2.0 COMPUTER SYSTEM 2.2 Number System and Representation

Learning Outcome At the end of this topic, students should be able to: 2.2.1 Binary Represent data in binary form 2.2.2 Hexadecimal Represent data in hexadecimal form 2.2.3 Conversion between binary and hexadecimal Convert from binary to hexadecimal Convert from hexadecimal to binary

Numbering System Introduction Concept A numbering system is a way of representing numbers. Common numbering system used is called the decimal numbering system / base 10 numbering system The decimal numbering system is the numbering system that represents all numbers using 10 symbols (0 – 9). Exp: 2310

Binary Numbering System 2.2.1 Concept The numbering system that has just two unique digits, 0 and 1, called bits. Exp: 10102 1 2 3 4 5 6 7 8 9 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001

Binary Numbering System 2.2.1 Why? Computer uses the binary numbering system to represent the electronic status of the bits in memory. It also is used for other purposes such as addressing the memory locations.

Hexadecimal Numbering System 2.2.2 Concept The numbering system that uses 16 different symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F). Exp: 53D16

Hexadecimal Numbering System 2.2.2 Why? Hexadecimal notation is a shorthand method for representing the binary digits stored in a computer. e.g.: 11001001101000012 – can easily be misread by people, hexadecimal notation groups binary digits into units of four, which in turn are represented by other symbols; i.e. C9A116

Hexadecimal Symbol Decimal Equivalent Binary Equivalent 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001 A 10 1010 B 11 1011 C 12 1100 D 13 1101 E 14 1110 F 15 1111

Conversion Between Binary and Hexadecimal 2.2.3 1. Decimal to Binary 2. Binary to Decimal 3. Decimal to Hexadecimal 4. Hexadecimal to Decimal 5. Binary to Hexadecimal 6. Hexadecimal to Binary

1. Decimal to Binary - (base 10 to base 2) Division – Remainder Method (Divide by 2, find the remainder) (2) Convert 15610 to binary 2 156 78 39 19 1 9 4 (1) Convert 2310 to binary 2 23 11 1 5 Ans : 101112 *Read the remainder from below Ans : 100111002

1. Decimal to Binary - (base 10 to base 2) Division – Remainder Method (Divide by 2, find the remainder) Try this: Convert 5610 to binary

2. Binary to Decimal - (base 2 to base 10) Place Value Method e.g. Convert 111012 to decimal Step 1 1 Step 2 16 (24 ) 8 (23 ) 4 (22) 2 (21) (20) Step 3 16 + 8 + 4 + 0 + X X X X X Place value = 2910 12

2. Binary to Decimal - (base 2 to base 10) Place Value Method Try this: Convert 1100112 to decimal Step 1 Step 2 Step 3 13

3. Decimal to Hexadecimal - (base 10 to base 16) Division – Remainder Method (Divide by 16, find the remainder) e.g. Convert 7710 to hex 16 77 4 13 = D Read the remainder as hex number = 4D16 14

3. Decimal to Hexadecimal - (base 10 to base 16) Division – Remainder Method (Divide by 16, find the remainder) Try this: Convert 9110 to hex 15

4. Hexadecimal to Decimal - (base 16 to base 10) Place Value Method e.g. Convert 4D16 to decimal Step 1 4 D Step 2 13 Step 3 16 (161) 1 (160) Step 4 64 + Decimal equivalent X X Place value = 7710 16

4. Hexadecimal to Decimal - (base 16 to base 10) Place Value Method Try this: Convert AF1016 to decimal Step 1 Step 2 Step 3 Step 4 17

5. Binary to Hexadecimal - (base 2 to base 16) Step 1: Convert Binary to Decimal Place value method Step 2: Convert Decimal to Hex Division – Remainder Method (Divide by 16, find the remainder) 18

5. Binary to Hexadecimal - (base 2 to base 16) Step 1: Place Value Method e.g. Convert 101012 to decimal Step 1 1 Step 2 16 (24 ) 8 (23 ) 4 (22) 2 (21) (20) Step 3 16 + 0 + 4 + X X X X X Place Value = 2110 19

5. Binary to Hexadecimal - (base 2 to base 16) Step 2: Division – Remainder Method (Divide by 16, find the remainder) e.g. Convert 2110 to hex 16 21 1 5 Read the remainder as hex number = 1516 20

5. Binary to Hexadecimal - (base 2 to base 16) Try this: Convert 111010012 to hex 1. Step 1 ? 2. Step 2 ? 21

5. Binary to Hexadecimal - (simplify method) 4 Bits = 1 Hex digit e.g. Convert 10101111012 to hex Group to 4 bits (R to L) Step 1 0010 1011 1101 Step 2 8421 Step 3 2 11 13 Step 4 B D Place value Decimal equivalent Hex equivalent = 2BD16 22

6. Hexadecimal to Binary - (base 16 to base 2) Step 1: Convert Hexadecimal to Decimal Place Value Method Step 2: Convert Decimal to Binary Division – Remainder Method (Divide by 2, find the remainder) 23

6. Hexadecimal to Binary - (base 16 to base 2) Step 1: Place Value Method e.g. Convert F116 to decimal Step 1 F 1 Step 2 15 Step 3 16 (161) (160) Step 4 240 + Decimal equivalent X X Place Value = 24110 24

6. Hexadecimal to Binary - (base 16 to base 2) Step 2: Division – Remainder Method (Divide by 2, find the remainder) e.g. Convert 24110 to binary 2 241 120 1 60 30 15 7 3 *Read the remainder from below = 111100012 25

6. Hexadecimal to Binary - (base 16 to base 2) Try this: Convert 1E16 to binary 1. Step 1 ? 2. Step 2 ? 30 11110 26

6. Hexadecimal to Binary - (simplify method) 1 Hex digit = 4 Bits e.g. Convert 9AF16 to binary Break up each digit Step 1 9 A F Step 2 10 15 Step 3 8421 Step 4 1001 1010 1111 Decimal equivalent Place value Group all digits = 1001101011112 27

TO BASE FROM BASE 2 10 (Decimal) 16 (Hexadecimal) Place Value Method (Binary) 10 (Decimal) 16 (Hexadecimal) Place Value Method Simplify Method (4 Bits = 1 Hex digit) - 28

TO BASE FROM BASE 2 (Binary) 10 (Decimal) 16 (Hexa decimal) Simplify Method (1 Hex digit = 4 Bits) Place Value Method - 29

TO BASE FROM BASE 2 (Binary) 10 (Decimal) 16 (Hexadecimal) – Division – Remainder Method (Divide by 2, find the remainder) – (Divide by 16, find the remainder) - 30

TRY This… Convert hexadecimal AF116 to binary number.

TRY This… Express 4010 in binary number.

TRY This… Convert the decimal number 86010 to hexadecimal number.

TRY This… Convert 110101012 to hexadecimal number.