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Number systems

Number systems. Converting numbers between binary, octal, decimal, hexadecimal (the easy way). Small numbers are easy to convert. But it helps to have a system for converting larger numbers to avoid errors. 12 10 = C 16. 5 10 -> 101 2. 1100 2 = 12 10. DEMONSTRATE.

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Number systems

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  1. Number systems Converting numbers between binary, octal, decimal, hexadecimal (the easy way)

  2. Small numbers are easy to convert • But it helps to have a system for converting larger numbers to avoid errors. 1210 = C16 510 -> 1012 11002 = 1210

  3. DEMONSTRATE Converting from base 10 (decimal)to base 2 (binary) example number = 42 • Write the powers of 2 in a row starting on the RIGHT side with a 1 • Keep doubling (*2) until you get to something greater than your number (42) 64 32 16 8 4 2 1 This is too big 1 0 1 1 3. Write a 1 underneath if that place value is used, 0 if not. subtract to find out what is left. 0 0 42 -32 ---- 10 10 - 8 ---- 2 2 -2 ---- 0 Watch Read your answer from left to right The number in binary is 101010

  4. DO TOGETHER Converting from base 10 (decimal)to base 2 (binary) example number = 7053 • write the powers of 2 in a row until you get to something > the number 8192 4096 2048 1024 512 256 128 64 32 16 8 4 2 1 Too big 0 1 1 0 1 1 1 0 0 0 1 1 1 7053 -4096 ------- 2957 2957 -2048 ------- 909 909 - 512 ------- 397 397 -256 ------ 141 141 -128 ------- 13 13 - 8 ----- 5 5 -4 --- 1 1 -1 --- 0 Do this together the number in binary is 1101110001101

  5. STUDENT’S TURN Do this one 15010 binary • 1 • 2 • 4 • 8 • 16 • 32 • 64 • 128 • 256 1 0 Too big 1 0 0 1 0 1 Click to see each digit that is needed. The answer is: 10010110

  6. To convert binary to decimal the number in binary is 10111001101 • Write the powers of 2 below each digit and only add the values with a 1 above them. • 0 1 1 1 0 0 1 1 0 1 • 1024 512 256 128 64 32 16 8 4 2 1 Start at the right and double each number 1024 + 256+128+64 + 8 + 4 + 1 = 1,485 Watch

  7. Your turn. Convert 1000100112 to decimal • 1 0 0 0 1 0 0 1 1 • 256 128 64 32 16 8 4 2 1 • 256 + 16 + 2+1 = • 275 • …. And now, for more about number systems.

  8. Part 2 • Number Systems

  9. Quick review • What’s 41 in binary? • 32 16 8 4 2 1 • 1 0 1 0 0 1 The answer is: 101001

  10. Quick Review: binary to decimal • 10011012 decimal • 64 + 8 + 4 + 1 • =77

  11. An Introduction toHexadecimal • 16 digits • Use letters when you run out of single digits • 0 1 2 3 4 5 6 7 8 9 A B C D E F • SO… 1110 = ?16 • B16 • 1510 = ? • F16 • 1610 = ? • 1016

  12. from base 10 to base 16 (decimal tohexadecimal) example number = 7053 • write the powers of 16 in a row until you get to one > the number • divide the number by each power of 16 and write the answer and save the remainder 65,536 4,096 256 16 1 • Too high • 7053/4096 = 1 R 2957 • 2957/256 = 11 R 141 • 141/16 = 8 R 13 • 13 ones • the numbers in hex are: • 1 2 3 4 5 6 7 8 9 A B C D E F (A=10…. F=15) • So your number is 1 11 8 13 = 1B8D16 Watch

  13. Do this one • 96210 hexadecimal • 3C216 • This is 3*256 + C(10)*16 + 2

  14. from hexadecimal (base 16) back to decimal Watch 1B8D16 • Write the number across a row. Write the powers of 16 below it. Multiply. Then add the products. • 1 B 8 D • =(1X4096)+ • (11*256)+ • (8*16)+(13*1) = • 4096 + 2816 + 128 + 13 = 7053 4096 256 16 1

  15. Do this one • A10E16 decimal • 41230

  16. Octal • Base 8 • Uses 8 different digits • 0 1 2 3 4 5 6 7

  17. from base 10 to base 8(decimal tooctal) example number = 7053 • write the powers of 8 in a row until you get to one > the number • divide the number by each power of 8 • write the answer and save the remainder • 32768 4096 512 64 8 1 • too high • 7053/4096 = 1 R 2957 • 2957/512 = 5 R 397 • 397/64 = 6 R 13 • 13/8 = 1 R 5 • = 5 ones • so your number in octal is 156158 Watch

  18. Do this one: • 94610 octal • 16628

  19. from octal (base 8) back to decimal 156158 • write the number • write the powers of 8 below it and multiply. then add the products. • 1 5 6 1 5 • 4096 512 64 8 1 • 1 *4096 = 4096 • 5 * 512 = 2560 • 6 * 64 = 384 • 1* 8 = 8 • 5 * 1 = 5 • added together = 7053 Watch

  20. Do this one • 20458 • 106110

  21. 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 Binary  hex  octal • If you can count from 1 to 15 in binary you have it made

  22. Binary to hexadecimal and hex to binary Watch • 4 binary digits correspond to 1 hexadecimal digit • Start grouping digits on the RIGHT side • 0000 0 • 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 To convert binary 1101011110 to hex Binary  Hexadecimal 11 0101 1110 3 5 E 35E16 Write this down the side of your paper. Hex  Binary 28D1 10 1000110100012

  23. Practice Hex  Binary  Hex • Convert E5816 to Binary • 111001011000 • Convert 110010110 to Hexadecimal • 196

  24. binary to octal and octal to binary • 3 binary digits correspond to 1 octal digit • 000 0 • 001 1 • 010 2 • 011 3 • 100 4 • 101 5 • 110 6 • 111 7 Binary to octal 10110011 10 110 011 263 • Octal to binary • 451 • 100 101 001 • 101001 Watch

  25. Practice Octal  Binary  Octal • Convert 3078 to Binary • 11000111 • Convert 110010110 to Octal • 646

  26. octal to hex and hex to octal. • Convert to binary, regroup and convert to other base. Octal to binary to hex 4518 100 101 001 100101001 1 0010 1001 12916 Watch

  27. Practice Octal  Hex • Convert 3078 to Hex • 11 000 111 first in binary • 11000111 • 1100 0111 divide into groups of 4 • 12 7 • C716

  28. Practice Hex  Octal • Convert 2B1D16 to Octal • 10 1011 0001 1101 first in binary • 10101100011101 • 10 101 100 011 101 divide into groups of 3 • 2 5 4 3 5 • 254358

  29. The End

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