Notice that we find two groups of 12 at the 10s
position (that is, two groups of 120), and three groups
of 12 at the units position. Thus: 276 ÷ 12 = 23. The
standard algorithm taught to school children is nothing
more than a recording system for this process of finding groups of twelve. Notice too that one does not need
to know the type of machine, that is, the base of the
number system in which one is working in order to
compute a long-division problem. If we simply write
the base number as x, and work with a 1 ← x machine,
then the same computation provides a method for
dividing polynomials. In our example, we see that:
(2x
2 + 7x + 6) ÷ (x + 2) = 2x + 3
Thus the division of polynomials can be regarded as a
computation of long division. There is a technical difficulty with this: a polynomial may have negative coefficients, and each negative coefficient would correspond
to a negative number of pennies in a cell. If one is willing to accept such quantities, then the number-base
machine model continues to work. (Note, in this
extended model, that one can insert into any cell an
equal number of positive and negative pennies without
changing the system. Indeed, it might be necessary to
do this in order to find the desired groups of pennies.)
The process of long division might produce non
zero remainders. For example, in base 5, dividing 1432
by 13 yields the answer 110 with a remainder of 2
units. (In base 10, this reads: 242 ÷ 8 = 30 with a
remainder of 2.) If one is willing to work with negative
powers of five, and “unfire” a group of five pennies,
one can continue the long division process to compute,
in base 5, that 1432 ÷ 13 = 110.1111…
See also BABYLONIAN MATHEMATICS; BINARY NUMBERS; DIGIT; MAYAN MATHEMATICS; NESTED MULTIPLICATION; ZERO.
base of a polygon/polyhedron The base of a triangle, or of any POLYGON, is the lowest side of the figure,
usually drawn as a horizontal edge parallel to the bottom of the page. Of course other edges may be considered the base if one reorients the figure. The base of a
POLYHEDRON, such as a cube or a pyramid, is the lowest FACE of the figure. It is the face on which the figure
would stand if it were placed on a tabletop.
The highest point of a geometric figure opposite
the base is called the APEX of the figure, and the distance from the base to the apex is called the height of
the figure.
basis See LINEARLY DEPENDENT AND INDEPENDENT.
Bayes, Rev. Thomas (1702–1761) English Probability, Theology Born in London, England, (the exact
date of his birth is not known), theologian and mathematician Reverend Thomas Bayes is best remembered
for his influential article “An Essay Towards Solving a
Problem in the Doctrine of Chances,” published posthumously in 1763, that outlines fundamental principles of
PROBABILITY theory. Bayes developed innovative techniques and approaches in the theory of statistical inference, many of which were deemed controversial at the
time. His essay sparked much further research in the
field and was profoundly influential. The work also
contains the famous theorem that today bears his name.
An ordained minister who served the community of
Tunbridge Wells, Kent, England, Bayes also pursued
mathematics as an outside interest. As far as historians
can determine, he published only two works during his
lifetime. One was a theological essay in 1731 entitled
“Divine Benevolence, or an Attempt to Prove that the
Principal End of the Divine Providence and Government
is the Happiness of His Creatures.” The other was a
mathematical piece that he published anonymously in
1736, “Introduction to the Doctrine of Fluxions, and a
38 base of a polygon/polyhedron
Long division base five
position (that is, two groups of 120), and three groups
of 12 at the units position. Thus: 276 ÷ 12 = 23. The
standard algorithm taught to school children is nothing
more than a recording system for this process of finding groups of twelve. Notice too that one does not need
to know the type of machine, that is, the base of the
number system in which one is working in order to
compute a long-division problem. If we simply write
the base number as x, and work with a 1 ← x machine,
then the same computation provides a method for
dividing polynomials. In our example, we see that:
(2x
2 + 7x + 6) ÷ (x + 2) = 2x + 3
Thus the division of polynomials can be regarded as a
computation of long division. There is a technical difficulty with this: a polynomial may have negative coefficients, and each negative coefficient would correspond
to a negative number of pennies in a cell. If one is willing to accept such quantities, then the number-base
machine model continues to work. (Note, in this
extended model, that one can insert into any cell an
equal number of positive and negative pennies without
changing the system. Indeed, it might be necessary to
do this in order to find the desired groups of pennies.)
The process of long division might produce non
zero remainders. For example, in base 5, dividing 1432
by 13 yields the answer 110 with a remainder of 2
units. (In base 10, this reads: 242 ÷ 8 = 30 with a
remainder of 2.) If one is willing to work with negative
powers of five, and “unfire” a group of five pennies,
one can continue the long division process to compute,
in base 5, that 1432 ÷ 13 = 110.1111…
See also BABYLONIAN MATHEMATICS; BINARY NUMBERS; DIGIT; MAYAN MATHEMATICS; NESTED MULTIPLICATION; ZERO.
base of a polygon/polyhedron The base of a triangle, or of any POLYGON, is the lowest side of the figure,
usually drawn as a horizontal edge parallel to the bottom of the page. Of course other edges may be considered the base if one reorients the figure. The base of a
POLYHEDRON, such as a cube or a pyramid, is the lowest FACE of the figure. It is the face on which the figure
would stand if it were placed on a tabletop.
The highest point of a geometric figure opposite
the base is called the APEX of the figure, and the distance from the base to the apex is called the height of
the figure.
basis See LINEARLY DEPENDENT AND INDEPENDENT.
Bayes, Rev. Thomas (1702–1761) English Probability, Theology Born in London, England, (the exact
date of his birth is not known), theologian and mathematician Reverend Thomas Bayes is best remembered
for his influential article “An Essay Towards Solving a
Problem in the Doctrine of Chances,” published posthumously in 1763, that outlines fundamental principles of
PROBABILITY theory. Bayes developed innovative techniques and approaches in the theory of statistical inference, many of which were deemed controversial at the
time. His essay sparked much further research in the
field and was profoundly influential. The work also
contains the famous theorem that today bears his name.
An ordained minister who served the community of
Tunbridge Wells, Kent, England, Bayes also pursued
mathematics as an outside interest. As far as historians
can determine, he published only two works during his
lifetime. One was a theological essay in 1731 entitled
“Divine Benevolence, or an Attempt to Prove that the
Principal End of the Divine Providence and Government
is the Happiness of His Creatures.” The other was a
mathematical piece that he published anonymously in
1736, “Introduction to the Doctrine of Fluxions, and a
38 base of a polygon/polyhedron
Long division base five