Proof by Contradiction: Odd and Its Square

We have had several examples on proof by contradiction in this blog like irrationality of square root of 3 and the sum of root of 2 and root of 3. In this post, we will have another detailed example.

In proof by contradiction, we want to change the statement “if P then Q” to “if NOT Q then NOT P.” These two statements are equivalent, so if we can prove the latter, then we have proved the former. Consider the following theorem.

Theorem: For any integer x, if x^2 is odd, then x is odd.

We can assign the statements above to P (the hypothesis) and Q (the conclusion) as follows.

P: x^2 is odd

Q: x is odd.

If we are going to change this statement to “if NOT Q then NOT P,” then, we have to find the opposite of P and opposite of Q.  Continue reading…

 

A Geometric Representation of the Distributive Property

The distributive property of multiplication over addition states that for all real numbers a, b and c, then

a(b + c) = ab + ac.

In this short post, we are going to see the visual representation or ‘visual proof’ of this property where it is represented as area. However, one limitation of this representation is it does not represent negative values for a, b or c. This means that this is only good for positive numbers.

A Geometric Representation of  the Distributive Property

In the diagram below, two rectangles (red and blue) are placed adjacently. The have the same height a. The red rectangle has width b while the blue rectangle  has width cContinue reading…

 

Proof That Only 3 Regular Polygons Tessellate

Tessellation is the tiling a surface (in this case a flat surface) without gaps and overlaps. This concept is very important since many of the floors and walls are tiled nowadays. But tessellation is a lot more than that. It actually obeys the law of mathematics particularly angles.

The plane can be tiled with just one polygon as shown below. Some of such types of polygons are squares, triangles, and hexagons.

Can you think of other shapes that can tile the plane individually?

Continue reading…

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