CS255 Sylabus
Numeric values
• Numeric value are something intrinsic

One would ideally represent "numeric values" in a universal manner, such as:

• This is obviously very clumsy (try larger values :-)

• So humanoids have invented many different representations for numerical values

(This practice is obviously very important for their survival...)

• The most popular representation NOW (that was not always the case !) for numerical value is the decimal number system

This system is based on the following ten familiar looking symbols:

 ``` 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ```

I am sure you are thoroughly familiar with this decimal number system, in fact, so familiar that you do not even think about what decimal numbers actually mean...

• There are other representations for numerical values invented by humanoids.

A famous example is the number system invented by a class of humanoids that we call Romans

Their number system goes like:

 ``` I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, .... ```

Chinese numbers:

BTW, notice there is no symbol for ZERO. Chinese character for ZERO is:

A not-so-famous system is the Egyptian number system:

 = 1 = 10 = 100 = 1000 = 10000 = 100000 = 1000000

Note:

 Repesentations for numbers is an example of a code !!!

There are many other humanoids who have invented their own representation systems for numerical values, among others: Greeks (they use the Greek alphabet), Chinese (I'll show you in class...), etc.

Here is a copy of a page from a book of my 6 yr old first grader (in 2003) that show a number of number systems used in the other cultures: click here

• It is important to know that:

• A value does not depends on the representation system used:

• If you see that there are 4 students in the classroom, no matter how you represent this number, there will be 4 students, no more and no less

• As I mentioned above: a value is an intrinsic property....

• Roman Arithmetic....

• How can I use Roman numerals to do arithmetic problems?

 Let's start with an addition problem: 23 + 58. In Roman numerals, that's XXIII + LVIII. We'll begin by writing the two numbers next to each other: XXIII LVIII. Next, we rearrange the letters so that the numerals are in descending order: LXXVIIIIII. Now we have six I's, so we'll rewrite them as VI: LXXVVI. The two Vs are the same as an X, so we simplify again and get LXXXI, or 81, as our final answer. (We can check this answer using Arabic numerals.)

• Complex arithmetic in Roman era:

• When Romans wanted to do complicated arithmetic problems, they used a special counting board or an abacus:

An abacus represents values using a positional representation:

 The right most column has weight = 1 The second right most column has weight = 10 And so on.

Notice that this is an encoding method !!!

It is an agreement on how to represent a value

• Positional representation system

• Positional (value) representation:

• A position representation system uses the same symbol to represent different values

• The value that is represented by a certain symbol depends on:

 The symbol itself, and The position in which that symbox is found !!!

• Example:

 The symbol 1 in the number 111 represents the value * (= 1 dot). The symbol 1 in the number 111 represents the value ********** (= 10 dot).

In contrast:

 The symbol V will represent the value ***** (5) not matter where you find it in a Roman number !!!

• When humans started to use positional system (based on 10), we can teach children to add any two numbers by:

• The base 10 addition table:

 ``` | 1 2 3 4 5 6 7 8 9 ----+------------------------------------- 1 | 2 3 4 5 6 7 8 9 10 2 | 3 4 5 6 7 8 9 10 11 3 | 4 5 6 7 8 9 10 11 12 4 | 5 6 7 8 9 10 11 12 13 5 | 6 7 8 9 10 11 12 13 14 6 | 7 8 9 10 11 12 13 14 15 7 | 8 9 10 11 12 13 14 15 16 8 | 9 10 11 12 13 14 15 16 17 9 | 10 11 12 13 14 15 16 17 18 ```

 Add digits from right to left When the sum of two digits exceeds 9, write down the the right most digit and add the carry to the next position of the sum

After learning these techniques, the positional systen enable a ordinary humans to become a human calculater !!!

(In contrast, a Roman fellow will need to use an abacus !!!)

• Note:

 You have memorized these rules in elementary school