# 1.1 Number systems

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**Counting sheep using stones**

Primitive people used to count their sheep using stones, such that each stone would represent a sheep.

So for example, if we have 10 sheep, we can represent this by 10 stones, i.e.:

This might be simple enough, but usually, for even a fairly small flock of sheep, there may be hundreds. Using this method makes it almost impossible to represent, then!

Because of the clumsiness of this system, some improvements were made to the system.

**Egyptian number system**

Egyptians used certain symbols to represent 1, 10, 100, 1000 (and also beyond).

These include:

- representing the number “1”
- representing the number “10”
- representing the number “100”
- representing the number “1000”

So for example, the number is equal to

Notice the order in which the Egyptian numbers are written make no difference.

That is, we can rewrite the number above (2311) as , back-to-forth, which would still be the same number! In fact, you can write the symbols in any order and it would make no difference! This is unlike “2311” which will be different if written “1132”.

The Egyptian number system however, was still relatively clumsy, so was introduced the Roman system.

**Roman number system**

The Roman system, rather than using strange symbols, uses letters from the English alphabet to represent various numbers, including:

**I**representing the number “1”**V**representing the number “5”**X**representing the number “10”**L**representing the number “50”**C**representing the number “100”**D**representing the number “500”**M**representing the number “100”

Note that starting with the Roman number system, order is now important.

In particular, if a smaller symbol precedes a larger one, you need to subtract it from the larger one.

For example, instead of writing the letter “4” as **IIII** (4 I’s), we write it as **IV**, which means

Applying what we know, the number 19 would be broken down into its units, 10+9. 10 is **X**, and 9 is **IX**, or together, 19 is **XIX**.

Another example, 149, which is made up of 100+40+9. Now we know 100 is **C**. 40 is 50-10, which is **XL**. Also, 9 is 10-1, or **IX**. Putting these together, is **CXLIX**.

To solve Roman numerals, you must look at each unit/tens/hundreds/thousands separately.

Going backwards, if we are given **CDIX**, we know **C** is 100, **D** is 500, **I** is 1 and **X** is 10. Note **C** is less than **D**, and **I** is less than **X**, meaning we must the first listed from the second listed. So **CD** together is 500-100=400, and **IX** together is 10-1=9. Together, these are

**Hindu/Arabic number system**

The number system we use is called the Hindu/Arabic number, which is separated into the units/tens/hundreds/thousands/etc columns.

For example, 12345 is 12,345, or “Twelve thousand, three hundred and forty five”.

Like the Roman system (which order is important due to the use of subtraction), in the Hindu/Arabic system, the position (place value) is important.

Therefore, the Arabic number system uses the zero (0).

The effect is that the number “5” is different from “50”, which is again different from “500”.

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