490 surd

6

ζ

90

ζ

This gives, for example, and .

surd A numerical expression containing irrational

numbers that arise solely from the operation of taking

square or higher roots is called a surd. For example, √

–

5,

2 – 3

√

–

7, and are all surds. (However, the irrational

number e+ ln 2, for instance, contains no roots and so

is not a surd.) A pure surd contains only irrational root

terms (such as √

–

11 + 4

√

–

13, for instance) and a mixed

surd contains both rational and irrational terms (such

as 2 – 3

√

–

7, for instance).

The conjugate of a surd that is the sum of two

terms is the difference of those same two terms. For

example, the conjugate of 2√

–

3 + 4√

–

5 is 2√

–

3 – 4√

–

5. If

only square roots are involved, then the product of a

surd and its conjugate is always a rational number.

This observation is utilized in the process of

RATIO

-

NALIZING THE DENOMINATOR

of a complicated ratio-

nal expression.

Surds of the form , where a, b, and care

integers, with bnot a perfect square, are sometimes

called quadratic surds. The term surd is rarely used

today and is generally considered obsolete.

syllogism See

ARGUMENT

.

symmetry A figure or an expression is said to be

symmetrical if parts of it can be interchanged without

changing the figure or the expression as a whole. For

example, a geometric figure such as the letter “W” can

be divided into two parts, the left half and the right

half, which can be interchanged via a

REFLECTION

about a vertical line to leave the overall shape of the

figure unchanged. We say that this figure has reflection

symmetry about a vertical line. In the same way, the

letter “C” has reflection symmetry about a horizontal

line and the letter “S” has

ROTATION

symmetry about

its center point. The figure of a

CIRCLE

has reflection

symmetry about any line that passes through its center

and rotational symmetry through any degree of turning

about its center. A

CUBE

has reflection symmetry about

any plane through its center parallel to one of its faces.

A geometric figure is said to have n-fold rotational

symmetry about an axis of rotation if it is symmetrical

about a rotation of about that axis. For example,

a regular pentagon has five-fold rotational symmetry

about a line through its center

PERPENDICULAR

to the

page on which the figure is drawn. A

TETRAHEDRON

has three-fold rotational symmetry about an axis that

passes through one of its vertices, and two-fold rota-

tional symmetry about an axis that passes through the

midpoints of two opposite edges.

The equation x2+ xy + y2is symmetrical in the

variables xand y, meaning that interchanging the two

variables yields a new equation equivalent to the origi-

nal. In general, a function fof several variables is said

to be totally symmetric if interchanging any two vari-

ables does not change the function: f(x1,…,xi,…,

xj,…,xn) = f(x1,…, xj,…, xi,…, xn) for all values iand j.

The set of all

PERMUTATION

s of nletters x1,x2,…,xn

forms a

GROUP

called the nth order symmetric group.

Any

POLYNOMIAL

equation p(x) = 0 is symmetrical with

respect to permutations of the roots of the equation. If

the polynomial is completely factored (as is possible

according to the

FUNDAMENTAL THEOREM OF ALGE

-

BRA

), then any rearrangement of the roots of the equa-

tion (x– α1)(x– α2) … (x– αn) = 0 does not alter the

equation.

A

FUNCTION

y= f(x) whose graph is symmetrical

about the y-axis via a reflection is said to be even.

One that is symmetrical about a rotation of 180°

about the origin is said to be odd. Even functions sat-

isfy f(–x) = f(x) for all values x, and odd functions

f(–x) = –f(x).

A square

MATRIX

Ais said to be symmetrical if, for

each appropriate value iand j, the entry in the ith row

and jth column matches the entry in the jth row and ith

column: Aij = Aji.

A

RELATION

is said to be symmetrical if xbeing

related to ymeans that yis also related to x. For exam-

ple, “is a sibling of” is a symmetrical relation among

people. (If Lashana is Terell’s sibling, then Terell is also

Lashana’s sibling.) The relation, “is a sister of,” how-

ever, is not symmetrical.

A figure or expression that is not symmetrical is

called asymmetrical.

See also

EVEN AND ODD FUNCTIONS

;

FRIEZE

PATTERN

.

360°

——

n