3.3: Trading and Place Value

You will need:Base Two Models (Material Card 3), Calculator

You will be exploring more about place value in this exercise set. We'll start off by working with the Base Two Models on Material Card 3, which has a model set of cups (C), pints (P), quarts (Q), half-gallons (H), gallons (G) and double-gallons (D). Cut them out and use them to do exercises 1 - 4. Note that we'll be using triangles for our models except for a large square which will represent the double-gallon. By working with these models, it should be clear how many cups are in a pint, how many pints are in a quart, etc.

Exercise 1

Fill in the blanks with the correct numeral.

1. There are _____ cups in a pint.
2. There are _____ pints in a quart.
3. There are _____ quarts in a half-gallon.
4. There are _____ half-gallons in a gallon.
5. There are _____ gallons in a double-gallon.

Exercise 2

Suppose you were sick for nine days and a friend came over three times a day, each time leaving you a cup of soup. Suppose you never ate any of the soup, but instead kept putting each cup in the refrigerator in your garage. After those nine days were up, your refrigerator was stocked with all these individual cups of soup. You decide to consolidate these into as few containers as possible. You have five different kinds of containers –those holding cups, pints, quarts, half-gallons and gallons. Fill in each blank with the correct numeral.

1. Using your model cups, set aside three cups a day for nine days into a pile. There are _____ cups in this pile.
2. Trade in the cups for as many pints as you can. There are now _____ pint(s) and _____ cup(s) of soup. This is a total of _____ containers of soup.
3. Trade in the pints for as many quarts as you can. There are now _____ quart(s), _____ pint(s) and _____ cup(s) of soup. This is a total of _____ containers of soup.
4. Trade in the quarts for as many half-gallons as you can. There are now _____ half-gallon(s), _____ quart(s), _____ pint(s) and _____ cup(s) of soup. This is a total of _____ containers of soup.
5. Trade in the half-gallons for as many gallons as you can. There are now _____ gallon(s), _____ half-gallon(s), _____ quart(s), _____ pint(s) and _____ cup(s) of soup. This is a total of _____ containers of soup
6. Write down the information from part e in a consolidated fashion by filling in each space in the chart with the correct numeral.

   G	H	Q	P	C

7. Originally, from part (a), there were _____ cups of soup. By consolidating (see part e and f), the number of containers was reduced to only _____ containers of soup!

Exercise 3

Repeat exercise 2, but change the circumstances so that the friend only brings over two cups of soup a day for nine days.

a. Using your model cups, set aside two cups a day for nine days into a pile. There are _____ cups in this pile

b. Trade in the cups for as many pints as you can. There are now _____ pint(s) and _____ cup(s) of soup. This is a total of _____ containers of soup

c. Trade in the pints for as many quarts as you can. There are now _____ quart(s), _____ pint(s) and _____ cup(s) of soup. This is a total of _____ containers of soup.

d. Trade in the quarts for as many half-gallons as you can. There are now _____ half-gallon(s), _____ quart(s), _____ pint(s) and _____ cup(s) of soup. This is a total of _____ containers of soup

e. Trade in the half-gallons for as many gallons as you can. There are now _____ gallon(s), _____ half-gallon(s), _____ quart(s), _____ pint(s) and _____ cup(s) of soup. This is a total of _____ containers of soup

f. Write down the information from part e by filling in each space in the chart with the correct number G H Q P C

Exercise 4

Repeat exercise 2 one more time, but change the circumstances so that the friend brings over five cups of soup a day for nine days. Also, let's assume you also own a 2 gallon container, which we'll refer to as a double-gallon.

a. Using your model cups, set aside five cups a day for nine days into a pile. There are _____ cups in this pile

b. Trade in the cups for as many pints as you can. There are now _____ pint(s) and _____ cup(s) of soup. This is a total of _____ containers of soup

c. Trade in the pints for as many quarts as you can. There are now _____ quart(s), _____ pint(s) and _____ cup(s) of soup. This is a total of _____ containers of soup

d. Trade in the quarts for as many half-gallons as you can. There are now _____ half-gallon(s), _____ quart(s), _____ pint(s) and _____ cup(s) of soup. This is a total of _____ containers of soup

e. Trade in the half-gallons for as many gallons as you can. There are now _____ gallon(s), _____ half-gallon(s), _____ quart(s), _____ pint(s) and _____ cup(s) of soup. This is a total of _____ containers of soup

f. Trade in the gallons for as many double-gallons as you can. There are now _____ double-gallon(s), _____ gallons, _____ half-gallon(s), _____ quart(s), _____ pint(s) and _____ cup(s) of soup. This is a total of _____ containers of soup

g. Write down the information from part f by filling in each space in the chart with the correct number. G H Q P C

In the previous exercises, exchanges or trades were being made any time you had two of one container. You were basically employing the grouping technique we discussed back in Exercise Set 1 (look back at the bottom of page 8 of that exercise set). When employing the grouping technique, we are working in a particular base, depending on how many it takes to form a group. In the exercises you just completed, you were working in Base Two. The final answers that you obtained in Exercises 2, 3 and 4 are the way to express the original number of cups as a Base Two numeral. In exercise 4, five cups of soup for nine days translates to 45 in our Base Ten system (four groups of ten and five units). In Base Two, the final answer of 1 double gallon, 0 gallons, 1 half-gallon, 1 quart, 0 pints and 1 cup is written as a Base Two numeral like this: \(101101_{\text{two}}\). In other words, \(27_{\text{ten}} = 11011_{\text{two}}\). On the next pages, I'll explain in detail what this numeral means and how the place value system works in various bases. But first, there are some important points you need to know about how to write, read and say numerals in different bases.

To write a numeral in a given base, notice that the base is written out in words to the right and a little below the numeral, as in \(11011_{\text{two}}\). The only time you do not have to write the base is when it is a Base Ten numeral! A numeral without the base explicitly written out is assumed to be a Base Ten numeral.

• In exercise 3, the Base Ten numeral 18 (two cups for nine days) is written as \(10010_{\text{two}}\).
• In exercise 4, the Base Ten numeral 45 (five cups for nine days) is written as \(101101_{\text{two}}\).

It is imperative that you read and say these numerals in different bases correctly. It's much easier to make mistakes if you say the numeral wrong and much easier to avoid mistakes if you say the numeral correctly. The following paragraph explains the proper way to read or say the numerals.

In Base Ten, we have abbreviated ways of saying numerals out loud. For instance, "13" is read "thirteen". But, "\(13_{\text{five}}\)" is read "One, three, base five" and "\(246201_{\text{eight}}\)" is read "two, four, six, two, zero, one, base eight." It's extremely important that you learn to read and say "\(13_{\text{five}}\)" as "one, three, base five" AND NOT AS "thirteen, base five"! THERE IS NO THIRTEEN IN BASE FIVE!!! When you say thirteen, it refers to the base ten numeral meaning one group of ten and three ones. Although I haven't yet explained what we mean when we say and write "\(13_{\text{five}}\)", or "one, three, base five", it's important that you practice saying these numerals correctly from the start. Even though when we read it, we say the word "base", do not write the word "base".

The following are incorrect: \(^{13}5.......^{13}\text{Base } 5........^{13}\text{Base five}\)

Exercise 5

Write out in words how to say each of the following numerals.

1. \(302_{\text{six}}\) _____
2. \(1011_{\text{two}}\) _____
3. \(435_{\text{seven}}\) _____

Exercise 6

Write the given numerals

1. "Five, zero, one, six, base eight" _____
2. "one, zero, one, zero, zero, one, base two" _____

When you see a Base Two numeral like \(11011_{\text{two}}\) from exercise 2, you need to understand what each place value stands for. In all bases, the rightmost place value is the units or 1's place. Next, the number name "two" to the right of the numeral tells you this is a Base Two numeral, which means each place value as you move left increases by a multiple of 2. To the right is a chart showing the first eight place values in Base Two. They are written in Base Ten.

128

64

32

16

8

4

2

1

To check that the Base Two numeral \(11011_{\text{two}}\) really is \(27_{\text{ten}}\), put it into a chart with just five place values showing (since it's a five-digit numeral) and check the total value of the numeral. Look at the numeral written to the right. The place value for each digit is written below each digit of the numeral. The numeral is written in bold so it does not get confused with the place values. You check this exactly the way you did for the Mayan numerals:

\[\begin{align*} \text{(1 group of 16)} + \text{(1 group of 8)} + \text{(0 groups of 4)} + \text{(1 group of 2)} + \text{(1 group of 1)} &= (\mathbf{1} \times 16) + (\mathbf{1} \times 8) + (\mathbf{0} \times 4) + (\mathbf{1} \times 2) + (\mathbf{1} \times 1) \\[4pt] &= 16 + 8 + 2 + 1 \\[4pt] &= \mathbf{27}.\end{align*} \nonumber \]

All right! It really works! In fact, it's really easy to check the arithmetic for a Base Two numeral because for each place value containing a 1, that place value gets added to the total, whereas those place values containing a zero do not get added to the total. There's no multiplication you have to do –you can skip directly to the addition: 16 + 8 + 2 + 1 = 27.

\[\boxed{\frac{\mathbf{1}}{16} \frac{\mathbf{1}}{8} \frac{\mathbf{0}}{4} \frac{\mathbf{1}}{2} \frac{\mathbf{0}}{1} \ \mathbf{ two}}\nonumber \]

Let's convert \(\mathbf{10 \ 011 \ 010}_{\mathbf{two}}\) to Base Ten. Again, we'll write it with the place values shown under each digit and add up the place values containing a 1. This numeral converts to 128 + 16 + 8 + 2 = 154.

\[\boxed{\frac{\mathbf{1}}{128} \frac{\mathbf{0}}{64} \frac{\mathbf{0}}{32} \frac{\mathbf{1}}{16} \frac{\mathbf{1}}{8} \frac{\mathbf{0}}{4} \frac{\mathbf{1}}{2} \frac{\mathbf{0}}{1} \ \mathbf{ two}}\nonumber \]

Exercise 7

Fill in the chart to show what values are missing in this Base Two place value chart.

128

64

32

16

8

4

2

1

Exercise 8

Convert each Base Two numeral to Base Ten.

a. \(10 \ 011_{\text{two}}\)

b. \(1 \ 000 \ 001_{\text{two}}\)

c. \(111 \ 111_{\text{two}}\)

Let's convert some numerals in other bases. Consider the Base Five numeral, \(24_{\text{five}}\). First, we need to establish the place values in Base Five just like we did for Base Two.

Exercise 9

Fill in the chart to show what values are missing in this Base Five place value chart.

25

1

In Base Five, \(\mathbf{24}_{\mathbf{five}}\) is a two digit numeral. Note there are 2 groups of 5 and 4 units or \(2 \times 5 + 4 \times 1 = 10 + 4\) = 14.

\[\boxed{\frac{\mathbf{2}}{5} \frac{\mathbf{4}}{1} \ \mathbf{ five}}\nonumber \]

Exercise 10

Convert \(\mathbf{13}_{\mathbf{five}}\) to a Base Ten numeral. Remember not to read this as "thirteen"!!!

Here is how to convert \(\mathbf{31204}_{\mathbf{five}}\) to Base Ten. Look below to understand this computation:

\[\begin{align*} \mathbf{3} \times 625 + \mathbf{1} \times 125 + \mathbf{2} \times 25 + \mathbf{0} \times 5 + \mathbf{4} \times 1 &= 1875 + 125 + 50 + 4 \\[4pt] &= \mathbf{2054}. \end{align*} \nonumber \]

\[\boxed{\frac{\mathbf{3}}{625} \frac{\mathbf{1}}{125} \frac{\mathbf{2}}{25} \frac{\mathbf{0}}{5} \frac{\mathbf{4}}{1} \ \mathbf{ five}} \nonumber \]

Let's take the Base Three numerals \(\mathbf{1 \ 002 \ 021}_{\mathbf{three}}\) and \(\mathbf{22122}_{\mathbf{three}}\) and convert to Base Ten. First, we should make a place value chart for Base Three.

Exercise 11

Fill in the missing place values for Base Three.

243

9

To convert \(\mathbf{1 \ 002 \ 021}_{\mathbf{three}}\), look at the above chart to understand this computation:

\[\mathbf{1} \times 729 + \mathbf{2} \times 27 + \mathbf{2} \times 3 + \mathbf{1} = 729 + 54 + 6 + 1 = \mathbf{790}. \nonumber \]

To convert \(\mathbf{22122}_{\mathbf{three}}\), look at the above chart to understand this computation:

\[\mathbf{2} \times 81 + \mathbf{2} \times 27 + \mathbf{1} \times 9 + \mathbf{2} \times 3 + \mathbf{2} = 162 + 54 + 9 + 6 + 2 = \mathbf{233}. \nonumber \]

Exercise 12

Convert the following Base Three numerals to Base Ten.

1. \(200 \ 112_{\text{three}}\) = ____
2. \(12 \ 002 \ 110_{\text{three}}\) = ____
3. \(1 \ 111 \ 111_{\text{three}}\) = ____

Exercise 13

Figure out the base in each place value chart and fill in the missing place values for each

a. Base ____ 36

b. Base ____ 7

c. Base ____ 81

d. Base ____ 64

e. Base ____ 64

f. Base ____ 10000

g. Base ____ 12

h. Base ____ 11

Another way to write out the values in a place value chart for a given base is by using exponents. Each place value is a power of the base. When writing it out this way, the actual Base Ten values are not explicitly written out. The following is another way to write a place value chart for Base Seven:

\(7^{11}\)

\(7^{10}\)

\(7^{9}\)

\(7^{8}\)

\(7^{7}\)

\(7^{6}\)

\(7^{5}\)

\(7^{4}\)

\(7^{3}\)

\(7^{2}\)

\(7^{1}\)

\(7^{0}\)

Did you remember that 70 = 1? Any nonzero number raised to the zero power equals 1. For example, \(3^{0} = 1, 15^{0} =1, 1^{0} = 1, 10^{0} = 1, 546^{0} = 1\) and so on.

Exercise 14

Don't panic, but it's time to generalize just a little about the place value system for any given base. Consider Base n –yes, it's a variable just like in algebra! Look at the place value chart below and fill in the missing place values for Base n.

\(n^{11}\)

\(n^{7}\)

\(n^{3}\)

\(n^{0}\)

If you are converting very large numbers, it might be useful to write out a place value chart using the base written to a given exponent to save time and space.

For instance, let's say you were asked to convert the numeral \(100 \ 200 \ 020 \ 000 \ 100_{\text{three}}\) to Base Ten. This represents a very large number. But there are a lot of zeros for most of the place values. In fact, this fifteen digit numeral only has four nonzero digits. So why bother writing out a chart with all those place values? Instead, we know each place value is a power of three, since this is a Base Three numeral. In Base Ten, this numeral is:

\[\begin{align*} 1 \times 3^{14} + 2 \times 3^{11} + 2 \times 3^{7} + 1 \times 3^{2} &= 1 \times 4782969 + 2 \times 177147 + 2 \times 2187 + 1 \times 9 \\[4pt] &= 4782969+ 354294 + 4374 + 9 \\[4pt] &= 5,141,646 \end{align*} \nonumber \]

Exercise 15

\(100 \ 200 \ 020 \ 000 \ 100_{\text{three}}\) can be written as \(1 \times 3^{14} + 2 \times 3^{11} + 2 \times 3^{7} + 1 \times 3^{2}\). This is called expanded notation. Pay close attention to the fact that the 1 in the third place from the right is 3 to the second (not the third) power. Write the following numerals in expanded notation:

1. \(3 \ 000 \ 600 \ 020 \ 000 \ 000_{\text{eight}}\) ____
2. \(3 \ 000 \ 040 \ 020 \ 000 \ 000_{\text{five}}\) ____
3. \(400 \ 030 \ 000 \ 000 \ 002_{\text{eleven}}\) ____
4. \(100 \ 100 \ 000 \ 010 \ 000_{\text{two}}\) ____

Exercise 16

Refer to the numeral, \(3 \ 040 \ 001 \ 000 \ 002_{\text{nine}}\). For each nonzero digit in the numeral, write the place value of that digit as a power of nine. Do not write out the actual Base Ten numeral.

1. 1 ____
2. 2 ____
3. 3 ____
4. 4 ____

If a number is written in expanded notation, you can reverse the process and write it as a number in the base used. For instance, \(4 \times 7^{13} + 5 \times 7^{10} + 2 \times 7^{8} + 3 \times 7^{3} + 6 \times 7^{0}\) can be written as the Base Seven numeral \(40 \ 050 \ 200 \ 003 \ 006_{\text{seven}}\). NOTE: A number in the ninth place from the right is really the eighth power of the base, as shown by the placement of 2 in the numeral \(40 \ 050 \ 200 \ 003 \ 006_{\text{seven}}\).

Exercise 17

Rewrite the following numbers that are in expanded notation to a numeral in the base indicated:

1. \(2 \times 5^{9} + 4 \times 5^{8} + 1 \times 5^{6} + 3 \times 5^{2}\) in Base Five ____
2. \(3 \times 8^{10} + 4 \times 8^{7} + 1 \times 8^{3} + 6 \times 8^{2}\) in Base Eight ____
3. \(7 \times 12^{9} + 8 \times 12^{5} + 4 \times 12^{3}\) in Base Twelve ____
4. \(2 \times 3^{8} + 1 \times 3^{6} + 1 \times 3^{4} + 2 \times 3^{2}\) in Base Three ____

Exercise 18

a. If a Base Six numeral has eighteen digits, what is the place value of the first (leftmost) digit written as a power of six? ____

b. If a Base Nine numeral has twenty digits, what is the place value of the first (leftmost) digit written as a power of nine? ____

Take another look at some numerals in some bases besides Base Ten:

\(1 \ 100 \ 101_{\text{two}}\)

\(212 \ 201_{\text{three}}\)

\(42101_{\text{seven}}\)

\(80034_{\text{nine}}\)

Did you notice that there are only 0's and 1's in Base Two numerals? That's because if there was a 2 or higher for one of the placeholders, it could be traded in for the next place value to the left. Using physical models at the beginning of this exercise set, two cups could be exchanged for a pint, two pints could be exchanged for a quart, etc. Although we only went up to the double-gallon which contained 32 cups, there is really no limit to the number of place values a numeral has in a place value system. Also, one doesn't have to come up with a new name for each place value (as in "heel bone" or "scroll" in Egyptian).

Think about any Base Ten numeral. There are ten possible digits (or symbols) for each placeholder in the numeral 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. There is no separate single digit for the number ten (10); it is made up of a 1 and 0, which means one group of ten and zero units! In Base Two, there are only two possible digits for each place value in the numeral: 0 and 1. There is no need or use for the symbol (or digit) "2" when writing a numeral in Base Two.

Exercise 19

How many possible digits can each place value in Base Three have? _____ What are those possible digits? ____

Exercise 20

How many possible digits can each place value in Base Four have? _____ What are those possible digits? ____

Exercise 21

How many possible digits can each place value in Base Five have? _____ What are those possible digits? ____

Exercise 22

How many possible digits can each place value in Base Six have? _____ What are those possible digits? ____

Exercise 23

How many possible digits can each place value in Base Eight have? _____ What are those possible digits? ____

Exercise 24

Can the digit (symbol) "7" be used in a Base Seven numeral? _____ Explain your answer.

Exercise 25

a. What is the lowest base in which the digit "6" may appear? _____

b. What is the highest base in which the digit "6" may appear? _____

Do you see a pattern developing? Just wait until you see what happens in Base Eleven, Twelve and Thirteen!

Exercise 26

How many possible digits for each place value do you need for Base Eleven? _____

Exercise 27

How many possible digits for each place value do you need for Base Twelve? _____

Exercise 28

How many possible digits for each place value do you need for Base Thirteen? _____

In Base Eleven, you need eleven different symbols (or digits) for possible placeholders. The problem we encounter is that we only have these ten recognizable digits –0,1,2,3,4,5,6,7,8 and 9. Remember that 10 is made up of two separate digits –it isn't a single symbol!! So we need to introduce a new symbol to represent ten in Base Eleven and higher. The convention is to use the letter "T". Similarly, we use the letter "E" to represent the number eleven in Base Twelve and higher and the letter "W" to represent the number twelve in Base Thirteen and higher. Since the second place value from the right represents the base you are working in, you never need a separate symbol to represent the value of the base or any number higher than the base. That's why there are only 0's and 1's and no 2's in Base Two and why there are only 0's, 1's, 2's, 3's and 4's and no 5's in Base Five, etc.

The next few problems are worked exactly like those you have done so far, except the bases are higher than ten. You will need to remember the values of the new "digits" T, E and W when working these problems.

144

12

1

Let's convert the numeral \(TE5_{\text{twelve}}\) to Base Ten. To remember the place values, we might want to draw a Base Twelve place value chart for three digits as shown to the right.

Then, \(TE5_{\text{twelve}} = 10 \times 144 + 11 \times 12 + 5 \times 1 = 1440 + 132 + 5 = \mathbf{1577}\).

Pay close attention to the difference between \(2T9_{\text{eleven}}\) and \(2109_{\text{eleven}}\).

To convert \(2T9_{\text{eleven}}\) to Base Ten, first consider the Base Eleven place value chart shown.

1331

121

11

1

\(2T9_{\text{eleven}} = 2 \times 121 + 10 \times 11 + 9 \times 1 = 242 + 110 + 9 = \underline{361}\).

Now, let's convert \(2109_{\text{eleven}}\) to Base Ten. Looking at the Base Eleven place value chart, \(2109_{\text{eleven}}\) represents \(2 \times 1331 + 1 \times 121 + 0 11 + 9 \times 1 = 2662 + 121 + 0 + 9 = 2792\)

If someone told another person to convert "two, ten, nine, base eleven" as opposed to "two, T, nine, base eleven," it might not be clear what they meant. One person might write \(2109_{\text{eleven}}\) (which would convert to 2792 in Base Ten, as illustrated above) whereas another person might write \(2T9_{\text{eleven}}\) (which would convert to 361 in Base Ten, as illustrated above). It's important to remember to use the "T" to represent the number ten in Base Eleven and higher or it might be written as "10" which is two separate place values. \(2T9_{\text{eleven}}\) is a three digit numeral and it is clear that there is a ten in the middle digit's place value. On the other hand, \(2109_{\text{eleven}}\) is a four digit numeral –that is not a ten in the middle –and it represents a completely different number than \(2T9_{\text{eleven}}\) !

Let's try another one. We will convert \(TEW_{\text{thirteen}}\) to Base Ten. First, we should fill in a place value chart for Base Thirteen.

2197

169

13

1

Then, \(TEW_{\text{thirteen}} = 10 \times 169 + 11 \times 13 + 12 \times 1 = 1690 + 143 + 12 = \underline{1845}\)

Exercise 29

Convert each of the following to a Base Ten numeral. Show your work.

a. \(47E_{\text{twelve}}\) =	b. \(T74_{\text{eleven}}\) =	c. \(TTT_{\text{thirteen}}\) =
d. \(2034_{\text{twelve}}\) =	e. \(1025_{\text{eleven}}\) =	f. \(1028_{\text{thirteen}}\) =

Exercise 30

Convert each of the following to a Base Ten numeral. Show work

a. \(110_{\text{two}}\) =	e. \(110_{\text{six}}\) =	i. \(110_{\text{eleven}}\) =
b. \(110_{\text{three}}\) =	f. \(110_{\text{seven}}\) =	j. \(110_{\text{twelve}}\) =
c. \(110_{\text{four}}\) =	g. \(110_{\text{eight}}\) =	k. \(110_{\text{thirteen}}\) =
d. \(110_{\text{five}}\) =	h. \(110_{\text{nine}}\) =	l. \(110_{\text{twenty}}\) =

Exercise 31

Without doing any computations, circle which numeral has the larger value. Explain your reasoning. Try to be mathematically clear in your explanation. Notice the sequence of digits is the same for each numeral. Only the base is different.

[Hint: It has to do with the place values.]

\[13 \ 201 \ 154 \ 320 \ 050 \ 146_{\text{eight}} \ \mathbf{OR} \ 13 \ 201 \ 154 \ 320 \ 050 \ 146_{\text{eleven}} \nonumber \]

Exercise 32

Write \(4000T00000E000_{\text{twelve}}\) in expanded notation

Exercise 33

Write \(6 \times 13^{14} + 10 \times 13^{11} + 11 \times 13^{7} + 12 \times 13^{2}\) as a base thirteen number in standard form