congruence Two numbers a and b are said to be congruent modulo N, for some positive integer N, if a and
b leave the same remainder when divided by N. We
write: a ≡ b (mod N). For example, since 16 and 21 are
both 1 more than a multiple of 5, we have 16 ≡
21(mod5). Also 14 ≡ 8(mod3) and 28 ≡ 0(mod7).
One can equivalently interpret the statement a ≡
b(modN) to mean: “the difference a – b is divisible by
N.” Since –2 and 16, for instance, differ by a multiple
of 9, we have –2 ≡ 16(mod9). Two numbers that are
not congruent modulo N are called incongruent modulo N.
One can add, subtract, and multiply two congruences of the same modulus N in the same manner
one adds, subtracts, and multiplies ordinary quantities.
For example, noting that 14 ≡ 4(mod10) and 23 ≡
3(mod10), we do indeed have:
14 + 23 ≡ 4 + 3(mod10)
14 – 23 ≡ 4 – 3(mod10)
14 × 23 ≡ 4 × 3(mod10)
Unfortunately the process of division is not preserved
under congruence. For example, 14 ÷ 2 is not congruent to 4 ÷ 2 modulo 10. A careful study of MODULAR
ARITHMETIC explains under which circumstances division is permissible.
The arithmetic of congruence naturally occurs in
any cyclic phenomenon. For example, finding the day
of the week for a given date requires working with
congruences modulo 7, and the arithmetic for counting hours as they pass works with congruences modulo 24 or modulo 12. (This leads to the study of
CLOCK MATH.)
Certain DIVISIBILITY RULES can be explained via
congruences. For example, since 10 ≡ 1 (mod9), any
power of 10 is also congruent to 1 modulo 9: 10
n
≡ 1
n
= 1 (mod9). Consequently, any number is congruent
modulo 9 to the sum of its digits. For example,
486 = 4 × 10
2 + 8 × 10 + 6 × 1
≡ 4 × 1 + 8 × 1 + 6 × 1(mod9)
= 4 + 8 + 6(mod9)
Since 4 + 8 + 6 is a multiple of 9, it follows that 486 is
divisible by 9.
See also CASTING OUT NINES; DAYS-OF-THE-WEEK
FORMULA.
congruent figures Two geometric figures are congruent if they are the same shape and size. More precisely,
two POLYGONS are congruent if, under some correspondence between sides and vertices, corresponding sidelengths are equal and corresponding interior angles are
equal. Two different squares with the same side-length,
for example, are congruent figures.
Note that two plane figures can be congruent without being identical: one figure may be the mirror image
of the other. Two figures are called directly congruent if
one can be brought into coincidence with the other by
rotating and translating the figure in the plane and
oppositely congruent if one must also apply a reflection. Two identical squares, for example, are directly
congruent no matter where on the plane they are
placed. Two scalene triangles with matching sidelengths might or might not be directly congruent. There
are a number of geometric tests to determine whether
or not two triangles are congruent as given by the
AAA/AAS/ASA/SAS/SSS rules.
In three-dimensional space, two solids are directly
congruent if they are identical. If each is the mirror
image of the other, they are oppositely congruent.
The term congruent is sometimes applied to other
geometric constructs to mean “the same.” For example, two line segments are congruent if they have equal
length, or two ANGLEs are congruent if they have equal
measure.
See also SIMILAR FIGURES.
conic sections Slicing a right circular CONE with a
plane that does not pass through the vertex of the cone
produces curves called the conic sections, or simply
conics. If the slicing plane is parallel to a straight line
that generates the cone, then the resulting conic is a
PARABOLA. Otherwise, if the slicing plane passes
through just one nappe of the cone, the curve produced
is either a CIRCLE or an ELLIPSE, or a HYPERBOLA if the
slicing plane cuts both nappes.
If we think of the cone as light rays emanating
from a light source held at the vertex, then the shadow
cast by a circular ring onto a sheet of card will be a
conic section; the particular conic produced depends on
the angle at which the card is held. The open ring at
the top of a lampshade, for example, casts a hyperbolic
shadow on the wall. In the same way, shadows cast by
solid balls are conic sections.
92 congruence
b leave the same remainder when divided by N. We
write: a ≡ b (mod N). For example, since 16 and 21 are
both 1 more than a multiple of 5, we have 16 ≡
21(mod5). Also 14 ≡ 8(mod3) and 28 ≡ 0(mod7).
One can equivalently interpret the statement a ≡
b(modN) to mean: “the difference a – b is divisible by
N.” Since –2 and 16, for instance, differ by a multiple
of 9, we have –2 ≡ 16(mod9). Two numbers that are
not congruent modulo N are called incongruent modulo N.
One can add, subtract, and multiply two congruences of the same modulus N in the same manner
one adds, subtracts, and multiplies ordinary quantities.
For example, noting that 14 ≡ 4(mod10) and 23 ≡
3(mod10), we do indeed have:
14 + 23 ≡ 4 + 3(mod10)
14 – 23 ≡ 4 – 3(mod10)
14 × 23 ≡ 4 × 3(mod10)
Unfortunately the process of division is not preserved
under congruence. For example, 14 ÷ 2 is not congruent to 4 ÷ 2 modulo 10. A careful study of MODULAR
ARITHMETIC explains under which circumstances division is permissible.
The arithmetic of congruence naturally occurs in
any cyclic phenomenon. For example, finding the day
of the week for a given date requires working with
congruences modulo 7, and the arithmetic for counting hours as they pass works with congruences modulo 24 or modulo 12. (This leads to the study of
CLOCK MATH.)
Certain DIVISIBILITY RULES can be explained via
congruences. For example, since 10 ≡ 1 (mod9), any
power of 10 is also congruent to 1 modulo 9: 10
n
≡ 1
n
= 1 (mod9). Consequently, any number is congruent
modulo 9 to the sum of its digits. For example,
486 = 4 × 10
2 + 8 × 10 + 6 × 1
≡ 4 × 1 + 8 × 1 + 6 × 1(mod9)
= 4 + 8 + 6(mod9)
Since 4 + 8 + 6 is a multiple of 9, it follows that 486 is
divisible by 9.
See also CASTING OUT NINES; DAYS-OF-THE-WEEK
FORMULA.
congruent figures Two geometric figures are congruent if they are the same shape and size. More precisely,
two POLYGONS are congruent if, under some correspondence between sides and vertices, corresponding sidelengths are equal and corresponding interior angles are
equal. Two different squares with the same side-length,
for example, are congruent figures.
Note that two plane figures can be congruent without being identical: one figure may be the mirror image
of the other. Two figures are called directly congruent if
one can be brought into coincidence with the other by
rotating and translating the figure in the plane and
oppositely congruent if one must also apply a reflection. Two identical squares, for example, are directly
congruent no matter where on the plane they are
placed. Two scalene triangles with matching sidelengths might or might not be directly congruent. There
are a number of geometric tests to determine whether
or not two triangles are congruent as given by the
AAA/AAS/ASA/SAS/SSS rules.
In three-dimensional space, two solids are directly
congruent if they are identical. If each is the mirror
image of the other, they are oppositely congruent.
The term congruent is sometimes applied to other
geometric constructs to mean “the same.” For example, two line segments are congruent if they have equal
length, or two ANGLEs are congruent if they have equal
measure.
See also SIMILAR FIGURES.
conic sections Slicing a right circular CONE with a
plane that does not pass through the vertex of the cone
produces curves called the conic sections, or simply
conics. If the slicing plane is parallel to a straight line
that generates the cone, then the resulting conic is a
PARABOLA. Otherwise, if the slicing plane passes
through just one nappe of the cone, the curve produced
is either a CIRCLE or an ELLIPSE, or a HYPERBOLA if the
slicing plane cuts both nappes.
If we think of the cone as light rays emanating
from a light source held at the vertex, then the shadow
cast by a circular ring onto a sheet of card will be a
conic section; the particular conic produced depends on
the angle at which the card is held. The open ring at
the top of a lampshade, for example, casts a hyperbolic
shadow on the wall. In the same way, shadows cast by
solid balls are conic sections.
92 congruence