Why does the presenter switch from examples like $6\times 8$ and $2\times 4$ to an algebraic proof?

Conditions

• The goal is to prove a universal statement about all even numbers.
• Numerical examples are used only to build intuition.
• Algebra is necessary for a general proof.

Reasoning, step by step

1. Check specific numerical examples like $6\times8=48$ and $2\times4=8$.
2. Observe that these examples support the claim but only for particular cases.
3. Recognize that the actual target is a universal statement about any two even numbers.
4. Conclude that the argument must move from arithmetic examples to algebra to prove the statement for every even number.

Example

The presenter writes `$6 \times 8 = 48$ ✓` and `$2 \times 4 = 8$ ✓` on the right side of the page. The speaker says, "say if I multiply six by eight, I get forty-eight ... two by four is eight ... and eight is also even so it works for that one." Then he adds, "But we want to prove for every even number, when you multiply any two even numbers, you get another even number. So we're going to have to use algebra."

Common misconceptions

• Thinking that verifying a few cases constitutes a general proof.
• Believing that algebra is unnecessary if examples seem to work.

Watch the explanation

Explore next

• What exactly is assumed in the proof by contradiction for this theorem?
• How are the arbitrary even numbers represented algebraically in the proof?
• How is the product of two even numbers rewritten to show it is twice another quantity?