Simple complex analysis question

Click For Summary
SUMMARY

In complex analysis, the relationship iy = yi, where y is a real number, is established through the definition of complex numbers as ordered pairs (a, b). The operations of addition and multiplication for complex numbers are defined as (a, b) + (c, d) = (a + c, b + d) and (a, b)(c, d) = (ac - bd, bc + ad). This definition allows for the verification that (a, 0)(0, 1) = (0, a) = (0, 1)(a, 0), confirming the commutative property of multiplication in complex numbers.

PREREQUISITES
  • Understanding of complex numbers as ordered pairs
  • Familiarity with basic operations of addition and multiplication in mathematics
  • Knowledge of real numbers and their representation in complex form
  • Basic principles of complex analysis
NEXT STEPS
  • Study the properties of complex number multiplication in detail
  • Learn about the geometric interpretation of complex numbers
  • Explore the axioms of complex analysis and their implications
  • Investigate the applications of complex numbers in various fields of mathematics
USEFUL FOR

Students of mathematics, particularly those studying complex analysis, educators teaching mathematical concepts, and anyone interested in the foundational properties of complex numbers.

McLaren Rulez
Messages
289
Reaction score
3
Hi,

In complex analysis, is it an axiom that iy=yi where y is real? Or can this result be proved somehow? Thank you.
 
Physics news on Phys.org
Well, it depends on how you define the product. The most elementary way to define complex numbers is by defining the to be couples (a,b) with addition

(a,b)+(c,d)=(a+c,b+d)
(a,b)(c,d)=(ac-bd,bc+ad)

Here a real number is of the form (a,0) and i is (0,1). It can now easily be checked tht

(a,0)(0,1)=(0,a)=(0,1)(a,0).
 
Thank you micromass. That is a nice way of looking at it
 

Similar threads

Undergrad Newton-Raphson Method With Complex Numbers
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Analysis of functions of a single variable via multivariable calculus?
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Why are analyticity and convergence related in complex analysis?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Proving Continuous Functions in Smooth Infinitesimal Analysis
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Several Questions About Smooth Infinitesimal Analysis
  • · Replies 1 ·
Replies
1
Views
3K
Is f(z) =(1+z)/(1-z) a real function?
  • · Replies 17 ·
Replies
17
Views
4K
Undergrad Can Functions Have Complex Poles Beyond Infinitesimal Limits?
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Questions on ##\mathbb{R}##
  • · Replies 5 ·
Replies
5
Views
2K
High School Simple Derivation Of Euler's Formula And Applications
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Question about uniform convergence in a proof
  • · Replies 1 ·
Replies
1
Views
3K