Isomorphism between groups of real numbers

In summary, there exists an isomorphism between the additive group of real numbers and the multiplicative group of positive real numbers, represented by the exponential function y=e^x. This isomorphism allows for a bijection between the two sets, despite both being uncountably infinite. The familiarity of the exponential function can help in understanding the concreteness of this isomorphism. Additionally, the fact that e^0=1 serves as an example of how homomorphisms map the identity to the identity.
  • #1
blahblah8724
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Apparently there is an isomorphism between the additive group [itex](ℝ,+)[/itex] of real numbers and the multiplicative group [itex](ℝ_{>0},×)[/itex] of positive real numbers.

But I thought that the reals were uncountably infinite and so don't understand how you could define a bijection between them?!

Thanks for your help!
 
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  • #2
The isomorphism is [itex]f(x)=e^x[/itex].

But I thought that the reals were uncountably infinite and so don't understand how you could define a bijection between them?!

Could you explain more about what's bothering you??
 
  • #3
There exist a bijection between two sets if and only if they have the same cardinality (that is, essentiallt, the definition of "cardinality"). The fact that two sets are both uncountably infinite doesn't mean they do not have the same cardinality.
 
  • #4
blahblah8724 said:
Apparently there is an isomorphism between the additive group [itex](ℝ,+)[/itex] of real numbers and the multiplicative group [itex](ℝ_{>0},×)[/itex] of positive real numbers.

Yes, and you are already very familiar with it: It's just the exponential function y = e^x. For all real numbers a and b we have e^(a+b) = e^a * e^b. That fits the definition of a homomorphism. Then note that e*x is a bijection between the additive reals and the multiplicative positive reals.

Does that help in terms of seeing the concreteness and familiarity of this isomorphism?

ps -- That's why e^0 = 1. A homomorphism always maps the identity to the identity. I vividly remember being in my first abstract algebra class and slogging through homomorphisms and normal subgroups ... then they mentioned that exp and log are isomorphisms ... and I got this AHA moment -- this stuff is actually about something!

pps -- I see Micromass already mentioned e^x. Hopefully I was able to add some detail for the OP's benefit.
 
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  • #5


I can confirm that there is indeed an isomorphism between the additive group of real numbers (ℝ,+) and the multiplicative group of positive real numbers (ℝ_{>0},×). This is known as the logarithmic isomorphism and it is defined as f(x) = e^x, where e is the base of the natural logarithm. This function maps each element in the additive group to its corresponding element in the multiplicative group, and vice versa.

You are correct in stating that the real numbers are uncountably infinite, which means that there is no way to list or count all of the real numbers. However, this does not prevent us from defining an isomorphism between two groups. In fact, isomorphisms between infinite groups are often used in mathematics to study their properties and relationships.

The logarithmic isomorphism is a powerful tool that allows us to translate operations and structures from one group to another, providing a deeper understanding of the relationships between them. It also allows us to perform calculations and proofs in one group that can then be applied to the other group.

In conclusion, the existence of an isomorphism between the additive and multiplicative groups of real numbers is a fundamental concept in mathematics and has been extensively studied and proven. It is a valuable tool for understanding and analyzing the properties of these groups and their elements.
 

1. What is an isomorphism between groups of real numbers?

Isomorphism between groups of real numbers is a bijective homomorphism, meaning it is a one-to-one and onto mapping between two groups that preserves the group structure. In other words, the two groups have the same algebraic properties and can be considered equivalent.

2. How do you determine if two groups of real numbers are isomorphic?

To determine if two groups of real numbers are isomorphic, you must first check if they have the same number of elements. Then, check if there is a function that maps the elements of one group to the other in a way that preserves the group structure, such as preserving the operation and identity element.

3. Why is isomorphism between groups of real numbers important?

Isomorphism between groups of real numbers is important because it allows us to study one group by understanding its isomorphic counterpart. This can simplify complex mathematical structures and make them easier to analyze.

4. Can two groups of real numbers be isomorphic if they have different operations?

No, two groups of real numbers cannot be isomorphic if they have different operations. Isomorphisms only exist between groups with the same operation, as this is a key component in preserving the group structure.

5. How can isomorphism between groups of real numbers be used in real-world applications?

Isomorphism between groups of real numbers has various applications in fields such as cryptography, computer science, and physics. For example, isomorphism can be used to efficiently solve complex equations or to encrypt information in a secure manner.

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