Transcript Whole numbers and numeration - Pacific Lutheran University

Whole numbers and numeration
Math 123
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.
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.
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?
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.
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.
• 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.
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?
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
• 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.
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.
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.
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.
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).
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.
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, bahgi-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-gipah, tah-gi-loh, ...
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.
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.
• 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.