Chapter 3

1. Chapter 3 Whole Numbers Section 3.8 Place Value and Algorithms in Other Bases

2. 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.

3. 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

4. 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 203five 2110three 3210four 73462nine The digit 2 is in the 81 = 8’s place The digit 2 is in the 52 = 25’s place The digit 2 is in the 33 = 27’s place The digit 2 is in the 42 = 16’s place 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.

5. 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.

6. 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 Notice that when the numbers convert they stay in the same order. 23676 211five 1300four 112 23677 212five 1301four 113 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. 1302four

7. Lets convert some of these other numbers to base 10. 2012three 274eight Converting a number to a different base 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. remainders remainders quotients quotients 2467 2012three

8. 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. 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 base 10 Converting to base 10 1 1 1 1 1 3 2 0 1four 3 1 2 0four

9. 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. 0 1 1 1 0 1 0 1 5 5 0 6 2 1 1 0 3 1 2 0four 6 6 1 6 Try the following addition problems in the given bases. You have to figure out the basic addition facts as you are doing the problems. 0 1 1 1 1 0 1 0 3 2 0 0 3 3 1 5 4 3 1 0five 1 3 4 1 5six

10. 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. 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 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.

11. 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. 1 1 1 1 0 1 2 2 2 0 2 0 1 1 1 3 2 0 0 0 1 3 1 2 3 3 2 211332four 1 1 The lattice to the right demonstrates how to do the base 7 multiplication problem 26seven 34seven. 0 2 1 6 4 3 1 3 1 3 1 3 1313seven