# Interactive Real Analysis

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## 1.4. Natural Numbers, Integers, and Rational Numbers

### Examples 1.4.3(b):

Let A be the set N x N and define an equivalence relation r on N x N and addition of the equivalence classes as follows:
1. (a,b) is related to (a’,b’) if a + b’ = a’ + b
2. [(a,b)] + [(a',b')] = [(a + a', b + b')]
3. [(a,b)] * [(a’, b’)] = [(a * b’ + b * a’, a * a’ + b * b’)]
Below are some examples for addition and multiplication of these equivalence classes.
If you add [(1,2)] + [(4, 6)] you would get the following:
• [(1,2)] + [(4, 6)] = [(1 + 4, 2 + 6)] = [(5, 8)]
By the above examples, that implies
• [(1,2)] contains all pairs whose difference y - x = 1
• [(4,6)] contains all pairs whose difference y - x = 2
• [(1,2)] + [(4,6)] contains all pairs whose difference y - x = 3
Adding [(3,1)] + [(1,3)] gives the following:
• [(3,1)] + [(1,3)] = [(3+1, 1+3)]
This is, by the above example, equivalent to the following:
• [(3,1)] contains all pairs whose difference y - x = -2
• [(1,3)] contains all pairs whose difference y - x = 2
• [(3,1)] + [(1,3)] contains all pairs whose difference y - x = 0.
Multiplying the equivalence classes [(5,4)] * [(7, 4)] we get the following:
• [(5,4)] * [(7, 4)] = [(5*4 + 4*7, 5*7 + 4*4)] = [(48,51)]
This is, by the above example, equivalent to the following:
• [(5,4)] contains all pairs whose difference y - x is -1
• [(7,4)] contains all pairs whose difference y - x is -3
• [(5,4)] * [(7, 4)] = [(48,51)] contains all pairs whose difference y - x = 3
Multiplying the equivalence classes [(1,2)] * [(2,1)] we get the following:
• [(1,2)] * [(2,1)] = [(1*1 + 2*2, 1*2 + 2*1)] = [(5,4)]
This is, by the above example, equivalent to the following:
• [(1,2)] contains all pairs whose difference y - x is 1
• [(2,1)] contains all pairs whose difference y - x is -1
• [(1,2)] * [(2,1)] = [(5,4)] contains all pairs whose difference y - x is -1.
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