Numbering Systems A number is a basic unit of mathematics. Numbers are used for counting, measuring, and comparing amounts. A numeral system is a set of symbols, or numerals, that are used to represent numbers. The most common number system uses 10 symbols called digits—0, 1, 2, 3, 4, 5, 6, 7, 8, and 9—and combinations of these digits. Numeral systems are classified by base. Base 2 - Binary Base 10 - Decimal Base 8 - Octal Base 16 - Hexadecimal base is the number of unique digits, including zero, that a numeral system uses to represent numbers. For example, for the decimal system the base is ten, because it uses the ten digits from 0 through 9. A base is rather simple to understand. The base is simply The highest value a position can reach in the system. Binary • Binary is a mathematical numbering system which is used a lot in computer science and used also for our mobile phones. Binary follows a system of base-2 numeral system, which is represented by two symbols; 1 and 0. • Digital systems such as digital clocks and phone signals, use binary. • Binary can be used to do mathematical calculations, basic calculations such as addition, subtraction and multiplication can be done as shown below. Addition Subtraction Multiplication • 1 + 1 = 0 (remainder 1 that • 0 - 0 = 0 • 0 x 0 = 0 is carried over to the next • 1 - 0 = 1 • 0 x 1 = 0 column) • 1 - 1 = 0 • 1 x 0 = 0 • 1 + 0 = 1 • 0 - 1 = 1 with a borrow from the • 1 x 1 = 1 • 0 + 0 = 0 next column Finger Binary Finger binary is a method created that allows you to count and show binary numbers using the fingers on your hands. You can use both of your hands but mostly the right hand is used. You can count from 0 to 31 using the fingers of a single hand, or from 0 through 1023 if you use both of your hands. Hexadecimal • The second number system I will discuss is hexadecimal. Hexadecimal is a number system like binary, but has a base, of 16 instead of 2. Hexadecimal uses 16 different symbols, which are shown below Hexadecimal • In order to convert binary into hexadecimal you have to first of all break the binary string in separate four bits. • Work out the decimal equivalent for each part of the new binary strings. • Using the table, give the hexadecimal equivalent • E.g. 110110100010 would be broken into 1101 1010 0010 • 1101 is 13 in decimal, 1010 is 10 in decimal, 0010 is 2 in decimal • Therefore the hexadecimal equivalent is DA2. • E.g. B (Hexadecimal) is 11 in decimal, what is 11 in binary? 1011. • E.g. E3 (Hexadecimal) is 14(E) and 3(3) in decimal, work each into binary separately. 14 is 1110, 3 is 0011. • - Therefore the binary equivalent is the two put together i.e. 11100011 Octal • One type of numbering system, besides binary and hexadecimal is The Octal numbering system. The Octal Numbering System is similar to the hexadecimal numbering system but in The Octal numbering system, a binary number is divided up into groups of only 3 bits, with each group or set of bits having a distinct value of between 000 (0) and 111 ( 4+2+1 = 7 ). • Octal numbering system only has a range of 8 digits, 0, 1, 2, 3, 4, 5, 6, and 7 making them a Base-8 numbering system. • Since there are only 8 distinct counting digits in the octal number system, 0-7, each digit has a value of ‘8’ , this starts from the smallest which is bit (LSB). Decimal Number 3-bit Binary Number Octal Number MSB Octal Number LSB 0 000 0 1 001 1 2 010 2 3 011 3 8 7 6 5 4 3 2 1 0 8 8 8 8 8 8 8 8 8 4 100 4 5 101 5 16M 2M 262k 32k 4k 512 64 8 1 6 110 6 7 111 7 8 001 000 10 (1+0) 9 001 001 11 (1+1) Continuing upwards in groups of three • Octal is base 8. Base 8 is where the only numbers you can use are zero thru to seven. i.e.: the decimal value for 1 is represented in octal as 1 but the octal value of 8 (Decimal) is shown as 10 the value of 9 (Decimal) is 11 in octal. Decimal Octal Decimal Octal Decimal Octal 1 1 11 13 30 36 2 2 12 14 40 50 3 3 13 15 50 62 4 4 14 16 60 74 5 5 15 17 70 106 6 6 16 20 80 120 7 7 17 21 90 132 8 10 18 22 100 144 9 11 19 23 500 764 10 12 20 24 1000 1750 .