Natural numbers arithmetic
What is a natural number
To start off, we will look at natural numbers. A natural number refers to numbers used for counting.
When counting we have 10 unique numbers to work with. In order, they are...
| number | symbol |
|---|---|
| zero | 0 |
| one | 1 |
| two | 2 |
| three | 3 |
| four | 4 |
| five | 5 |
| six | 6 |
| seven | 7 |
| eight | 8 |
| nine | 9 |
Note
zero refers to having 'none' of the thing you are counting. If you have zero apples, you don't have any apples.
If you have anything that can be counted in terms of whole numbers (\(1, 2, 3, \ldots\)) that number can be considered a natural number (I have 5 apples, the jar has 450 jelly beans, there is 6022140760000000000000001 atoms in 12 gram of carbon, etc...).
Once we have used up all the numbers, we add a new 'place' to the left, increment it by one, and then start over. Once that new place gets filled up, we repeat the process. Here is an animation that I think will make it a bit clearer.
The right most digit is called either the ones place or the units place. Moving to the left we call that the tens place, this is because every circle that is filled in this place refers to a full set of ten circles that have been filled in the ones place. Moving left again to the number in the third position, we refer to as the hundreds place, because similar to the tens, every circle in the hundreds place refers to ten circles filled in the tens place which refers to a hundred circles filled in the ones place. Below is a table of the first few places for later reference.
| (number)s place | numbers in from the right | value |
|---|---|---|
| one | 0 | 1 |
| ten | 1 | 10 |
| hundred | 2 | 100 |
| thousand | 3 | 1,000 |
| ten thousand | 4 | 10,000 |
| hundred thousand | 5 | 100,000 |
| million | 6 | 1,000,000 |
| ten million | 7 | 10,000,000 |
| hundred million | 8 | 100,000,000 |
| billion | 9 | 1,000,000,000 |
| ten billion | 10 | 10,000,000,000 |
| hundred billion | 11 | 100,000,000,000 |
| trillion | 12 | 1,000,000,000,000 |
Note on the commas
the comma's in the number are used to make it easier to figure out what your looking at. It can be very difficult to tell what a number like 10000000000 is, but with commas (10,000,000,000) we can quickly see that its ten billion. Both are the same numbers and are correct. Later on we will go over some alternate and better ways to show large numbers using something called 'scientific notation'
For the number
\[19,083,564\]we have
- \(4\) in the 1's place (representing 4 'units' of what we are counting)
- \(6\) in the 10's place (representing 6 sets of 10 of the units we are counting)
- \(5\) in the 100's place (representing 5 sets of 100 of the units we are counting)
- \(3\) in the 1000's place (" " '' 3 sets of 1000 of the unit)
- \(8\) in the 10,000's place ('" " " 8 sets of 10,000 of the unit)
- \(0\) in the 100,000's place (" " " 0 sets of 100,000 of the unit)
- \(9\) in the 1,000,000's place(" " " 9 sets of 1,000,000 of the unit)
- \(1\) in the 10,000,000's place(" " " 1 set of 10,000,000 of the unit)We could also say that there is a \(0\) in the 100,000,000's place, which would represent 0 sets of a hundred million of the unit. Technically we could write the number as something like
\[\ldots0,000,000,000,019,283,564\]with any amount of \(0\)'s preceding the \(19,083,564\) While this is not wrong, because we can always add more places and zeros, we generally don't show zero's to the left of a number. However there are examples where you would show the leading zero's such as in a date \((02/05/2000)\). It's fine to have them if it helps with clarity, however in most cases it doesn't make sense to add them.
If you notice the 100,000's (hundred thousands) place it's 0, but because we have a non-zero number to the left of it (\(9\)) it still contributes to the total number. You could also think of there being 90 sets of 100,000 (which is equivalent to 0 sets of 100,000 and 9 sets of 1,000,000).
We've already talked a bit about combining and breaking up sets of things, but lets formalize it and go over our first two operations, addition and subtraction.
--> Addition and Subtraction with Natural Numbers
<-- Arithmetic Introduction
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This is true for specifically carbon-12 and is the definition of the base unit 'mol' which is often used in chemistry ↩