Bases: base 10, base 2, base 8, 16, (computers), other bases. fractions in bases. Experiment with our calculators base 8 and 16 capabilities, then learn how to calculate in bases by hand. Note that base 10 isn't special, it's just convenient because we have 10 fingers. We could count to 1023 on our fingers if we used base 2 though!
Algorithm for converting from base 10 to base n:
What's this business with the quotient? Following this algorithm for the "trivial" case of n=10, we see what's going on. When we "divide X by n", we have to be sure that (after we convert it), we put it somewhere that gives it back its "real" value. Just like 10 is worth 10 times as much as 1, moving a digit to the left in base n increases the value of that digit by a factor of n. 175 = 17*10 + 5, so the rightmost digit is 5. 170 is just 17 * 10, so the ones digit on 17 is the same as the 10's digit of 170. 7, the remainder we get from dividing 17 by 10 ( 17 = 1*10 +7), belongs one digit to the left of the 5, in the 10's digit of our answer. 1 = 0*10 + 1), so 1 goes to the left of the 5. The knew quotient is 0, so we stop. As you'd expect, 175 is "175" in base 10. :)
Homework: Convert a dozen multiple-digit numbers between base 10 and various other bases. (Make up your own numbers.) Do 2 arithmetic problems, each problem in 3 different number bases. Do each problem two ways: once, by calculating the answer directly in the base, and again, by working the problem in base 10 and converting your addends/factors answer into other bases. You will use your homework directly tomorrow.