A Story of Numbers · Lesson 1 of 8
Reema’s Curiosity
“Explore why humans count, one-to-one mapping, number systems and numerals.”
• Explain why people needed ways to count and record quantities. • Use one-to-one mapping to compare and count collections. • Distinguish a number from the numeral used to represent it. • Explain why a useful number system needs a fixed ordered sequence. • Compare counting with objects, names and written symbols.
Reema’s Curiosity
Imagine finding a page filled with symbols that do not look like the digits 0–9. Would you still be able to tell which symbols represent one, ten or a hundred? People have faced this problem for thousands of years because different societies used different ways to record numbers.
Long before modern numerals became common, people still needed to know how much food they had, how many animals belonged to them, what had been traded, and how many days had passed. Counting came before our present way of writing numbers.
The number system used around the world today developed from the Indian system of ten digits, including 0. The chapter traces how ideas about counting and representation gradually became more efficient until numbers could be written and calculated with using a small set of symbols.
The Mechanism of Counting
Suppose you have a herd of cows but you are not allowed to use words such as eight or symbols such as 8. You can still keep track of the herd by pairing every cow with one stick.
If every cow can be paired with exactly one stick, then the collection of sticks records the size of the herd. If a cow is missing later, one stick will be left without a matching cow. This idea works even when no number name is spoken.
A pairing in which each object in one collection is matched with exactly one object in another collection, with no object used twice.
Problem
Yesterday you kept one stick for every cow. Today one stick is left after pairing all returning cows. What does that tell you?
- 1.Each stick represented one cow yesterday.
- 2.Today every returning cow is paired with one stick.
- 3.One stick is still unpaired.
- 4.So one cow from yesterday's herd has not returned.
Instead of physical objects, we can use an ordered sequence of sounds or names. For example, if the letters a, b, c, d, ... are used in order, each new object can be matched with the next letter. The last letter reached tells us the size of the collection in that system.
A third possibility is to use a fixed sequence of written symbols. Roman numerals such as I, II, III, IV, V are one historical example. The important idea is not which symbols are chosen; it is that everyone agrees on their order and meaning.
A standard ordered sequence of objects, names or written symbols that can be used to count and represent the size of a collection.
A written symbol or group of symbols used to represent a number.
The number is the quantity itself. A numeral is a written representation of that quantity. The same number can be written as 8 in the Hindu system or VIII in the Roman system.
Quiz
What makes one-to-one mapping useful for counting?
Which statement best describes a numeral?
Why is a sequence of only 26 letters limited if each number gets one letter?
Which is an example of one-to-one mapping?
Why does a number system need a fixed order?
Practice Problems
- Describe how you could check whether all animals in a herd returned without using number words or written numerals.
- Explain the difference between the number eight and the numerals 8 and VIII.
- Design a counting method using five different sounds. State its biggest limitation.
- Explain how sticks could be used to compare two herds and determine which has more animals.
- Create a small number system using your own written symbols for the first six numbers and explain how a user would know their order.
Key Takeaways
• Counting does not depend on modern digits; it depends on matching objects with an ordered sequence. • One-to-one mapping lets us compare and record quantities. • A number is a quantity; a numeral is a written representation of that number. • A number system needs a fixed order so that counting is consistent. • Physical objects, spoken names and written symbols can all serve as counting resources. • A useful number system must be extendable to larger quantities.
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Next · Lesson 2
Some Early Number Systems