24 Understand that a positional notation system can use any number of symbols as long as it uses at least 2. So you could have a base-3 system, or a base-17 system, or a base-1065 system, or whatever. The decision to use base-10 was really just an arbitrary historical decision, likely made because we have 10 fingers and 10 toes, so it seemed natural to people to use base-10. And in fact, some civilizations used other bases, like the ancient Babylonians used base-12, probably because it fit best with their calendar.
25 (positional notation using 8 symbols) octal base-8(positional notation using 8 symbols)So let's look at a positional notation system which uses something other than base-10. The base-8 system goes by the name octal.
26 Rather than contriving new symbols, we use the same Arabic numerals as we do in decimal; we just use 8 of them instead of all 10.
27 0710When you have the quantity seven and you wish to express one higher, the '7' cycles back to '0', and the digit to its left cycles to '1', so the quantity we call 'eight' in decimal is written as '1' '0' in octal.While it's very, very tempting to read '1' '0' as the quantity 'ten', it is rather, in octal, the same quantity which we write as '8' in decimal. (After a lifetime of habit, it's really hard to stop yourself from reflexively reading numerals as decimal numbers.)
117 recapEach digit is a coefficient of the number base raised to an increasing power.One technique works for all number base conversions.Shortcut conversions:decimal to binary (sum powers of 2)binary to decimal (subtract out powers of 2)octal and binary (1 octal digit = 3 binary digits)hex and binary (1 hex digit = 4 binary digits)Effectively, hex and octal serve as a compacter way of writing binary.
140 finite rational (rational with a fractional component which can be expressed with a finite number of digits)3/4-7/1138/201/3738/610.75-7.06.90.33…
141 a ratio with denominator 2a5b is finite in decimal a ratio with denominator 2a is finite in octal a ratio with denominator 2a is finite in hex a ratio with denominator 2a is finite in binaryall ratios which are finite in binary are also finite in decimalsome ratios which are finite in decimal are also finite in binary
142 rational as two integers 3/4Numerator:Denominator:-7/13Numerator:Denominator:
143 (the computing equivalent of radix-point notation) fixed-point(the computing equivalentof radix-point notation)57/8bInteger:Fraction: