Electric-Eye
Visualizing Cube Roots
How do you extract the cube root of 87?
One of the best ways of finding square roots for all numbers which is commonly taught, is called the Babylonian method. It resembles long division, with remainders that build at an angle under a long radical sign. To understand how it works, you can think of it in geometric terms. On each iteration a root which when squared will account for a larger and larger square area of the total area of the square is discovered. The remaining two-dimensional area to be resolved is then represented by two long rectangles, and a smaller square which border the square of the root discovered so far.
This image represents my attempt to reason by analogy from this geometric understanding of the square root algorithm, to understand how the cube-root algorithm must work.
Once I had this picture, it became obvious that the second and all subsequent steps of the cube root algorithm involves coming as close as possible to the area of 3 squat boxes, 3 long boxes and a cube.
Where in the square root algorithm digits are broken up into groups of 2, beginning at the decimal point, and going into each direction, they are in the cube root algorithm broken into groups of 3. Where in the square root algorithm you begin by finding the closest square that is less than or equal to a number represented by the first group of digits, in the cube root algorithm your first step is to find the cube that is equal to or less than the number represented by the first group of digits. You second step is to take that quantity, call it a and then find a single-digit quantity b, that when the following formula is applied, give an answer equal to or less than the remainder:
b^3 + 3( 10a^2 * b + b^2 * 10a)
or b cubed plus 3 times the expression 10a squared times b plus b squared times 10a.
b cubed is literally the cube of the digit chosen for b. This quantity is taken once in each iteration. With each iteration it becomes the most negligible quantity of them all.
The rest of the formula is multiplied by 3, because we are dealing with 3 identical sets of two different boxes. These represent the 3 undiscovered edge-boxes, and 3 undiscovered face-boxes of the cube discovered so far.
In both of these terms a is multiplied by 10, because we are doing the calculation for the sake of the next place-value digit. We are doing a calculation for one tenth the magnitude of the preceding digit. The last few digits of these numbers seem to always be zeros, except for the cube of b.
Both of these terms have a squared value multiplied by a value without an exponent. They describe a 3 dimensional area, one side of which is always square. They can be thought of as extruded from the face of one of the cubes. Terms with a squared in them are extruded from the faces of the discovered cube, terms with b squared in them can be thought of as extruded from the face of the cube of b, all the way to the edge of the cube of a.
With each iteration of the algorithm the significance of b shrinks to about a tenth of what it was in the previous iteration, and therefore terms that depend on b cubed or squared rapidly diminish in importance. After a couple of iterations it is more important to see if the digit chosen for b produces an acceptable result when multiplied against 30a^2, and move on if it isn't close. (For that matter, just look at 30a^2, and see if it is already large or small compared with the remainder.)
One important lesson of the cube root algorithm is that cubes are very touchy, the slightest change to a digit way off to the right can put you out of the ball-park for the answer you're trying to get. The calculations are so cumbersome and susceptible to error simply because they are so numerous when attempted by hand that you would quickly be satisfied with a gross approximation rather than five correct significant digits.
This method of finding a cube root will find the cube root of 87. Since 87 is not a perfect cube, unlike, say 27, 64, or 125, some other methods that can find cube roots, like prime factoring, will not work for 87.
Visualizing Cube Roots
How do you extract the cube root of 87?
One of the best ways of finding square roots for all numbers which is commonly taught, is called the Babylonian method. It resembles long division, with remainders that build at an angle under a long radical sign. To understand how it works, you can think of it in geometric terms. On each iteration a root which when squared will account for a larger and larger square area of the total area of the square is discovered. The remaining two-dimensional area to be resolved is then represented by two long rectangles, and a smaller square which border the square of the root discovered so far.
This image represents my attempt to reason by analogy from this geometric understanding of the square root algorithm, to understand how the cube-root algorithm must work.
Once I had this picture, it became obvious that the second and all subsequent steps of the cube root algorithm involves coming as close as possible to the area of 3 squat boxes, 3 long boxes and a cube.
Where in the square root algorithm digits are broken up into groups of 2, beginning at the decimal point, and going into each direction, they are in the cube root algorithm broken into groups of 3. Where in the square root algorithm you begin by finding the closest square that is less than or equal to a number represented by the first group of digits, in the cube root algorithm your first step is to find the cube that is equal to or less than the number represented by the first group of digits. You second step is to take that quantity, call it a and then find a single-digit quantity b, that when the following formula is applied, give an answer equal to or less than the remainder:
b^3 + 3( 10a^2 * b + b^2 * 10a)
or b cubed plus 3 times the expression 10a squared times b plus b squared times 10a.
b cubed is literally the cube of the digit chosen for b. This quantity is taken once in each iteration. With each iteration it becomes the most negligible quantity of them all.
The rest of the formula is multiplied by 3, because we are dealing with 3 identical sets of two different boxes. These represent the 3 undiscovered edge-boxes, and 3 undiscovered face-boxes of the cube discovered so far.
In both of these terms a is multiplied by 10, because we are doing the calculation for the sake of the next place-value digit. We are doing a calculation for one tenth the magnitude of the preceding digit. The last few digits of these numbers seem to always be zeros, except for the cube of b.
Both of these terms have a squared value multiplied by a value without an exponent. They describe a 3 dimensional area, one side of which is always square. They can be thought of as extruded from the face of one of the cubes. Terms with a squared in them are extruded from the faces of the discovered cube, terms with b squared in them can be thought of as extruded from the face of the cube of b, all the way to the edge of the cube of a.
With each iteration of the algorithm the significance of b shrinks to about a tenth of what it was in the previous iteration, and therefore terms that depend on b cubed or squared rapidly diminish in importance. After a couple of iterations it is more important to see if the digit chosen for b produces an acceptable result when multiplied against 30a^2, and move on if it isn't close. (For that matter, just look at 30a^2, and see if it is already large or small compared with the remainder.)
One important lesson of the cube root algorithm is that cubes are very touchy, the slightest change to a digit way off to the right can put you out of the ball-park for the answer you're trying to get. The calculations are so cumbersome and susceptible to error simply because they are so numerous when attempted by hand that you would quickly be satisfied with a gross approximation rather than five correct significant digits.
This method of finding a cube root will find the cube root of 87. Since 87 is not a perfect cube, unlike, say 27, 64, or 125, some other methods that can find cube roots, like prime factoring, will not work for 87.