Number Systems Different number systems Representation of numbers

Number Systems Different number systems Representation of numbers in binary Conversion between decimal and binary, Conversion between binary and hexadecimal Use of subscripts 2, 10 and 16 for bases

Number Systems �Decimal number system – Base 10 = 1, 2 , 3 4, 5, ect. . �Binary number system –Base 2 = 0001, 0010, 0011, ect… �Hexadecimal number system = Base 16 = 9, A, B, 4 C ect…

Decimal Number Systems �Decimal numbers are base 10 �They are made up of 10 numbers – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. �Combining the ten numbers will create units, tens, hundreds and thousands Hundreds Tens Units 341 102 101 100 300 40 1 300 + 40 +1 = 341

Split the following decimal numbers Hundreds Tens Units 550 Hundreds Tens 982 Units

Answers Hundreds Tens Units 550 102 101 100 500 + 50 + 0 = 55010 Hundreds Tens Units 982 101 100 900 80 2 900 + 80 + 2 = 98210

Binary Number System �Binary numbers are base 2 �Computer language �They are made up of 2 numbers – 1 and 0 Decimal Binary 010 02 510 1012 110 12 610 1102 210 102 710 1112 310 112 810 10002 410 1002 910 10012

Hexadecimal Number Systems �Hexadecimal numbers are base 16 �Computer memory locations �They are made up of 16 numbers Decimal Hex 010 016 510 516 1010 A 16 1510 F 16 110 116 610 616 1110 B 16 210 216 710 716 1210 C 16 310 316 810 816 1310 D 16 410 416 910 916 1410 E 16

Importance of Base numbers �Writing the base numbers is very important as; ◦ 1510 and 1516 are not the same number but without the base they would be both considered as the same number ◦ 1010 and 102 are not the same number as 102 represents 210

Complete the table Number 2010 2 A 16 10101012 10110 1516 1110001112 Number System

Answers Number System 2010 Decimal 2 A 16 Hexadecimal 10101012 Binary 10110 Decimal 1516 Hexadecimal 1110001112 Binary

Converting Binary to Decimal

Explanation 1. Write down the placement value on top of each number. 24 23 22 21 20 16 8 4 2 1 2. Write the values that are on (the ones with a one under them 3. Add the numbers together

Example �We want to convert 110012 to decimal 24 23 22 21 20 1 1 0 0 1 16 8 4 2 1 16 8 1 16 + 8 + 1 25

Working �Convert 1. 2. 3. 4. 5. the following to decimal 1010102 1110112 101010012 001112 1110101002

Answers �Convert 1. 2. 3. 4. 5. the following to decimal 1010102 = 4210 1110112 = 5910 101010012 = 16910 001112 = 10310 1110101002 = 46810

Converting Decimal to Binary

Method One 1. Write down the placement values of binary 124 64 32 16 8 4 2 1 2. Chose the numbers that add up to you decimal number 3. Put a 1 under the numbers used to add up to your decimal number

Example �Convert 124 64 0 0 4610 to binary 32 16 8 4 2 1 1 0 1 1 1 0 32 + 8 + 4 + 2 = 46 4610 = 001011102

Method Two �Divide the original number by 2 and write down the remainder even if it is 0 �Keep on dividing the decimal numbers by 2 until 1 is divided by 2 �Write down the remainders next to each other starting from the bottom moving upwards

Example �Convert 4610 to binary 46 / 2 = 23 r 0 23 / 2 = 11 r 1 11 / 2 = 5 r 1 5 / 2 = 2 r 1 2 / 2 = 1 r 0 1 / 2 = 0 r 1 Ans 4610 = 1011102

Working �Convert the following decimal numbers to binary 1. 1010 2. 6610 3. 12010 4. 3510 5. 8810

Answers �Convert the following decimal numbers to binary 1. 1010 = 10102 2. 6610 = 10000102 3. 12010 = 11110002 4. 3510 = 1000112 5. 8810 = 10110002

Converting Binary to Hexadecimal

Explanation �Split the binary number into groups of 4 1001110 = 0100 – 1110 �Write the 2 x on top of each number starting from the right �Add the numbers that are on �Write down the totals, 0 1 0 if a 0 total 1 is 1 1 0 larger than 9, convert 23 22 it 21 20 23 22 21 20 to the hex letter 8 4 2 1 NOTE: when we do not have enough bits lefts to create a group of 4 we add 0 s 4 14 4 E 16

Example �Convert 11001112 in Hexadecinal 0 1 1 0 0 1 1 1 23 22 21 20 8 4 2 1 6 7 6716

Working �Convert the following into Hexadecimal 1. 2. 3. 4. 5. 1110101002 11101112 1010102 11100012

Working �Convert the following into Hexadecimal 1. 2. 3. 4. 5. 1110101002 = 1 D 416 11101112 = 7716 1010102 = 2 A 16 1112 = 716 11100012 = 7116

Converting Hexadecimal to Binary

Explanation Write each individual number in the hexadecimal number eg B 416 2. Write the binary placement values for each hex number 3. List 1 s under the placement values that are on B = 11 4 1. Write the split 23 22 21 20 binary number 8 4 2 1 as one whole 1 0 1 0 0 number 101101002 4.

Example �Convert 2 C 16 to binary 2 C = 12 23 22 21 20 8 4 2 8 2 1 0 0 1 1 001011002 4 1 0 0

Working �Convert binary 1. 2. 3. 4. 5. AB 16 F 716 1516 CC 16 2216 the following hex numbers to

Answers �Convert the following hex numbers to binary 1. 2. 3. 4. 5. AB 16 = 101010112 F 716 = 111101112 1516 = 000101012 CC 16 = 11002 2216 = 00102

Converting Decimal to Hexadecimal

Method One �Divide the decimal number by 16 taking note of the remainders �Keep on dividing the whole number by 16 until the whole number obtained is 0. �Write down the remainders next to each other starting from the bottom, greater than 9 to 465 /changing 16 = 29 numbers r 1 ANS = 1 D 116 29 /letters 16 = 1 r 13 1 / 16 = 0 r 1

Example �Convert 80010 to hexadecimal 800 / 16 = 50 r 0 50 / 16 = 3 r 2 3 / 16 = 0 r 3 ANS = 32016

Method Two Convert the decimal number to binary 2. Convert the binary number to hexadecimal Eg, changing 45610 to hexadecimal 1.

Example �Convert 80010 to hexadecimal 512 256 128 64 1 1 0 0 32 16 8 4 2 1 1 0 0 0 512 + 256 + 32 = 80010 = 11001000002 0 0 1 1 23 22 21 20 8 4 2 1 3 0 0 1 0 23 22 21 20 8 4 2 1 2 32016 0 0 23 22 21 20 8 4 2 1 0

Working �Convert the following to Hexadecimal numbers 1. 34010 2. 11910 3. 6610 4. 2510 5. 11110

Answers �Convert the following to Hexadecimal numbers 1. 34010 = 15416 2. 11910 = 7716 3. 6610 = 4216 4. 2510 = 1916 5. 11110 = 6 F 16

Converting Hexadecimal to Decimal

Explanation �Writing down the placement values on top of each number starting with 160 �Multiply the top value with the hexadecimal number. �Add all the results Converting 43 A 16 to decimal 162 161 160 256 16 1 4 3 A (256 x 4) (16 x 3) (1 x 10) 1024 48 10 =1024+48+10 =108210