How are the arbitrary even numbers represented algebraically in the proof?

Conditions

• The numbers $x$ and $y$ are assumed to be even.
• $c$ and $p$ are natural numbers ($\mathbb{N}$).
• The definition of an even number is applied: any even number is twice another number.

Reasoning, step by step

1. Introduce two arbitrary even numbers, $x$ and $y$.
2. Apply the definition of evenness: any even number can be written as twice some other number.
3. Write $x = 2c$ with $c \in \mathbb{N}$.
4. Write $y = 2p$ with $p \in \mathbb{N}$.
5. Use these representations to substitute into the product expression.

Example

The presenter writes `$x = 2c c \in N$.` and `$y = 2p p \in N$.` The speaker says, "any even number can be written as twice some other number" and gives the example $10=2(5)$.

Common misconceptions

• Thinking that $c$ and $p$ must be even themselves.
• Assuming the representation applies to odd numbers.
• Confusing the multipliers $c$ and $p$ with the even numbers $x$ and $y$.

Watch the explanation

Explore next

• How is the product of two even numbers rewritten to show it is twice another quantity?
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• Why does the presenter switch from examples like $6\times 8$ and $2\times 4$ to an algebraic proof?