Whole numbers and numeration

1. Whole numbers and numeration Math 123

2. 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 Learn how to count in these bases. Become acquainted with the blocks. They are crucial for understanding place value systems, as well as operations with whole numbers.

3. Base 5 • Try to transfer what you just learned to base 5. Learn how to count in this base. • What comes after 245, 4445, 12345? • What comes before 405, 3005, 123405? • Is there 50 in base 5? Use blocks or draw.

4. What is a base? • What is going on when we go from 245 to 305, both in terms of blocks and in terms of numbers? How is this similar to going from 29 to 30 in base 10? • What is a long called in every base? • No matter which base you are in, you will say that you are in base 10. Why?

5. Homework assignment • Your first group assignment will be to do some research on ancient numeration systems, and talk either about • History of counting • A particular numeration system. • Here we will only briefly talk about properties of some ancient numeration systems.

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 Having worked in bases 2, 3, 4, 5, 6, 7, and 10, which all have place value, think about the following questions: • 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. 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.

14. Difficulties with place value • Examples: • Twenty-nine, twenty-ten, twenty-eleven • Twenty-nine, thirty-one • Children do not necessarily understand the concept of tens and ones; for example, it may not be clear that eleven is ten plus one • Difficulties with operations (you will see many examples of this).

15. Place value and operations • Think of children’s strategies you saw on Friday. Do these children understand place value? How does that help them add and subtract more easily? Give examples.

16. 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, ...

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.