1. Whole numbers and numeration Math 123 September 17-19, 2008

2. Number vs. numeral A number is an idea that represents a quantity. A numeral is a symbol representing the number. Children need to learn that whether they count 3 fish or 3 cookies, the quantity in both cases is still 3. The book likes to explain things using sets. For later discussion, this definition is important: the number of elements of set A is written as n(A).

3. Manipulatives I am going to let you play with base blocks. Each group will get a different base to work with, but in any case. the names for the blocks in front of you are: Unit Long Flat Block Become acquainted with these blocks. They are crucial for understanding place value systems, as well as operations with whole numbers.

4. Base 6 Try to transfer what you just learned to base 6. Learn how to count in this base. What comes after 25 6, 555 6, 1235 6 ? What comes before 40 6, 300 6, 12340 6 ? Use blocks or draw.

5. Ancient numeration systems Write 9, 23, 453, 1231 in Egyptian, Roman, Babylonian and Mayan numerals. Which ones are easy? hard? similar? very different from our system? How are they similar? How are they different?

6. Properties of ancient numeration systems Note: from the historical perspective, it is fascinating to learn different number systems from the past and see how they led to the system we use today. The Egyptian system is additive since the values for various numerals are added together. If our system were additive, the number 34 would be read as 3+4 = 7.

7. The Roman numeration system is subtractive, since for example IV is read as V - I, which is 4. Similarly, XL is 40 etc. If our system were subtractive, 15 could be read as 5 - 1 = 4. The Babylonian numeration system is a place value system, like ours. We will return to place value in a moment. The Mayan system was the first to introduce zero.

8. Place value Discuss in your groups: Which properties does a place value numeration system have? What are the advantages of this type of system? What is the base of a system? Why do we use a base 10 system?

9. Properties of place value systems No tallies. Any amount can be expressed using a finite number of digits (ten in the case of our system). The value of each successive place to the left is (base)*the value of the previous place. In our system the base is 10. The values of the places are: … 100,000 10,000 1000 100 10 1

10. Expanded form: every number can be decomposed into the sum of values from each place. In the case of our system: 234 = 2*100 + 3 *10 + 4*1. The concept of zero.

11. Why base 10? Because we have ten fingers. It is actually not the most convenient base for computation. Base 8 or 16 would be more convenient.

12. What is the base? The easiest way to think about it: the number of units in a long. It is the number of units you trade in for the next place value, the long.

13. Recall Alphabetia This was a base 5 system. Here, every quantity can be written using 0, 1, 2, 3, 4. The values of the consecutive place values are: 5 4 =625 5 3 =123 5 2 =25 5 1 =5 5 0 =1. The number 234 in base 5 is equal to 2*25+2*5+4.

14. Why study different bases? Because you have been using the base 10 system for 15+ years. When you use the base 5 system, your experience is similar to the experience of a five-year old. Furthermore, properties of place value systems can be better seen in an unfamiliar system. Base 2 and base 16 are commonly used in computer science.

15. Some problems about place value The following shows an ancient number system that has place value. Enough information has been uncovered to be able to count in this system. If the following sequence begins at zero (i.e. “loh” = zero), can you determine the base of this system? loh, bah, noh, tah, goh, pah, bah-gi-loh, bah-gi-bah, bah- gi-noh, bah-gi-tah, bah-gi-goh, bah-gi-pah, noh-gi-loh, noh-gi-bah, noh-gi-noh, noh-gi-tah, noh-gi-goh, noh-gi- pah, tah-gi-loh,...

16. Which of the following, if any, is a place value system: a)A, B, C, BA, BB, BC, BAA, BAB, BAC, BBA, BBB, BBC… b)A, B, C, AA, AB, AC, AAA, AAB, AAC… c)A, B, C, AB, AC, BC, ABC…? Could you fix the ones that aren’t so they become place value systems?

17. Another ancient system has been discovered. Individually, the symbol # represents what we call “2” and @ represents what we call “5”. Together, though, # @ represents what we would call 21. If it is believed this system has place value, determine its base.

18. Confusing? How is it that 25 in base 6 is equal to 21 in base 10? How can two different numbers be equal? It is important to remember the properties of place value systems, in particular the expanded form. In base 6, 25 means 2*6 +5; in base 10, 21 means 2*10+1. It just to happens that both represent the same quantity. They are different representations of the same quantity.

19. I like to think of this in terms of manipulatives. In any base, 25 means 2 longs and 5 units. The only difference is how long a long is. In base 6, one long is 6 units, that is, we trade 6 units for one long. In base 10, we trade ten units for one long. This is why 25 represents a different quantity in the different bases.