1.4 A Pinch of History

From Mesopotamia to Brahmagupta

Some of the earliest tables of squares and cubes were inscribed on Mesopotamian clay tablets nearly 3,700 years ago. Scribes used them to read off square and cube roots when surveying land or laying out buildings, and one famous tablet even records a remarkably accurate value for the diagonal of a square (essentially \sqrt{2}).

In ancient India the Sanskrit word varga named both a square figure (with its area) and the second power, while ghana named both the solid cube and the third power. Aryabhata (around 499 CE) described a square as “a figure of four equal sides, and the number giving its area.”

Why do we call \sqrt{\ } a “root”? The Sanskrit mula meant the root of a plant, and also a basis or origin. Varga-mula (the origin of the square) meant square root, and ghana-mula meant cube root. The idea passed into Arabic (jidhr) and Latin (radix), both words for a plant’s root, which is why English settled on “root.” Brahmagupta (628 CE) summed it up neatly: the root of a square is that of which it is a square.

Figure it Out: Cubes & Reasoning

Practice

  1. Find the cube roots of 21952 and 13824.
  2. What is the smallest number by which 1536 must be multiplied to make a perfect cube?
  3. State true or false, with reasons:   (i) The cube of an even number is always even.   (ii) A perfect cube can end in the digit 2.   (iii) The cube of a one-digit number is always a one- or two-digit number.   (iv) If a number is divisible by 3, then its cube is divisible by 27.   (v) Every perfect cube is also a perfect square.
  4. Using the last-digit trick, guess the cube roots of 2197, 6859, 24389 and 50653.
  5. Which is greatest?   (i) 50^3 - 49^3   (ii) 38^3 - 37^3   (iii) 50^2 - 49^2   (iv) 38^2 - 37^2
  1. 21952 = 28^3, so \sqrt[3]{21952} = \mathbf{28}; and 13824 = 24^3, so \sqrt[3]{13824} = \mathbf{24}.
  2. 1536 = 2^9 \times 3. The power of 2 is already a multiple of 3, but the single 3 needs two more to become 3^3. Multiply by 3^2 = \mathbf{9}: 1536\times 9 = 13824 = 24^3.
    1. True, an even number stays even when cubed. (ii) True, for example 8^3 = 512, which ends in 2. (iii) False, 5^3 = 125 already has three digits. (iv) True, if n = 3k then n^3 = 27k^3, a multiple of 27. (v) False, 8 = 2^3 is a cube but not a perfect square.
  4. Last digits give the units of each root: 2197\to\mathbf{13}, 6859\to\mathbf{19}, 24389\to\mathbf{29}, 50653\to\mathbf{37}. (The size of the number fixes the tens digit; the last-digit swap rule fixes the units.)
  5. (i) 50^3 - 49^3 = 7351 is the greatest. (For comparison: 38^3-37^3 = 4219, 50^2-49^2 = 99, 38^2-37^2 = 75.)