Transcript Whole numbers and numeration - Pacific Lutheran University

Whole numbers and numeration
Math 123
September 17-19, 2008
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).
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.
Base 6
• Try to transfer what you just learned to base
6. Learn how to count in this base.
• What comes after 256, 5556, 12356?
• What comes before 406, 3006, 123406?
Use blocks or draw.
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?
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
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?
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.
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:
54 =625 53 =123 52 =25 51 =5 50 =1.
The number 234 in base 5 is equal to
2*25+2*5+4.
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.
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, ...
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?
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.