Example 5
When one is given a scary equation like this: f(x)=319one should not panic. This equation can be sorted out in a few steps if
you know that f(x)=319 =
f(x)=316×33. Always reduce the
exponents into 2n sequence exponents and single-digit
remainder exponents that do not belong to the 2nsequence. The value 316 can be calculated in 4
multiplication operations as shown in table 1 in example 4. The value
33 will be calculated in 2 multiplication operations.
Therefore the total number of multiplication operations will 4+2=6. The
value 316×33 itself will take 1
multiplication operation to calculate. Therefore if you know that
m19 can be divided into m16 and
m3 then you will know that it takes only 4+2+1=7
multiplication operations to get the answer instead of doing all the 16
multiplication operations that is used in the normal calculations.
Knowing how a particular number is divided into its 2nsequence components is extremely important and one can also use a table
of 2n values to make work easier. For example, with
enough practice or with a 2n table nearby, it is not
difficult at all to realize that 163 can be broken into its
2n components like this 128,32 and 3. The number 3 is
not a 2n component, it is just the remainder.
Therefore with enough practice or with a table nearby it is not
difficult for one to discover that m163 =
m128×m32×m3.
m128 requires 7 multiplication operations.
m32 requires 5 multiplication operations, however we
do not have to recalculate m32 since we already
calculated it when we were calculating m128. This
means we can just pick the values for m32 from our
calculations of m128. Therefore zero multiplication
operation is necessary to calculate m32 because it is
already calculated. The value m3 requires 2
multiplication operations. Remember that
m128×m32×m3 itself
requires 2 multiplication operation. Therefore the total number of
multiplication operations required is 7+2+2=11. So instead of
multiplying m by itself 162 times, we can just use the ngazi method
which requires only 11 multiplication operations to get the answer.
Next, The author will go ahead and carry out the actual calculations to
prove that one need only 7 multiplication operations to calculate
f(x)=m19.
First of all, we assume that the mathematician who is trying to use this
method is proficient in converting m19 to its
2n components. Now let us solve an actual problem.