Multiplication -- prove this one by induction

In summary, the conversation discusses proving by induction that n x (m++) = (n x m) + n for any natural numbers n and m. The base case and assumption are given, but the proof is stuck at n++(m++)=((n++)m) + n++. The notation m++ is believed to be shorthand for m + 1. It is suggested to use + 1 instead to better utilize normal arithmetic axioms. The use of axioms is also questioned.
  • #1
rb120134
9
0
Homework Statement
Prove by induction that for any natural numbers n and m , n x (m++)= (n x m) + n
Relevant Equations
n x (m++) = (n x m) +n
Prove by induction that for any natural numbers n and m , n x (m++)= (n x m) + n

The base case, n=0 gives 0 x m++=(0 x m) +0 gives 0=0
Now assume n x (m++) = (n x m) +n
For n++ we get

n++(m++)=((n++)m) + n++

from this point I am stuck, how can I prove both sides are the same?
 
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  • #2
What do you mean by m++ ? In mathematics, this means nothing. Do you mean the programming language C convention? But then, actually n * (m++) == n *m is true, while n * (++m) == n * m + n is true.
 
  • #3
PAllen said:
What do you mean by m++ ? In mathematics, this means nothing. Do you mean the programming language C convention? But then, actually n * (m++) == n *m is true, while n * (++m) == n * m + n is true.
The OP posted a similar question using this notation in another thread (https://www.physicsforums.com/threads/proof-multiplication-is-commutative.782057/#post-6537515).

I believe it is less sophisticated notation than is used in C et al, and has nothing to do with pre- or post-increment. I believe that the notation m++ is just shorthand for m + 1.
 
  • #4
Mark44 said:
The OP posted a similar question using this notation in another thread (https://www.physicsforums.com/threads/proof-multiplication-is-commutative.782057/#post-6537515).

I believe it is less sophisticated notation than is used in C et al, and has nothing to do with pre- or post-increment. I believe that the notation m++ is just shorthand for m + 1.
shorthand in what way? Same number of characters, and + needs right pinky reach as well as left pinky shift, while 1 just needs left pinky reach (at least on my keyboard).
 
  • #5
PAllen said:
shorthand in what way?
OK, maybe samehand...
 
  • #6
Can you say what axioms you are allowed to use? I would also recommend explicitly writing + 1 to better use normal arithmetic axioms.
 
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1. What is induction in the context of multiplication?

Induction is a mathematical proof technique used to prove that a statement is true for all natural numbers. In the context of multiplication, it is used to prove that a certain property holds true for all integers.

2. How does induction work for proving multiplication?

Induction works by first proving that the statement is true for a base case, typically n=1. Then, it is assumed that the statement is true for some arbitrary integer k. Using this assumption, the statement is then proved to be true for the next integer, k+1. This process is repeated until the statement is proved to be true for all natural numbers.

3. Can you give an example of using induction to prove a multiplication statement?

One example is proving that the product of any two even numbers is also even. First, we prove that the statement is true for the base case n=1 (2*2=4). Then, we assume that the statement is true for some arbitrary even number k. Using this assumption, we can show that the product of k and k+2 is also even (k*(k+2)=2k(k+1)). Since k and k+2 are both even, their product is also even. Therefore, the statement is true for all even numbers.

4. Is induction the only way to prove multiplication statements?

No, there are other proof techniques such as direct proof, proof by contradiction, and proof by contrapositive. However, induction is a commonly used technique for proving statements about multiplication because it is relatively straightforward and efficient.

5. Can induction be used to prove statements about other mathematical operations besides multiplication?

Yes, induction can be used to prove statements about any mathematical operation as long as the statement can be expressed in terms of natural numbers. However, the specific steps of the induction process may vary depending on the operation being used.

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