Transcript Section 3.8

Chapter 3
Whole Numbers
Section 3.8
Place Value and Algorithms in Other
Bases
In this section we will talk about how to count, write, add, subtract, and multiply
numbers in other bases. We will also talk about how you convert from one base to
another. This is not part of the early childhood curriculum but we will do this for
several reasons.
1. This will show you how much mathematics you have already learned and
have committed to memory.
2. You will get some perspective on what you need to learn when you (and the
students you will be teaching) are first learning to count, add, subtract and
multiply.
3. It has been a long time since you were in elementary school an you will be
reminded of the thought processes you need to go through to learn how to do
arithmetic.
Other Number Bases
To write numbers in other number bases we use the same principles as base 10
but not with all the symbols of base 10 and with different place values than base
10. The table on the next slide gives the other number bases the symbols they use
and their place values.
Base
Symbols
2
0,1
3
0,1,2
4
0,1,2,3
5
Place Values as Numbers
Place Values as Powers
… , 16, 8, 4, 2, 1
… , 24, 23, 22, 21, 1
… , 81, 27, 9, 3, 1
… , 34, 33, 32, 31, 1
… , 256, 64, 16, 4, 1
… , 44, 43, 42, 41, 1
0,1,2,3,4
… , 125, 25, 5, 1
… , 53, 52, 51, 1
6
0,1,2,3,4,5
… , 216, 36, 6, 1
… , 63, 62, 61, 1
7
0,1,2,3,4,5,6
… , 343, 49, 7, 1
… , 73, 72, 71, 1
8
0,1,2,3,4,5,6,7
… , 512, 64, 8, 1
…,8 ,8 ,8 ,1
9
0,1,2,3,4,5,6,7,8
… , 729, 81, 9, 1
… , 93, 92, 91, 1
10
0,1,2,3,4,5,6,7,8,9
… , 1000, 100, 10, 1
… , 103, 102, 101, 1
3
2
1
Writing Numbers in Other Bases
A number in another base is written using only the digits for that base. The base is
written as a subscripted word after it (except base 10).
For Example:
1032four is a legitimate base four number “Read 1-0-3-2 base four”
1542four is not a legitimate base four number not allowed 4 or 5
Place Values
The place values for each number in a different base start with the ones place as
the right most digit and go up by the next higher power of the base as you move to
the left.
Example:
What is the place value of the digit 2 in each of numbers below?
17526eight
The digit 2 is in the 81 = 8’s place
203five
The digit 2 is in the 52 = 25’s place
2110three
The digit 2 is in the 3 = 27’s place
3210four
The digit 2 is in the 42 = 16’s place
73462nine
3
The digit 2 is in the 90 = 1’s place
Counting
The next slide shows the first 17 base four numbers along with what they are in
base 10 and how they are represented with base four Dienes Blocks.
Base
Four
Base
Ten
Dienes
Blocks
Base
Four
Base
Ten
Dienes
Blocks
1four
1
1 unit
21four
9
1 unit
2 longs
2four
2
2 units
22four
10
2 units
2 longs
3four
3
3 units
23four
11
3 units
2 longs
10four
4
30four
12
11four
5
1 unit
1 long
31four
13
1 unit
3 longs
12four
6
2 units
1 long
32four
14
2 units
3 longs
13four
7
3 units
1 long
33four
15
3 units
3 longs
20four
8
100four
16
1 long
2 longs
3 longs
1 flat
Notice that the numbers in go in order just like in base 10 but only using the
symbols 0, 1, 2, 3. The base 4 Dienes blocks represent 1, 4, 16 values.
We can use this different number system to illustrate what it is like to try to learn to
count. Give the three numbers that come before and the three numbers that come
after each of the numbers below.
23675
210five
1233four
111
23676
211five
1300four
112
23677
23678
212five
213five
1301four
1302four
113
114
23679
214five
1303four
115
23680
220five
1310four
116
23681
221five
1311four
117
Converting a number to base 10
This process is a combination of
multiplication and addition. You multiply
each digit by its place value and add up
the results. Convert 1302four to base 10.
Notice that
when the
numbers
convert
they stay in
the same
order.
1302four
21=
2
04=
0
3  16 =
48
1  64 =
+ 64
114
Lets convert some of these other numbers to base 10.
2012three
274eight
21=
2
41=
4
13=
3
87=
56
09=
0
2  64 =
+ 128
2  27 =
+ 54
188
59
quotients
remainders
246710= 246
r 7
593= 19
r2
24610= 24
r 6
193= 6
r1
2410= 2
r 4
63= 2
r0
23= 0
r2
quotients
To convert a number from base
10 to a different base you keep
dividing by the base keeping
tract of the quotients and
remainders then reversing the
remainders you got. The
examples to the right first show
how to convert a base 10
number 2467 to base 10. Then
how you convert 59 to base
three. (Notice 59 agree with
what we got for the base three
number above.
210= 0
2467
remainders
Converting a number to a different base
r 2
2012three
Adding Numbers in Different Bases
Adding numbers in different bases requires the need to have learned the basic
addition facts in another base. The table below give the basic addition facts for
base four.
+
0four
1four
2four
3four
0four
0four
1four
2four
3four
1four
1four
2four
3four
10four
2four
2four
3four
10four
11four
3four
3four
10four
11four
12four
The reasoning for how we have gotten
some of the entries is shown below.
2four + 2four = 4 (base 10) = 10four
2four + 3four = 5 (base 10) = 11four
3four + 3four = 6 (base 10) = 12four
Below is shown how the standard addition algorithm is applied to solve
addition problems in base four.
Converting to
Converting to
base 10
base 10
1 1 1
1
1
1 2 0 3four
+
1 3 3 2four
3 2 0 1four
+
9
9
1
2
6
2
2
5
2 3 1 2four
+
2 0 2four
3 1 2 0four
1
+
2
8
2
3
4
1
6
Addition of Numbers Using the Lattice Method
Another way to organize the addition of numbers is to use the lattice method. It
works similar to how you use it with multiplication but you fill in the addition facts
in the correct columns. The first problem shows how to use this in base 10 to add
849+5767 and the second shows how it is used in base 4 to add.
+
5
0
8
4
9
7
6
7
1
1
5
0
0
6
+
1
5
6
2
1
2four
2
0
2four
1
6
1
3
0
2
3
6
1
1
1
1
0
2
0four
Try the following addition problems in the given bases. You have to figure out
the basic addition facts as you are doing the problems.
+
2
4
1
3five
1
3
4
2five
0
1
1
3
4
1
2
3
+
0
1
4
0
5
2six
5
3
2
3six
1
0
0
0five
1
3
1
3
0
3
4
1
1
5
5six
Multiplying Numbers in Different Bases
Multiplying numbers in different bases requires the need to have learned both the
basic addition and the basic multiplication facts in another base. The table below
give the basic addition facts for base four.

0four
1four
2four
3four
0four
0four
0four
0four
0four
1four
0four
1four
2four
3four
2four
0four
2four
10four
12four
3four
0four
3four
12four
21four
The examples to the right show how
to use the standard partial products
algorithm in different bases. The first
shows how to multiply 23four  31four.
The second shows how to multiply
1322four  3four.
The reasoning for how we have gotten
some of the entries is shown below.
2four  2four = 4 (base 10) = 10four
2four  3four = 6 (base 10) = 12four
3four  3four = 9 (base 10) = 21four
23four
1322four
 31four
 3four
3four
13
12four
32
20four
1  20
120four
3  20
210four
30  3
2100four
3  300
1200four
30  20
2033four
3000four 3  1000
11232four
Multiplication of Numbers in Other Bases Using the Lattice Method
The lattice method for multiplication can be used to organize how numbers are
multiplied. It relies on using the basic multiplication facts. Below to the right we
show how to do the base four multiplication problem 312four  231four. I have given
the base 4 basic multiplication facts below to the right.

0four
1four
0four
1four
2four
3four
0four
0four
0four
0four
0four
1four
2four
3four
3
1
1
2
2four
0four
2four
10four
12four
1
3four
0four
3four
12four
21four
1
2
1
1
1
0
1
2
2
2
0
1
1
0
0
3
0
2
0
3
1
3
2
3
2
3
1
2
211332four
2
The lattice to the right
demonstrates how to do the
base 7 multiplication problem
26seven  34seven.
6
1
1
3
1
0
2
6
1
4
3
1
1
3
3
4
3
1313seven