The cube-root of a number x is a number b satisfying b3 = x. A number x is said to be a perfect cube if there is an integer b such that b3 = x. If a perfect cube has only three or four digits, it is not inconvenient to try to get its cube-root by looking at cubes of integers smaller than 22 - the cube of 22 exceeds 10000.
If the number x is not a perfect cube, the method is still useful in that we can find two consecutive integers between which the cube-root lies, but we are unable to state the cube-root with greater precision.
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Electronic calculators became publicly available around 1970 A.D. Before that, the method used for finding cube-root employed the concept of "common logarithms". Common logarithms refer to logarithms where the base is 10.
What is logarithm?
Logarithm, discovered by Napier, is associated with the powers of numbers. If b3 = x then 3 = logb x. Thus since 32 = 9, log39 = 2. Written another way, since Ö9 = 3, or 91/2 = 3, we get log93 = 1/2.
If you fix a base, then all positive numbers can be expressed in terms of powers if that base. Thus every positive number can be expressed as a power of 10. Then the logarithm of a positive number is the power to which 10 should be raised to get 10.
A booklet sold by various publishers listed the logarithms in base 10 of numbers from 0.0000 to 0.9999. The listing was all that was needed to specify, with four-digit precision, the logarithms of all numbers since every number can be written in scientific notation as the product of a power of 10 and a decimal between 1.000 and 9.999, inclusive.
Logarithms are suitable for not only finding cube-root, but any other real root. Why?
A result in logarithm in any base b is that if bn = x, then
For instance, to find the cube-root of 10000, we ask for b such that b3 = x. That is x1/4 = b.
is between two
with having to find the cube-root of a number, we mainly resort to one of three things:
(1) The electronic calulator
Eulcid, Book I of the Elements, Proposition 48
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Last updated - June 18, 2003