Any COMPOSITE NUMBER smaller than N has at least
one prime factor smaller than the square root of N. (A
number k smaller than N cannot factor as k = a × b with
both a and b larger than √
–
N.) This shows that when
performing this procedure, one need only delete multiples of prime numbers smaller than the square root
of N. For example, to find all the prime numbers
from 2 to 100, delete only the multiples of 2, 3, 5 and
7 from the list.This observation simplifies the procedure considerably.
In the 1800s, Polish astronomer Yakov Kulik used
this method to find all the prime numbers between 1
and 100,000,000. It took him over 20 years to complete the task. Unfortunately, the library to which Kulik
gave his manuscript lost the pages that listed the primes
he discovered between 12,642,000 and 22,852,800.
significant figures See ERROR.
similar figures Two geometric figures are similar if
they are the same shape but not necessarily the same
size. (Mirror images are allowed.) More precisely, two
POLYGONS are similar if, under some correspondence
between their sides and vertices, corresponding interior
angles are equal and corresponding sides differ in ratio
by a constant factor. The constant of proportionality is
called the scale factor.
As an example, any two squares are similar. If the
side-length of one square is double the side-length of
the other, say, then the two figures have scale factor 2.
Any figure enlarged or reduced in size, with the aid of
a photocopier say, produces a new figure similar to
the first.
Two TRIANGLES are similar if:
1. The three interior angles of one triangle match the
interiorangles of the other.
This follows, since the LAW OF SINES shows that corresponding sides will also have the same ratio. This is
often called the AAA rule.
2. Three sides of one triangle are proportional, by the
same scale factor, to the three sides of the other.
This time the LAW OF SINES shows that the three corresponding angles are equal.
3. An angle of one triangle is equal to an angle of the
other, and the sides forming the angle in one are
proportional to the same sides in the other.
The LAW OF COSINES shows that the third sides of the
triangles are in the same proportion.
Identifying similar triangles is often the key step in
proving geometric results. The CIRCLE THEOREMS,
PTOLEMY’S THEOREM, and the SECANT theorem, for
example, demonstrate this.
Any two circles are similar. This observation
explains why the value π is the same for all circles.
Two figures that are similar with a scale factor of
1 are called CONGRUENT FIGURES. A figure that is
composed of parts similar to the entire figure is a
FRACTAL.
See also AAA/AAS/ASA/SAS/SSS; EUCLIDEAN GEOMETRY.
simple interest See INTEREST.
Simpson, Thomas (1710–1761) British Calculus
Born on August 20, 1710, in Leicestershire, England,
Thomas Simpson is remembered in mathematics solely
for the rule that wrongly bears his name.
With limited education, Simpson began his career
as a professional astrologer and confidence man. After
an unfortunate incident in 1733 from which he was
obliged to leave his home county of Leicestershire,
Simpson accepted an evening teaching position in
Derby. His interest in mathematics then developed.
Simpson published mathematical articles in the
Ladies’ Diary and soon developed a reputation as a
capable scholar in the field of calculus. In 1737 he
wrote A New Treatise of Fluxions and followed this
work with the release of four more influential texts
over the following 6 years. His writing garnered him
considerable fame in England at the time. In 1743 he
was appointed second mathematical master at the
Royal Military Academy and in 1735 was elected as a
fellow of the ROYAL SOCIETY.
Later in his career, while still working as a mathematics teacher, Simpson also published three best-selling
elementary textbooks: Algebra (1745), with 10 English
editions; Geometry (1747), with six English editions,
five French editions, and one Dutch edition; and Trigonometry (1748), with five English editions.
Simpson, Thomas 463
one prime factor smaller than the square root of N. (A
number k smaller than N cannot factor as k = a × b with
both a and b larger than √
–
N.) This shows that when
performing this procedure, one need only delete multiples of prime numbers smaller than the square root
of N. For example, to find all the prime numbers
from 2 to 100, delete only the multiples of 2, 3, 5 and
7 from the list.This observation simplifies the procedure considerably.
In the 1800s, Polish astronomer Yakov Kulik used
this method to find all the prime numbers between 1
and 100,000,000. It took him over 20 years to complete the task. Unfortunately, the library to which Kulik
gave his manuscript lost the pages that listed the primes
he discovered between 12,642,000 and 22,852,800.
significant figures See ERROR.
similar figures Two geometric figures are similar if
they are the same shape but not necessarily the same
size. (Mirror images are allowed.) More precisely, two
POLYGONS are similar if, under some correspondence
between their sides and vertices, corresponding interior
angles are equal and corresponding sides differ in ratio
by a constant factor. The constant of proportionality is
called the scale factor.
As an example, any two squares are similar. If the
side-length of one square is double the side-length of
the other, say, then the two figures have scale factor 2.
Any figure enlarged or reduced in size, with the aid of
a photocopier say, produces a new figure similar to
the first.
Two TRIANGLES are similar if:
1. The three interior angles of one triangle match the
interiorangles of the other.
This follows, since the LAW OF SINES shows that corresponding sides will also have the same ratio. This is
often called the AAA rule.
2. Three sides of one triangle are proportional, by the
same scale factor, to the three sides of the other.
This time the LAW OF SINES shows that the three corresponding angles are equal.
3. An angle of one triangle is equal to an angle of the
other, and the sides forming the angle in one are
proportional to the same sides in the other.
The LAW OF COSINES shows that the third sides of the
triangles are in the same proportion.
Identifying similar triangles is often the key step in
proving geometric results. The CIRCLE THEOREMS,
PTOLEMY’S THEOREM, and the SECANT theorem, for
example, demonstrate this.
Any two circles are similar. This observation
explains why the value π is the same for all circles.
Two figures that are similar with a scale factor of
1 are called CONGRUENT FIGURES. A figure that is
composed of parts similar to the entire figure is a
FRACTAL.
See also AAA/AAS/ASA/SAS/SSS; EUCLIDEAN GEOMETRY.
simple interest See INTEREST.
Simpson, Thomas (1710–1761) British Calculus
Born on August 20, 1710, in Leicestershire, England,
Thomas Simpson is remembered in mathematics solely
for the rule that wrongly bears his name.
With limited education, Simpson began his career
as a professional astrologer and confidence man. After
an unfortunate incident in 1733 from which he was
obliged to leave his home county of Leicestershire,
Simpson accepted an evening teaching position in
Derby. His interest in mathematics then developed.
Simpson published mathematical articles in the
Ladies’ Diary and soon developed a reputation as a
capable scholar in the field of calculus. In 1737 he
wrote A New Treatise of Fluxions and followed this
work with the release of four more influential texts
over the following 6 years. His writing garnered him
considerable fame in England at the time. In 1743 he
was appointed second mathematical master at the
Royal Military Academy and in 1735 was elected as a
fellow of the ROYAL SOCIETY.
Later in his career, while still working as a mathematics teacher, Simpson also published three best-selling
elementary textbooks: Algebra (1745), with 10 English
editions; Geometry (1747), with six English editions,
five French editions, and one Dutch edition; and Trigonometry (1748), with five English editions.
Simpson, Thomas 463