Horner's Method for Converting an Integer from Base b to Base 10

​
base
2
3
4
5
6
7
8
9
degree 4
2
3
4
5
new number
steps
0
1
2
3
4
10110
2
1
0
1
1
0
2
1
​
Horner's method for expressing a polynomial reduces the number of multiplications and results in greater numerical stability by potentially avoiding the subtraction of large numbers. It is based on successive factorization to eliminate powers of
x
greater than 1.
Suppose
P(x)=
a
n
n
x
+
a
n-
1
n-1
x
+…+
a
0
; then the method gives
P(x)=((
a
n
x+
a
n-1
)x+…)x+
a
0
.
As an application of Horner's method, if
a
n
a
n-1
…
a
0
b
is an integer in base
b
, then
P(b)
is the corresponding integer in base 10.

External Links

Horner's Rule (Wolfram MathWorld)
Synthetic Division (Ruffini's Rule)
Horner's Method
Ruffini-Horner Algorithm for Complex Arguments
Ruffini-Horner Method for Polynomials with Rational Roots
Ruffini-Horner Method for a Polynomial in Powers of
Graphical Application of Horner's Method
Basic Arithmetic in Other Bases
Trick to Multiply by
Adding in a Given Base

Permanent Citation

Izidor Hafner
​
​"Horner's Method for Converting an Integer from Base b to Base 10"​
​http://demonstrations.wolfram.com/HornersMethodForConvertingAnIntegerFromBaseBToBase10/​
​Wolfram Demonstrations Project​
​Published: January 24, 2019