Before one is introduced to the subjects of geometry or arithmetic, one has to have experience using numbers to describe amounts. In order to properly understand numbers one must be motivated by the utility of them. In search of the utility of numbers, one of the resources is efficiency. Numbers reduce the writing time when describing amounts. This is a subtle but useful tool in the pursuit of communicating effectively and efficiently to others.

Imagine this scenario. Everyday for several days I go on the internet and save pictures I like from google. Each picture I save under the same name, "picture". Then some time goes by and I need to access a particular picture. How do I access that picture? The answer is "not easily". One would have to personally review each document in sequence until they see the one they recognize.

Now consider the same situation of downloading pictures, but instead of calling each the same name I label them individually and uniquely. When the time comes to recall one of these pictures, I guarantee that finding it will happen in a much quicker time frame.

The way we think and memorize is similar. Going through our days we go from experience to experience, creating a timeline of life events, each a picture uploaded to our brains, each a file pretty much identical to any other. What makes certain thoughts more memorable is proper labeling.

Try telling a story about a particular life event (family gathering, meeting with friends, etc...) to a peer without naming any of the people, the place, or the objects present in the memory. Then go and tell the exact same story to another peer, this time including all of the names of people, the place, and the objects around. Which one of these peers understand the story better? 

Without the invention of numbers, amounts would be estimated based on relations figured through past experience. The amount counted would be more/less/ or equal to the usual amount. Recalling an amount without numbers can be done, but again, "not easily".

Numbers condense information for means of recording amounts. In the simplest language, numbers give each amount a unique and independent name. The utility of naming amounts in the benefit of effectively and efficiently recalling amounts, for means of communication with another or with future self.

NUMBERS

When delving into the heart of mathematics, we consider ones first encounter with it. Given a picture or environment of otherwise known objects, one can be asked to keep track of the amount of a particular item.

Consider this group of seeds. Assuming one can identify a single seed independent from the rest, then one can count the number of seeds in the bunch. 

The act of associating a group of objects with a number is a task deeply rooted in mathematics. One of the greatest utilities of relating groups of objects to a number is the efficiency in the communication of amount. In addition, numbers can applied to anything, not just seeds. In this way, numbers are the language of amount.

When there aren't any seeds in the picture, we say there are "zero" seeds. If there is only a single seed, we say there is "one" seed. Adding a seed into the picture, makes "two" seeds. Next, "three", then "four"... 

How to Count - Whole Numbers

It is the goal of mathematics to simplify. Without the invention of names for amounts it would take a long time to communicate how many you have. When writing the number each name has a symbol. Writing "1" instead of "one", "2" instead of two, etc... reduces the amount of time in the communication of amount as well.

The technique of reducing objects to numbers is common to all mathematical practices. It's simple but also valuable when keeping track of different amounts, amount changes, measurement, and much more. The act of symbolically representing a specific number of objects is copied over and over again throughout ones study of mathematics.

In sequence, numbers zero to nine are written in the following way:

What about amounts larger than nine? When speaking the number, each number has its own name. If we add one to nine, we get the number ten. Add one to ten, we get eleven. Counting up from eleven, we have: twelve, thirteen, fourteen...

One may see a pattern emerge as enumeration continues. Besides eleven and twelve, each of the numbers have a "teen" in common. The "teen" in the number means one group of ten, the prefix tells us the number of additional units. For instance, thirteen is short for "three-teen", meaning three and one group of ten. This continues until we reach twenty.

Twenty is interpreted as two groups of ten. For additional units less than ten, we add the number to the end of the amount of ten groups. If we consider twenty-three for example, the twenty represents two groups of ten and the three represents three additional units.

The system of counting we use groups tens as the amounts get larger. Naming these amounts follow the same pattern: "number of tens"-ty-"number of ones less than nine". 

The numbers between twenty and ninety-nine follow the exact same naming sequence mentioned above. Thirty-one, thirty-two, thirty-three,... Fourty-one, fourty-two, fourty-three,... Fifty-one, fifty-two, fifty-three,... and so on...

When writing numbers which extend beyond nine we follow a similar but different pattern. Since we only have ten symbols to write numbers (including zero), we repeatedly use them to express larger amounts. 

Continuing with the pattern of grouping in units of ten, to write larger amounts we add a place to left. In this way, numbers larger than nine have two digits. The first signifies the number of tens and the second, the number of additional ones.

When both the tens and the ones position reach nine, just like when we reached nine before, a new place is invented to the left of the previous digits. Instead of calling ten groups of ten, "tenty", a new name is invented. Ten groups of ten is called a hundred.

Now instead of grouping numbers by ten, we group by hundreds. 

When numbers above ninety-nine are spoken, we say the number of hundreds first, then we add an "and" to count additional units. For instance, to count a number which is fifty-six units above a hundred we say "one-hundred and fifty-six." See below for some more examples of enumerating with the hundreds,

Enumerating further, once all existing place values are filled with nines a new place value is created to the left of the previous digits. Each time we create a new place value, we also come up with a new name. Adding a unit to nine hundred and ninety-nine we get a new grouping called the thousands.

Numbers in the thousands follow a similar sequence as before. We always name the largest grouping first, then the second largest, and so on until all place values have been accounted for down to the ones position. It is only within the hundreds which we add an "and" in between the hundreds and tens positions when speaking the number.