 1))));

0 1 0 1 swap dup .

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