Some groups yesterday observed that their work was going
on “in base 7” when working
on the 1/7 problem. This is mostly true; the possible remainders, from 0 to 6,
re present the
digits of base 7. A significant difference is that a number “in base 7” can have
several digits,
while the possible remainders in the 1
7 problem would all be single-digit numbers in base 7.
For example, in base 7, 5 + 4 = 12 (read as one seven and
two ones ). In the remainder
system, 5 + 4 = 2. We will be exploring different bases today, and remainder
systems
tomorrow. So, some of those loose ends we found yesterday probably won’t get
discussed
until tomorrow.
The division algorithm , used repeatedly, can make the
conversion from a base-ten number
to another base. Here’s how to convert the number 1234 to base seven:
Note that at each stage, the quotient becomes the next
divident, and the divisor is always
the new base. By reading the remainders in revserse, we can read the number in
the new
base: 1234 = (3412)7, a notation that means that 1234 is written as
3412 in base seven. The
“4” of this number actually represents 4(7^2) = 196 in base ten. Each digit has
a particular
place value.
1. Show, using place value, that (3412)7 is
equal to 1234 in base ten.
2. As practice, compute these values in base seven without leaving it: 3412 +
326;
3412 − 346; 3412 × 346; 3412 ÷ 22 (find quotient and remainder, both in base
seven).
In bases higher than ten, additional letters are used to
represent the single digits for “10”,
“11”, etc. Hexadecimal computer code is written in base 16, with the letters A
through F
representing 10 through 15.
3. Use the division algorithm to convert the base-ten
number 1234 to base two, to base
three, to base ten, to base sixteen.
Why ask to convert to base ten in Problem 3? Because it
can help you see why the
algorithm works!
4. Use the results from Problems 1 and 3 to explain why
the algorithm works. In
particular, why do we read the remainders in reverse? Why does the remainder “4”
in our example represent 4(7^2)?
5. Convert the base-ten number 65403 to base two, to base four, to base sixteen.
Do
you notice anything unusual going on?
6. Convert the base-seven number (34652)7 to base two, to base three,
to base sixteen.
Do this without converting to base ten first; in other words, carry out the
algorithm
using base- seven arithmetic . You may find your work in Problem 2 helpful.
7. Convert the base-sixteen number (7F09)16 to base four.
As in Session 1, long division can be carried out to find
the expansion of a fraction as a
decimal. In other bases, although it is a misnomer to call this expansion a
decimal, this is
still the commonly used term.
8. Write the base-ten numbers 3, 9, 27, 243, 1/2 , 1/3 ,
1/4 , 1/7 , 1/9 as decimals in base two and
in base three. You may find your work from yesterday helpful here. Don’t forget
that the fraction “1/7” in base two will not involve writing the numeral “7”.
9. Why are some of the fractions in Problem 8 easier to
write in one base or the other?
10. Find the period of repetition for each repeating
decimal you found in Problem 8. Is
any of this related to yesterday’s work?
11. Write the base-ten fraction 1/7 as a decimal in each
base between two and ten, using
long division in the new base. Do the expansions have anything in common? How
can you tell when a decimal expansion is about to repeat? As an extension, write
the prime fractions from Session 1 in each base between two and ten, looking for
similiarities and differences in the expansions.
Yesterday, a participant suggested that any rational
number (a number in the form p/q ,
where p and q are integers, and q is non zero ) must have a terminating or
repeating
decimal.
12. Can you explain why, using what you’ve learned about
remainders?
Is the converse true? That is, can every terminating or repeating decimal be
written as a
rational number? Here’s a method used by many algebra texts for converting
repeating
decimals into fractions. Suppose we want to convert
to a fraction. Let
. Then
Subtract the top equation from the bottom to obtain
999x = 324
This may or may not result in a simplified fraction , but
it will result in a rational number.
13. Try this method with the following decimals,
especially the last one (which is an
“eventually repeating” decimal):
14. Does this explain why every repeating decimal
expansion represents a rational
number (a fraction in the form m/n )? Explain why or why not.
15. Will every terminating decimal expansion represent a
rational number? Explain why
or why not.
16. Describe rules which will determine if a rational
number has a terminating,
repeating, or eventually repeating decimal expansion in base ten. How do these
rules
change for other bases?
