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Counting Systems

Understand the significance of the base 10 number system, its use in representing quantities, and learn how to convert to binary, octal, and hexadecimal systems. Discover the convenience of grouping binary numbers into sets of 3 or 4 digits.

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Counting Systems

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  1. Counting Systems

  2. Base 10 Number System Why do we use base 10?

  3. Base 10 Number System Why do we use base 10?

  4. Base 10 Number System Base 10 counting • Ten one digit numbers (0–9) • To expand beyond 1-digit, add a position on the left, representing the next power of ten. • Each position represents a power of ten (a positional number system). • E.g. 315,826 • Operations can take place at each position (e.g. adding two numbers by column with carry).

  5. Base 2 Number System Base 2 counting • Two one digit numbers (0–1) • To expand beyond 1-digit, add a position on the left, representing the next power of two.

  6. Base 2 Number System Base 2 counting • Two one digit numbers (0–1) • To expand beyond 1-digit, add a position on the left, representing the next power of two.

  7. Base 2 Number System Base 2 counting • Two one digit numbers (0–1) • To expand beyond 1-digit, add a position on the left, representing the next power of two.

  8. Base 2 Number System Base 2 counting • Two one digit numbers (0–1) • To expand beyond 1-digit, add a position on the left, representing the next power of two.

  9. Base 2 Number System Base 2 counting • Two one digit numbers (0–1) • To expand beyond 1-digit, add a position on the left, representing the next power of two.

  10. Base 2 Number System Base 2 counting • Two one digit numbers (0–1) • To expand beyond 1-digit, add a position on the left, representing the next power of two. Each position is a power of 2:

  11. Base 2 Number System Base 2 counting • Two one digit numbers (0–1) • To expand beyond 1-digit, add a position on the left, representing the next power of two. Alternating every other number.

  12. Base 2 Number System Base 2 counting • Two one digit numbers (0–1) • To expand beyond 1-digit, add a position on the left, representing the next power of two. Alternating in groups of 2

  13. Base 2 Number System Base 2 counting • Two one digit numbers (0–1) • To expand beyond 1-digit, add a position on the left, representing the next power of two. Alternating in groups of 4

  14. Base 2 Number System Base 2 counting • Leading zeros • Are insignificant, but often written to indicate the number of bits in a quantity. • e.g., 0110 = 110.

  15. Base 2 Number System Converting to and from binary • Base 10 to base 2 conversion: repeated division with remainders • Example: Convert 9210 to binary.

  16. Base 2 Number System Converting to and from binary • Base 10 to base 2 conversion: repeated division with remainders • Example: Convert 9210 to binary.

  17. Base 2 Number System Converting to and from binary • Base 10 to base 2 conversion: repeated division with remainders • Example: Convert 9210 to binary. Rewrite from the bottom up: 10111002

  18. Base 2 Number System Converting to and from binary • Base 2 to base 10 conversion: repeated multiplication and addition • Example: Convert 10111002 to decimal.

  19. Base 2 Number System Converting to and from binary • Base 2 to base 10 conversion: repeated multiplication and addition. • Example: Convert 10111002 to decimal. Add these up, base 10: 4+8+16+64=9210

  20. Base 8 and Base 16 Number Systems Binary is cumbersome • Long strings of 1’s and 0’s are hard to read. Group into sets or 3 (octal) or 4 (hexadecimal).

  21. Base 8 and Base 16 Number Systems Binary is cumbersome • Long strings of 1’s and 0’s are hard to read. Group into sets or 3 (octal) or 4 (hexadecimal).

  22. Base 8 and Base 16 Number Systems Binary is cumbersome • Long strings of 1’s and 0’s are hard to read. Group into sets or 3 (octal) or 4 (hexadecimal). • Example: Rewrite 1101111001012 as octal and hexadecimal. • Group by 3: 110 111 100 101  67458 • Group by 4: 1101 1110 0101  DE516

  23. Counting Systems Summary • Base 10 number systems are not universal. • Computers employ a base 2 (binary) system. • Number systems use positions representing powers of the base. • Converting from base 10 to another base involves division by the base and examining the remainders. • Converting from another base to base 10 involves multiplying by a power of the base and summing. • Octal and Hexadecimal are convenience groupings of binary numbers.

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