Can Mathematical Induction Prove Theorems for Negative Numbers?

kntsy
Messages
80
Reaction score
0
Is the principle of mathematical induction unable to prove theorem of negative numbers?
 
Mathematics news on Phys.org


Why?
If you can prove that if it holds for -n (with n a positive integer), it must hold for -(n + 1), then it is completely equivalent to induction over the positive numbers.
 
  • Like
Likes matqkks


In fact, the Principle of Mathematical Induction allows us to prove results for Rational Numbers, or any countable set (sets with a bijection to the natural numbers). In practice this can be very hard to do though as it depends on you choosing a "nice" enough bijective function for the particular problem.
 

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Thread 'Fermat's Last Theorem'

Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
View full post »

Thread 'Useless continued fraction for 1'

I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
View full post »

Similar threads

A Proving Goldbach's conjecture by induction (or why it doesn't seem to work)
Replies
10
Views
2K
Question about mathematical induction
Replies
10
Views
2K
[ASK] Mathematical Induction: Prove 7^n-2^n is divisible by 5.
Replies
5
Views
2K
Proving combination is a natural number by induction
Replies
9
Views
3K
MHB Solve Prove by Induction: 3^2n+1 + 2^n-1 Divisible by 7
Replies
2
Views
2K
Insights Fermat's Last Theorem
2
Replies
91
Views
6K
B Is everything in math either an axiom or a theorem?
2
Replies
72
Views
7K
Prove the 2nd axiom of mathematical logic using the Deduction Theorem
Replies
1
Views
1K
Proving Relationships with Mathematical Induction
Replies
2
Views
2K
Mathematical Induction 4+11+14+21+....+(5n+(-1)^n
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top