Cardinal Numbers: Stories

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Numbers listed in order, such as first, second, third and fourth, are referred to as ordinal numbers. In other words, ordinal numbers are words that describe where in an order a number falls. Kindergartners spend a good deal of their math time dealing with ordinal numbers because they're part of a firm foundation in many other concepts, such as counting and sequencing, that the children will need as they get older.

Ordinal Numbers For Kids - Maths Concept For Grade 2 - Periwinkle

The story is about animals who are riding on the train to the zoo. There are one elephant, two hippos, three giraffes, etc. As you read this story to your students, talk about which animals are first, second, third and so on. When you get done reading, line up plastic zoo animals and ask the children which one is first, second or third. Then to check for understanding, you can also give them a drawing of a train and ask them to glue pictures of the animals in the right car. For example, glue the lions in the fifth car and the tigers in the second car.

After teaching children the ordinal numbers, have them practice by playing a memory game, suggests Sue Bowen, writing for the University of North Carolina.

Cardinal number - Wikipedia

Before the lesson make cards to use in the game. In his paper " On a Property of the Collection of All Real Algebraic Numbers ", Cantor proved that there exist higher-order cardinal numbers by showing that the set of real numbers has cardinality greater than that of N.


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His proof used an argument with nested intervals , but in an paper he proved the same result using his ingenious but simpler diagonal argument. This hypothesis has been found to be independent of the standard axioms of mathematical set theory; it can neither be proved nor disproved from the standard assumptions.

In informal use, a cardinal number is what is normally referred to as a counting number , provided that 0 is included: 0, 1, 2, They may be identified with the natural numbers beginning with 0.

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The counting numbers are exactly what can be defined formally as the finite cardinal numbers. Infinite cardinals only occur in higher-level mathematics and logic. More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence.

For finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can construct a set which has exactly the right size, e. However, when dealing with infinite sets it is essential to distinguish between the two — the two notions are in fact different for infinite sets.

Considering the position aspect leads to ordinal numbers , while the size aspect is generalized by the cardinal numbers described here.


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  • The intuition behind the formal definition of cardinal is the construction of a notion of the relative size or "bigness" of a set without reference to the kind of members which it has. For finite sets this is easy; one simply counts the number of elements a set has. In order to compare the sizes of larger sets, it is necessary to appeal to more subtle notions. A set Y is at least as big as a set X if there is an injective mapping from the elements of X to the elements of Y. An injective mapping identifies each element of the set X with a unique element of the set Y.

    Note the element d has no element mapping to it, but this is permitted as we only require an injective mapping, and not necessarily an injective and onto mapping.

    Ordinal Numbers Story Sequencing Activity Reception/Kindergarten

    The advantage of this notion is that it can be extended to infinite sets. We can then extend this to an equality-style relation. Two sets X and Y are said to have the same cardinality if there exists a bijection between X and Y. By the Schroeder—Bernstein theorem , this is equivalent to there being both an injective mapping from X to Y and an injective mapping from Y to X.

    This is called the von Neumann cardinal assignment ; for this definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement is the well-ordering principle. It is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.

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    The classic example used is that of the infinite hotel paradox, also called Hilbert's paradox of the Grand Hotel. Suppose you are an innkeeper at a hotel with an infinite number of rooms.


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    • The hotel is full, and then a new guest arrives. It is possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant. We can explicitly write a segment of this mapping:. When considering these large objects, we might also want to see if the notion of counting order coincides with that of cardinal defined above for these infinite sets. It happens that it doesn't; by considering the above example we can see that if some object "one greater than infinity" exists, then it must have the same cardinality as the infinite set we started out with.

      It is possible to use a different formal notion for number, called ordinals , based on the ideas of counting and considering each number in turn, and we discover that the notions of cardinality and ordinality are divergent once we move out of the finite numbers. It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described. This can be visualized using Cantor's diagonal argument ; classic questions of cardinality for instance the continuum hypothesis are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals.

      In more recent times mathematicians have been describing the properties of larger and larger cardinals. Since cardinality is such a common concept in mathematics, a variety of names are in use.

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      Cardinal numbers are numbers that describe quantity , like one pot, two kettles, or three pans. What is an ordinal number? Numbers can be confusing, and remembering ordinal numbers vs. Luckily, there is a simple mnemonic that can help. As long as you remember to associate ordinal numbers with order , you should have no trouble knowing what these words mean.

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