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Matrix Representation of the Addition Group

​
x
3
y
2
3 + 2 = 5
1
3
0
1
1
2
0
1
=
1
5
0
1
The real numbers with the operation of addition are commonly cited as an elementary example of a group. The requirements of closure, associativity, and the existence of an inverse are all fulfilled. It is well known that groups can be represented by matrices, with the group structure reflected in the corresponding matrix multiplication. So how can one reconcile representing addition by multiplication?
The answer is provided by a simple relation involving multiplication of 2×2 matrices:
1
x
0
1
1
y
0
1
=
1
x+y
0
1
.
In this Demonstration, values of
x
and
y
are limited to integers between
-10
and
10
for neatness, but the result applies to all real and complex numbers.
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