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Week 13: More Proofs without Words
Mathematics is abstract thought, mathematics is pure logic, mathematics
is creative art. All these statements are wrong, but they are all a little right,
and they are all nearer the mark than “mathematics is numbers” or
“mathematics is geometric shapes.”
Paul Halmos
Figurate Numbers (Continued)
We continue now with a few more “proofs without words” related to figurate numbers.
Exercise 1: The diagram to the left demonstrates a certain formula, which we
have never seen before, relating two consecutive triangular numbers to
another triangular number. What is the formula? Explain exactly how the
diagram demonstrates the formula.
Exercise 2: Give a formal (completely non-visual) proof of the formula
from Exercise 1.
Exercise 3: Give a “proof without words” that the sum of two
consecutive tetrahedral numbers is always a square
pyramidal number.
Infinite Sums
Proofs without words can be a powerful tool for grasping
the truth of many mathematical statements and formulas,
especially those involving limits, such as infinite series.
The proof without words shown to the right may be
the most famous of all of them.
Exercise 4: Consider the proof without words
shown to the right. Give a detailed explanation
of exactly how the diagram demonstrates the
fact that it claims to demonstrate.
1 1
1
1
+ 2+ 3+⋯ =
4 4
4
3
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Exercise 5: As with previous proofs without words, explain the one given below.
Exercise 6: The fact “proved” to the left can be
demonstrated just as well by drawing a line segment
(with length 1) and dividing it over and over again in
much the same way. Show this version of the proof
without words.
Exercise 7: Draw a new diagram like the one in the
previous exercise, but this time taking two-thirds of
the line segment, and then two-thirds of what
remains, and so on and so on. What infinite sum
formula does this new diagram prove?
𝟏 𝟏 𝟏 𝟏
𝟏
+ + +
+
+⋯ =𝟏
𝟐 𝟒 𝟖 𝟏𝟔 𝟑𝟐
Exercise 8: Create a new diagram, similar to the
previous one, to “prove” that 0.9999 … = 1.
Exercise 9: What does the diagram to the right
demonstrate about the sum 30 + 31 + 32 + … + 3𝑛 ?
Explain how the diagram shows this.
Exercise 10: Give a proof by induction for the
fact from Exercise 9.
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Geometry and Algebra
Perhaps the most famous theorem in all of mathematics is
the Pythagorean Theorem. Dozens of very different
proofs of the theorem have been devised, and many of
them can be given in the form of “proofs without words.”
The diagram to the right shows one of the most
straightforward and accessible proofs (without words) of
the Pythagorean Theorem 𝒂2 + 𝒃2 = 𝒄2 .
It is probably no surprise that a geometric diagram can
serve as a proof without words for a geometric theorem.
But there is also an ancient tradition of using simple
geometric diagrams to illustrate facts that are more purely in the realm of algebra. For
example, consider the following “proof without words” of a basic fact from algebra:
To complete the square 𝒙(𝒙 + 𝒃), or 𝒙2 + 𝒃𝒙, we can add (𝒃/𝟐)2 .
Some of the first cultures to discover the idea of “completing the square,” in order to solve a
quadratic equation, thought about the process in geometric terms, much like what is shown
above. The expression that we write in modern notation as 𝑥 2 + 𝑏𝑥 can be visualized as the
area of a rectangle with width 𝑥 and length 𝑥 + 𝑏. The sequence of three illustrations above
show how we can rearrange this area, and add a bit more, in order to form a square with
area (𝑥 + 𝑏/2)2 . This proof without words, along with the one to be considered next, is
available in animated form on myCMU.
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Here is a proof without words about a somewhat less familiar algebraic fact – or rather, a
fact about all positive real numbers:
The sum of a positive number and
its reciprocal is always at least 2.
Exercise 11: Using your basic knowledge of the areas of triangle and squares, and a little
algebra, explain in full detail exactly how the “proof without words” given earlier
demonstrates the Pythagorean Theorem.
Exercise 12: Explain in the fullest possible detail exactly how the diagram given directly
above demonstrates that the sum of a positive number and its reciprocal is always at least 2.
Exercise 13: Give a formal (completely non-visual) proof that the sum of a positive number
and its reciprocal is always at least 2.