Numbers and Number Systems

Natural Numbers

• {1, 2, 3, ....}

Whole Numbers

• {0, 1, 2, 3, ...}

Integers

• Either signed or unsigned, denoted by  referring to the set of all integers

Decimal (Base 10) Digits 0 - 9

• Utilizes the digits 0 - 9

Rational Numbers

• Terminating decimals, or repeating decimals (P / Q) Where P & Q are both integers, and Q != 0.
• 1/1 = 1 is a rational number, 2/1 is a rational number.

Irrational Numbers

• Neither terminate nor repeat, ie:  π (Pi)

Real numbers

• Rational Numbers + Irrational Numbers

Binary

• Base 2
• Converting Whole numbers to binary is easy.
• Steps are as follows:

1. Divide by 2, remainder represents LSB (Least Significant Bit)
2. Continue to divide by two, until division by 2 is not possible. 
3. The final remainder will represent the MSB (Most Significant Bit)
4. Written MSB on left, LSB on right.

Example: 25(base ten) : 25 / 2 = 12 r. 1[LSB], 12 / 2 = 6, r. 0, 6 / 2 = 3 r.0, 3/2 = 1 r.1, 1/2 = 0. r 1[MSB]
Written: 11001

• Like all systems utilizes place value.  Ie:  10110 = (1 * 2^4) + (0 * 2^3) + (1 *2^2) + (1 * 2^1) + (0 * 2^0) = 22

Hexadecimal

• Base 16
• Utilizes digits 0 - 9
• Expands, to include A, B, C, D, E, F - A representing 10, continuing respectively
• Utilizes the same place value system as used in our number systems across the board, however with a base of 16.
• Example: 2BD = (2 * 16^2) + (11 * 16^1) + (2 * 16^0)
• Converting Whole numbers to hexadecimal is identical to converting whole numbers to binary except a divisor of 16 is used.   Example:  589 / 16 = 36 r. 13, 36 / 16 = 2 r. 4, 2 / 16 = 0 r. 2 , written: 24D
• Converting hexadecimal to binary, divide the Hex Digits into 4 bit patterns (nibbles), 24D for example = 0010 / 0100 / 1101 -> 001001001101
• Each hex digit = nibble, 2 Hex digits = 1 byte