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Natural Number and Signed Number
Introduction ( Natural Number )
The concept of “Natural Number” is originated from how people count things, for example, one pencil, two pens, three apples, … and so on and so forth. Furthermore, in the language that we describe an ordered set, we say: 1st one, 2nd one, 3rd one, … Mathematically, we want to use a term to represent this concept and define it rigorously. For simplicity, we can say that the set of Natural Numbers ( denoted as N ) is
1, 2, 3, 4, 5, 6, 7, …
Or consider the set as
1, (1+1), (1+1+1), (1+1+1+1), (1+1+1+1+1), ….
In more advanced courses ( Group theory, Ring theory, Field theory ), the fundamental construction of the whole system will be established and with new meaning.
Addition ( Review )
We use the symbol “+” for the operation “addition”. We do not have the problem to understand
2=1+1 3=1+1+1 = 2+1 4=1+1+1+1 = 3+1 …
It can be considered as the definition of the symbols “2”, “3”, “4”, … How about the operation between those “symbols”? For example, 2+3 ? We know 2+3 = 5. But we should not take it for granted. We always want to go over the most fundamental steps to check if we can derive some new ideas:
2+3 = 2 + ( 1+1+1 ) = 2 + 1 + 1 + 1 = 3 + 1 + 1 = 4 + 1 = 5
We open up the bracket “( )”, and then we follow the definition of each symbol ( 2, 3, 4, 5, .. ) to get this result. From “2” to “3”, we use the operation “+” to increase something to jump to “3”; from “2” to “5”, it jumps more. Please also notice the action for “opening up the bracket”. Initially when we define the operation “+”, we only define the result for operating with the symbol “1”. To operate with more symbols, we would like to have more understanding.
The symbol “3” is defined as “1+1+1”; thus, the operation with symbol “3” can be considered as “+” operation with the symbol “1” for three times. That’s why we can open up the bracket to have the following step:
2+3 = 2 + ( 1+1+1 ) = 2 + 1 + 1 + 1
What if we would like to jump from the symbol “3” to symbol “2” ? We have another operation “Subtraction”.
Subtraction ( Review )
Up to this moment, we only have the symbols 1, 2, 3, 4, 5, … and the operation “+”. By using the operation “+” with 1, we can put those Natural Numbers in order according to how many times of operations with “1” are needed. Thus, we can list natural numbers in the following order
1,2,3,4,5, …
But operation “+” only allows you to move to one direction, for example, from 2 to 3. To move to another direction, we use the operation “Subtraction”. And we use the symbol “-“ for subtraction.
1=2-1 2=3-1 3=4-1 4=5-1 …
Similarly, we start with operation “-“ with 1. How about the result for 5-2? We know 5-2=3 . But we can try to write it in the following way:
5-2 = 5-(1+1) = 5-1-1 = 4-1 =3
Please notice the step for opening up the bracket. From previous definition, 2 is defined as (1+1). We think operation “-“ with “2” is to operate with “1” twice. That is why we have
5-2 = 5-(1+1) = 5-1-1
It does not work as operation “+”. Furthermore, with operation “+” among the natural numbers ( the symbols 1, 2,3,4,5, … ), the result is always another symbol in natural numbers. For subtraction “-“ , let us consider the following
1 – 1 =?
Obviously, it is not 1, or 2, or 3, or 4, …. It is not in our previous symbols. Hence, we introduce the symbol zero “0” and write it as
1 – 1 =0
Zero “0”
We introduce the symbol zero “0” so easily and write down the result for the previous question for 1 - 1 . But in history, it did not happen so naturally. For some reasons, zero did not show up until very late in history. One of the clues is: in roman numerals, there is no zero. Those who are interested in this topic can look up the associated literature for the introduction of zero.
And what is the result for the following
0 – 1 = ?
In other words, what is in front of 0 when we write down the sequence as follows ?, 0, 1, 2, 3, 4, 5, 6, 7, 8, …
And in front of that, there should exist a lot of “symbols” if we keep using the operation “-“ with 1. Thus, we realize that we need a class of symbols to represent those positions.
Negative Number
We use the following to introduce negative numbers:
0 – 1 = -1 (-1) – 1 = -2 (-2) – 1 = -3 (-3) – 1 = -4 … We have a whole new class of symbols “-1”, “-2”, “-3”, “-4”, “-5”, … And to write them in order with Natural Numbers, it is like
…, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, …
Please note that “-1”, “-2” … are just the symbols for the number should appear on that position up to this moment. Later on, we will find out it has a lot of convenience to represent in this way.
The introduction of “-1’, “-2” , “-3” .. can solve the issue to represent the result for the following questions
6 – 8 = -2 12 – 18 = -6 …
Due to the introduction of those “negative numbers”, we need to revisit operation “+” and operation “-“ . For example, what does it mean for
6+ 0 = ?
Back to our original definition, 0 = 1 -1 . Thus,
6 + 0 = 6 + 1 – 1 = 7 – 1 = 6
Another question is: 8+(-6) = ? (-6) = 0 – 1 – 1 – 1 – 1 – 1 – 1 So, 8+(-6) = 8 +0-1-1-1-1-1-1 = 8-1-1-1-1-1-1 = 7-1-1-1-1-1 … = 2
How about 8 – (-1) = ?
For operation “-“ with a positive number ( like 1, 2, 3, 4, 5, .. ) , we consider as moving to the position in front of the original number. Then the operation “-“ with a negative number should be considered as moving to opposite direction. Thus,
8 – (-1) = 8 + 1 = 9
In real scenario, you can consider the following problem: if you have 8 dollars, someone borrows 1 dollar from you and returns 2 dollars. How many dollars in your pocket after these actions? It can be represented as
8-1-(-2) = 7-(-2) = 9 or 8-1+2= 7+2 = 9
You can consider that the 2 dollars someone returns to you as (-2) dollars you lend to someone.
As a whole, we can consider the diagram above to label those numbers on a line. The arrow stands for the positive direction. Starting from the point 0, we mark the line by using the distance from 0 to 1 as “unit length” on both positive direction and negative direction. The operation “+” or “-“ associated with its operand is nothing about that the number of units should move along with positive direction or negative direction. This diagram will also be used for the introduction of “absolute value”.
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