Chapter 4 - Systems of Numeration
I think the beginning of the chapter is fun, but many people begin the old ”Why do we have to
learn this? When will I ever need to write Babylonian numbers in my “real” life?” The quick
answer is “ You probably won’t ever.” Still, it helps to better understand how our own system
works, and how some math was developed through history. It also gives us a chance to look at
the multicultural aspects of math and the people who work with it. To that end, you will be
asked to do a Biography project. (see the projects area)
Additionally, understanding how other bases work can help to understand how computers work
using the binary system. Understanding the idea that numbers can be represented in different
ways can lead to the idea of a cipher or code. Code breaking has been particularly important in
wartime, and much of our safety in commerce (especially electronic commerce) is based on
encryption. We also have lots of everyday coding such as Social Security numbers, bar codes,
telephone numbers and zip codes.
4.1 Additive, Multiplicative and Ciphered Systems of Numeration
OBJECTIVES - Students will be able to:
1. Define basic requirements of a numeration system
2. Translate from the Hindu-Arabic system to Egyptian, Roman, Greek and Chinese
systems.
3. Translate from Egyptian, Roman, Greek and Chinese systems to the Hindu-Arabic
system.
4. Use the basic principles to translate to and from unknown but clearly defined systems.
IMPORTANT Cultures
 Egyptian (c. 3000 BCE)
 Roman (Republic c. 580 BCE, empire 27BCE-395 CE)
 Ionic Greek (3000BCE, Greek 642-300 BCE, Hellenistic 408 BCE- 400CE)
 Traditional Chinese(c. 2000 BCE – estimates vary)
VOCABULARY –
 Additive system
 Ciphered system
 Deductive Reasoning
 Hindu Arabic system




Multiplicative system
Number
Numeral
Numeration system
COMMENTS A system of numeration consists of a set of numerals and a scheme or rule for combining
them to represent a number.
Additive systems- the number represented is the sum of the symbols.
With the Egyptian system “what you see is what you get” It is a purely additive. You simply
add up all the symbols. If you see three scrolls, five heels and a staff, you have 300+50+1 =
351. This is true even if written scroll, heel, heel, staff, scroll, scroll, heel, heel, heel. Order
does not matter.
Roman numerals do rely on order. They are read left to right. It is not a purely additive system;
it has both subtractive and multiplicative elements. In general you will not need more than 3 of
the same symbol. If a smaller numeral is placed in front of (to left of) a larger it indicates that
you should subtract the smaller from the larger. Thus IV means 1 less than 5. (or 5-1) VI is six
because it is in the proper order. No more than one smaller symbol may precede a larger one.
You cannot represent 8 by saying IIX. Also only powers of 10 can be subtracted and only from
the two numerals after them. Therefore, I (100 ) may be subtracted from V and X, X (101) may
be subtracted from L and C, and C (102) may be subtracted from D and M. But V, L and D may
not be subtracted and you may not subtract I from M or C, etc. In order that we may write even
larger numbers, and not have to use more than 3 of a symbol, we will use a bar over a numeral
to represent multiplication by 1000. Thus XIV would mean 14 multiplied by 1000, or 14 000.
Note: In some homework answers in previous editions of the book uses MMMM instead of IV .
We will use the bar.
Multiplicative systems have symbols for 1-9 and for powers of 10. the numeral placed before
the power of ten will be multiplied by it. So it might look like 4(100) 6(10) 3. This would mean
4X100+6X10+3.
Traditional Chinese is the representative system we will be learning. It is written vertically
and is read from the top down. This edition of the book introduces a symbol for zero which is
used when there are two or more “empty” places. I think there use is a little inconsistent. The
Chinese symbols tend to be particularly difficult for many people to draw. Please do not worry if
your numerals are not perfect. Mine are pretty poor, often. It takes a lot of practice to do them
well. We will hope that anyone of Chinese ancestry who can draw them will be patient and
forgiving. This is true of the Greek letters as well. (There will not be any ancient Egyptians
looking over our shoulders, so we can breathe easily over those! ☺)
Ciphered systems have numerals up to the base and for multiples of the base. The number
then is the sum of the numerals. Ionic Greek is the system we will use as an example. There is
also a multiplicative element as in the Roman numeral for numbers larger than 999 – if followed
by ( ´ ) a numeral is multiplied by 100. so ψ ´ would be 700 thousand. α ´ τ μ ε would be
1345. My type font is a bit different from the book’s. When a symbol does not look exactly the
same as you expect, chose the most reasonable one from the list.
Tables of symbols are available to you in the text and on a handout sheet. A copy of the sheet
will be available when you take your test, but you must know the rules for using them.