We don't know how number systems were invented, but here is a story of how it could have happened.

Once upon a time thousands of years ago two cavemen were talking. Oom made some grunts that meant "I caught this many rabbits yesterday." He held up two rabbit tails.

Uhm said, "I caught this many rabbits." He held up three fingers. Now here was something new: Instead of showing the rabbits or their tails, he used symbols to show the quantity. One finger stood for one rabbit, but it could also stand for one of anything. Everyone thought that was good idea and started using the finger number system. (The word "digit" comes from the word "finger>")

Erg got the idea that he would like to keep a written record of the rabbits he caught. He drew a sketch of a rabbit on his cave wall for each rabbit he caught. But his son, Merg, had a better idea. For each rabbit he would make a single straight mark on the cave wall. So for 3 rabbits he made three little marks.

The finger system and the marks worked fine for small numbers, but the day came when people needed larger numbers. Finally, someone got the idea of having different symbols for different numbers. It became the Roman Numeral System. An I stands for one, II stands for 2, V stands for 5, and X stands for ten. So they had to write XXXVIII for our 38.

Then we had three great inventions. One was the invention of the digits from 1 to 9. The second was the invention of 0 and the place value system, and the third was the decimal fraction system. After these inventions people could express any number no matter how large or small with only 10 digits including the zero. Here is a very large numeral that has only 1s and 0s: 1,100,101. These inventions were a big deal!

A number is the idea in our head of a count or measurement, and when we put it on paper it is a numeral such as XXV or 25, which consists of two digits. However, we will often use "number" for "numeral."
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PLACE VALUE

[You will need one paper dollar, three dimes, and 16 pennies.]

To help you understand place value, count out 12 cents in this way:

1. Count out 10 pennies.
2. Exchange the 10 pennies for one 10-cent piece or dime.
3. Count out 2 pennies and place them one above the other to the right of the dime.

Now write the numeral 12. Just as the dime equals 10 cents, the 1 in the numeral 12 means one set of ten pennies or ten of anything. The 2 means 2 ones or units of anything. Here is how we write 12 the long way, which we call expanded notation: 10 + 2.

So remember this: What a digit means depends upon its place in a numeral. Again, what does the digit 2 mean in 12? What does the digit 1 mean in 12?

Our U.S. money system is a decimal system. A dime equals 10 pennies, and a dollar equals 10 dimes.

EXAMPLE: Let's say that you buy an ice cream cone for one dollar and twenty-five cents or 125 cents.

You count out the money. You can think about the 125 in two ways:

As money: 1 dollar (hundred) + 2 dimes (tens) + 5 pennies (ones).
As a numeral written the long way: 100 + 20 + 5.

Write the numeral 243, and then write it the long way.

What is the value of the 3? Think......It is 3 ones because it is in the ONES place.
What is the value of the 4? Think......It is 4 tens or 40 because it is in the TENS place.
What is the value of the 2? Think......It is 2 hundred because it is in the HUNDREDS place.

CHALLENGE 1: Now count out 132 cents using only 2 pennies and other coins. What is the money value of the 2 in this number? What is the value of the 3? What is the value of the 1? Write this numeral, 132, the long way.

CHALLENGE 2: Write two hundred sixty-nine as a numeral. What is the value of the 2? What is the value of the 6? What is the value of the 9? Write 269 the long way.

CHALLENGE 3: In the numeral 50 can you explain why the zero is needed?

It is needed to hold the 5 in the tens place, so we call the zero in this situation a place-holder.

CHALLENGE 4: Write these numerals in a column so that the digits line up properly according to their place value: 10, 2, 3,000.
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OTHER NUMERATION SYSTEMS

[Cover this section when you think the learner is ready.]

The system we now use most of the time is the decimal system, which is a base ten system. The Latin word for ten is decem, which is the root in the word decimal. The reason the base ten system was chosen is probably that we have ten fingers, which ancient people used for counting.

You are probably wondering if we could have other systems with other bases. We can, and we do have such systems. Computers and calculators use the base two or binary system because they work with electrical circuits that are either off or on, which translates into the binary 0 or 1. When you enter a base ten numeral, it is changed to a binary. Computer programmers use a base 16 system called hexadecimal or hex for short. Base 16 requires six more digits, so they use the capital letters from A through F. Here are numerals in the five different systems. Compare the numerals in the base 10 and base 5 systems. For a fun challenge complete the numerals in the base 8 system.

Decimal   Base 5 System  Binary   Roman numeral   Base 8 System
   0           0          0000                          0
   1           1          0001       I                  1
   2           2          0010       II                 2
   3           3          0011       III                3
   4           4          0100       IV                 4
   5          10          0101       V                  5 
   6          11          0110       VI                 6 
   7          12          0111       VII                7
   8          13          1000       VIII                 
   9          14          1001       IX                 
  10          20          1010       X                  
  11          21          1011       XI                  
  12          22          1100       XII                
  13          23          1101       XIII               
  14          24          1110       XIV                 
  15          30          1111       XV                 
  16          31         10000       XVI               20
  17          32         10001       XVII              21 

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