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Mathematical Symbols

A brief article written for pupils about mathematical symbols.

Stone Age Counting

Stage: 1 Challenge Level: Challenge Level:1

There are no right and wrong answers to this problem; the drawings could be up to $25.000$ years old, so we cannot ask the people who drew them! What it does definitely show us, however, is that the people living all of those years ago had some sort of system for encoding information. This is what we need to communicate - we use codes all of the time. Numbers are codes, and so are letters; put together in a meaningful way, we can then decode this, and work out what the message is. Symbols can also be used to transmit information. For example, a smiley face shows that you are happy, a plus sign means that you add numbers together and so on.

Some great suggestions for the "meaning" of the drawings were submitted. If we can understand the code the cave men were using, we can then de-code this and read the message ...

Although this is posed as a mathematical problem, we must also bear in mind that the drawings may not be directly maths-related. Lauren, from the Princess Elizabeth School, suggested that the drawings could be a map of an area, possibly with directions. For example, the line on top of the vertical lines, joining them all together, could represent the direction you should walk in. The number of vertical lines, dipping down below, represent the number of steps to take in that direction. The "curvy" lines show a more complicated, curved path.

Lauren also made another suggestion:

It could also represent different musical notes: the longer the line the lower the note is (or maybe the other way around). The length of the line could also be related to the length of the time that the note is held for: the longer the note, the longer you hold it for.

Emma, from Village Elementary, suggested that the drawings could represent a family tree, or a collection of family trees. The long horizontal line that joins the shorter vertical lines could represent the mother or father. The vertical lines projecting down from this could represent the children. Single, unjoined lines could represent unmarried people with no children, and where lines meet, joining together different sections, this could represent relations joining the different family units.

Other people suggested that the drawings could be some form of early counting system. Joe, from Meridian Primary, and Lauren from Princess Elizabeth, thought that the lines could represent a certain quantity. In the simplest case, each line could mean "one". Then, to work out the total number, you would just count the number of lines. As Ewan, from Sofrydd Primary wrote, this a bit like the "tallying" that we use nowadays. Although it is simple, writing large numbers using simply "ones" may take a long time, and is very inefficent. So, perhaps combinations of these lines could mean different things? Perhaps some of the different symbols represent larger numbers? Read on ...!

The code could be more complicated; each line could mean two, or ten, or $100$. Or, as Grant from the Village School suggested, the lines could mean different quantities in different contexts. For example, when there are six lines joined together by a horizontal line across the top, this could represent $30$ or another number. A single line on its own could still mean "one", but when it is grouped together with other lines, it can mean something different.

This is actually similar to the way that the numbers, which we use today, work. A number $2$ on its own means "two". However, if the "$2$" is accompanied by other numbers (like having more lines in the cave drawings), its place (i.e. its context) in the number is important as it can mean different things in different places/contexts. In the number $236$, the "$2$" now means "two hundred". In the number $56721$, the "$2$" now means "twenty".

The numbers that we use most of the time today are known as Arabic numerals. The Arabic numerals are the ten digits: $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$. These digits are used either on their own, or in combinations, to make up any number that is desired.

There are many other numeral systems, and they each have their pros and cons. You may have come across Roman numerals before; these made up the numeral system in Ancient Rome. Today, in the UK, we use letters for words and separate digits for numbers. However, the Roman numeral system is based on letters of the Roman alphabet. This sort of system is still used today in languages such as Hebrew and Greek. In these systems, each letter also represents a number. The way that numbers are "built up" varies between the languages, as does the methods used to do calculations.

Sam, from Village Elementary, thought that the drawings in the cave may be some form of numeral system:

At first, I thought it may have been a way of tallies (four lines, then a slash through them represents "five"). But then I thought that it was more likely for them to be "cave man" numerals, like Roman numerals. That would better explain the drawings that don't look like the others. They could be numbers such as $10$, like the roman numeral for $10$ is X.

Different numeral systems have different advantages and disadvantages. As mentioned above, the Arabic numeral system uses the position of a number to encode different meanings. This is good because we need fewer numbers for our code (less to remember!). Also, it is quicker to write (and read) these numbers than, let's say, Roman numerals. Compare "$1998$" (Arabic) to MCMXCVIII (Roman)!

As well as being easier to write and read, the Arabic numeral system also means that calculations are easier to do. The "changeover" from the Roman to Arabic numeral system meant that scientists and mathematicians could do more complicated sums and come up with more complex theories.

Think about the advantages and disadvantages of using the system seen in the cave (if the drawings do indeed show some form of counting!).

What could the cave men be counting? Sam, from Village Elementary, had a suggestion:

I think it could be used to count how many spears you had, how much food you have (or need), or it may have even been to play a certain type of game.

Maia, from Culford was also thinking this way. She thought that the lines may be used to count sheep. Ethan thought that the lines may represent years, as part of a calendar.

Overall, there is no definite answer for what the drawings mean. People have different suggestions, or different versions of a similar suggestion. As long as the reasoning is good, then this is great! It is fantastic to have lots of different theories to choose from!