The additive identity is zero, written 0 as a digit. So if q represents any quantity, q+0=0+q=q. We will use this axiom later.
Each number system has a base b and the digit 1 followed by n zeroes represents the quantity bⁿ. “Follow” means “written to the right of”. When n=0 bⁿ=1, so in all number systems 1 represents the same quantity. When b is ten we have the decimal number system.
b¹=b, so n=1 gives the representation of b in its number system.
When b is ten, the base is written as 10 because 1 is followed by one zero.
The next power of ten is written 100, which is 10². We call this one hundred.
As the power of ten increases, the number of zeroes following 1 increases.
The position of the 1 digit moves stepwise to the left as the power of ten moves stepwise, so we can use its position as a placeholder, counting from the right.
These are the names of a few of the placeholder positions starting at 1:
1: position 1: one or unit
10: position 2: ten
100: position 3: hundred
1000: position 4: thousand
10000: position 5: ten thousand...
In the word description of the water area of Texas we can write the number as the sum of its components, using the above definitions:
Seven thousand: 7×1000=7000
Three hundred: 3×100=300
Sixty: 6×10=60
Five: 5×1=5
The sum:
7000+
300+
60+
5=
7365, using the additive identity axiom.
Traditionally, for readability, a comma is used to group digits in threes, starting from the one or unit position, so 7365 becomes 7,365.