# Numbers/Cardinal Numbers

Notice: Incomplete

Counting is a fundamental thing, and underpins a significant part of, if not all of mathematics. When we count, we use a reference set. For example, when we count out four things, we often say, '1', '2', '3', '4'. We create an injection ('1', '2', '3', and '4' all get mapped to different things; one-to-one) and surjection (all four objects are accounted for; onto), called a bijection (one-to-one and onto). We use the reference set {1,2,3,4}. This, however, presents a circular reasoning issue- notably, using the number 4 to define itself. In this resource, we'll use the reference set {0,1,2,3} instead.

## Number names

Reference set Hindu-Arabic numeral English name Notes
{} 0 Zero, nought, naught It took a long time to invent this concept. How can a set containing nothing be something?
{0} 1 One
{0,1} 2 Two
{0,1,2} 3 Three
{0,1,2,3} 4 Four
{0,1,2,3,4} 5 Five The number of digits on one hand. The digits on one hand can also be used as a reference
{0,1,2,3,4,5} 6 Six
{0,1,2,3,4,5,6} 7 Seven
{0,1,2,3,4,5,6,7} 8 Eight
{0,1,2,3,4,5,6,7,8} 9 Nine
{0,1,2,3,4,5,6,7,8,9} 10 Ten The number of digits on two hands, which can also be used as a reference. At this point, we recycle digits.
{0,1,2,3,4,...,7,8,9,10} 11 Eleven 'One left [from ten]'
{0,1,2,3,...,8,9,10,11} 12 Twelve 'Two left [from ten]'
{0,1,2,3,...,10,11,12} 13 Thirteen Three and ten
{0,1,2,3,...,11,12,13} 14 Fourteen Four and ten
{0,1,2,3,...,12,13,14} 15 Fifteen Five and ten
{0,1,2,3,...,13,14,15} 16 Sixteen Six and ten
{0,1,2,3,...,14,15,16} 17 Seventeen Seven and ten
{0,1,2,3,...,15,16,17} 18 Eighteen Eight and ten
{0,1,2,3,...,16,17,18} 19 Nineteen Nine and ten
{0,1,2,3,...,17,18,19} 20 Twenty Two tens. The number of digits on all hands and feet of the human body, another potential reference set.
{0,1,2,3,...,18,19,20} 21 Twenty-one Two tens and one
{0,1,2,3,...,19,20,21} 22 Twenty-two Two tens and two
{0,1,2,3,...,20,21,22} 23 Twenty-three Two tens and three
{0,1,2,3,...,21,22,23} 24 Twenty-four Two tens and four

In general, the reference set used here is the set of all numbers that come before it.

If you have a number, say 15 ({0,1,2,3,4,5,6,7,8,9,10,11,12,13,14}), the next number (16) contains 15 and all its elements ({0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}; italics used to emphasize the point)

We often use the reference set {1,2,3,4,5...} instead of {0,1,2,3,4...} as the last number in the reference set tell you the number of things.

To understand number names, notice this: Fifteen, or {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14} can be partitioned into {0,1,2,3,4,5,6,7,8,9} and {10,11,12,13,14}, the former being 10 and the latter looking like 5, or {0,1,2,3,4}, except that the numbers are increased by 10. Thus, five and 10; fifteen.

Thirty-seven, or {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36} can be partitioned into {0,1,2,3,4,5,6,7,8,9}, {10,11,12,13,14,15,16,17,18,19}, {20,21,22,23,24,25,26,27,28,29}, {30,31,32,33,34,35,36}, the first three of which being 10 and shifted versions, and the last of which looking like 7, or {0,1,2,3,4,5,6}, except that the numbers are shifted. Thus, three tens and seven; thirty-seven.

Of course, this is unending, I can keep adding the last number I made to the reference set, creating a new number. Once I keep going, I could construct {0,1,2,3,4,5,6,7,8...} and call it ${\displaystyle \omega }$. But {0,1,2,3,4,5..., ${\displaystyle \omega }$} doesn't give me anything new in terms of 'how many', because I can do this:

Member from first set Member from second set
0 ${\displaystyle \omega }$
1 0
2 1
3 2
4 3
5 4
6 5
... ...

They are however useful if you want to count the order, as opposed to the amount. Imagine if I'm a really bad runner in a race, and infinitely many people come before me, I am {0,1,2,3...} or ${\displaystyle \omega }$ in a race. If someone else came just after me, they'd be {0,1,2,3,4...${\displaystyle \omega }$} or ${\displaystyle \omega +1}$ in said race. The general term for this many things (${\displaystyle \omega }$/${\displaystyle \omega +1}$/${\displaystyle \omega +2}$ etc.) is called ${\displaystyle \aleph _{0}}$. It took even longer to invent the concept of ${\displaystyle \aleph _{0}}$ than to invent the concept of 0.

## Activity

1. Print out cards having numbers from 0 to 99 inclusive and give one copy of each card to each student in order. If you don't have time, choose a more reasonable ending number.
2. Write the students' names on each card, so that they don't get mixed up with each other's.
3. Tell them to start from 0 and work in order of the cards each time they count.
4. Tell them to map the cards to anything they desire, other students (provided permission), whiteboard markers, notebooks, notebook pages, toys, doors, words, letters, windows, etc.