Math 409409G History of Mathematics

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Math 409/409GHistory of Mathematics

• Babylonian Approximation of Square Roots

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• Before looking at how the Babylonians
  approximated square roots, lets first look at
  the type of problems that led them to a need to
  approximate square roots.

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Standard Babylonian Problem

• You want to construct a rectangle having a
  specified area and a specified diagonal. What are
  the sides of this rectangle?
• Given xy a
• x2 y2 d2
• Find x and y

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• Heres how the Babylonians solved the system of
  equations
• They recognized that
• So

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• In a similar fashion, expanding
• yields
• So the given system of equations
• is equivalent to the system

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• Adding these equations and solving for x gives
• and subtracting them yields

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Conclusion

• A rectangle having area a and diagonal d has
  sides

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Example

• What are the measures of the sides of a
  rectangle having area and whose
  diagonal measures ?

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Another example

• What are the measures of the sides of a
  rectangle having area and whose
  diagonal measures ?

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To approximate the
Babylonian way

• First find the largest perfect square a2 that is
  less than n.
• Then expressed n in the form
• And then used the formula

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Examples of using this formula

• 49 72 is the largest perfect square less than
  60, so
• Similarly,

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Conclusion

• What are the measures of the sides of a
  rectangle having area and whose
  diagonal measures ?

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Babylonian proof that

• Start with a square having side a.
• Attach a rectangle measuring b/a by a.
• The area of the resulting figure is a2 b.

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• Divide the smaller attached rectangle into two
  rectangles each measuring b/2a by a.
• Remove one of the b/2a by a rectangles, rotate
  it, and attach it to the bottom of the square
  having side a.

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• The area of the resulting figure is still a2
  b.
• Enclose this figure in a square.
• The sides of this square measure a b/2a.

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• The areas of these two figures are approximately
  equal, so

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• This ends the lesson on

Babylonian Approximation of Square Roots