Binary, Hexadecimal, and Base 10

8235455 656 DECIMAL SYSTEM OR Base 10 3547553 12875

The place values are powers of 10: 1’s  100 10’s  101 100’s  102 The decimal system is also referred to as base 10 A 10 in subscript is put beside any decimal number to show it is in base 10 (ex. 19310) Base 10 is the number system we normally use In base 10 the largest digit is 9 The place values are powers of 10: 1’s  100 10’s  101 100’s  102 1000’s  103 10,000’s  104 Etc.

1 1 1 1 1 1 BINARY 1 1 1 1 1 1 1

Binary is also referred to as base 2 A 2 in subscript is put beside any decimal number to show it is base 2 (ex. 1000112) In base 2, the only digits used are 1 and 0 The place values are powers of 2 first place value is 20  (1) second place value is 21  (2) third place value is 22  (4) fourth place value is 23  (8) fifth place value is 24  (16) Etc.

Binary Sample Place Value Chart 128 64 32 16 8 4 2 1 27 26 25 24 23 22 21 20

Converting Base 2 To Base 10 To convert base 2 (binary) to base 10 (decimal), Place the binary number from right to left in the place value chart. If the digit is a 1, then add the value of that place from the place value chart. If the digit is 0, add 0

16 8 4 2 1 1 1 2 = 2 = = = + 16 = 18 10

1 1 1 1 2 = = 4 = 8 = 16 = = + 64 = 92 10

Converting Base 10 To Base 2 To convert from decimal to binary we break up the decimal number into binary place values Find the largest place value you can divide the decimal number into Put a 1 in that place value Find out how much is left over, then see the next largest place value you can divide the leftover into (any number that can not divide into the number, put a zero) For example:

1024 512 256 128 64 32 16 8 4 2 1 1 1 1 13 = Remainders 5 1 1 1 1 1 1 114 = 50 18 2 2 2 1 1 1 1 420 = 164 36 36 4 4 4 1 1 1 1 1 1 1 1694 = 670 158 158 30 30 30 14 6 2

27A29B 83B38A298E 2784B47A HEXADECIMAL F83B9464 34B123E3 635F048B

Hexadecimal is also referred to as base 16 A 16 in subscript is put beside any hexadecimal number to show it is base 16 (ex. 836F89E16) In base 16, the digits 0-9 are used, as well as the letters A-F to represent the numbers 10-15 A=10 B=11 C=12 D=13 E=14 F=15 The place values are powers of 16 The first place value is 160  (1) The second place value is 161  (16) The third place value is 162  (256) The fourth place value is 163  (4096) Etc…

Base 10 vs. Base 16 1 2 3 4 5 6 7 8 9 10 A 11 B 12 C 13 D 14 E 15 F

Hexadecimal Place Value Chart 1048576 65536 4096 256 16 1 165 164 163 162 161 160

Converting Base 16 to Base 10 To convert a number from hexadecimal to decimal, find the value of each digit in the number, then multiply it by the value in the place value chart Then add all these values to get a total For example:

4096 256 16 1 8 3 2 D 16 2 13 = 32 1 = 768 = 1 + 32768 = 3 3 5 8 1 10

4096 256 16 1 2 A F 16 15 = = 2560 = + 8192 = 1 7 6 7 10

Converting Base 10 to Base 16 To convert a number from decimal to hexadecimal, find the largest place value that can fit into the number Find out how many times it can go into the number and write that digit If the digit is larger than 9, convert it to its corresponding letter Find the remainder (by subtracting the previous amount by the new amount) then repeat the above steps until there is no remainder For example:

65536 4096 256 16 1 3 3 5 8 1 8 3 2 D = 10 32768 768 32 13 Remainder 813 45 13

65536 4096 256 16 1 2 4 8 3 1 6 F F = 10 24576 240 15 Remainder 255 255 15

Converting Base 16 To Base 2 To convert from hexadecimal to binary, take each digit in the base 16 number and write out the binary equivalent Each digit must have four binary digits after converting it Combine all the binary numbers you just wrote out

Base 2 vs. Base 16 0000 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 8 1001 9 1010 A 1011 B 1100 C 1101 D 1110 E 1111 F

318 = 16 3 0011 1 0001 8 1000 0011 11 0001 1000 2

Converting Base 2 To Base 16 To convert binary to hexadecimal, first break the binary number into groups of four digits Convert these groups to decimal numbers Use these numbers as the digits for the hexadecimal number If the number is greater than 9, change it its corresponding letter (A - F)

0011 1101 1000 2 3 13 8 D 3 D 8 16