rounding 451

results. (Arithmetic was performed on a counting

board such as an

ABACUS

.)

The system of Roman numerals remained popular

in Western Europe until the 17th century. Although the

system was eventually replaced by the H

INDU

-A

RABIC

NUMERAL

system we use today, it still remains a tradi-

tion to use Roman numerals for clock faces, for the

inscription of dates on buildings, and for copyright

data on films and books, for instance.

See also

BASE OF A NUMBER SYSTEM

;

DECIMAL REP

-

RESENTATION

; E

GYPTIAN MATHEMATICS

;

NUMBER

.

root (zero) Any value of a variable in an equation that

satisfies the equation is called a root of the equation. For

example, the equation 2x+ 1 = 7 has root x= 3, and

the equation a3= 8 has root a= 2. As any equation

with a variable xcan be written in the form f(x) = 0 for

some

FUNCTION

f, the roots of an equation are some-

times called the “zeros” of the associated function f.

If f(x) is a polynomial, then the equation f(x) = 0 is

called a polynomial equation. The

QUADRATIC

formula

shows that a quadratic equation ax2+ bx = c= 0 has

two roots given by:

The

FACTOR THEOREM

shows that if x= ris a root

of a polynomial equation f(x) = 0, then (x– r) is a factor

of f(x).The root ris called a simple root if (x– r) is a

factor of f(x),but (x– r)2is not; a double root if (x– r)2

is a factor of f(x),but (x– r)3is not; and, in general, an

nth-order root if (x– r)ndivides f(x),but (x– r)n+1 does

not. For instance, x= 2 is a double root of the equation

2x3– 9x2+ 12x– 4 = 0, and x= 1/2 is a simple root.

(We have 2x3– 9x2+ 12x– 4 = (x– 2)2(2x– 1).) The

FUNDAMENTAL THEOREM OF ALGEBRA

asserts that a

polynomial equation of degree nhas precisely n(possi-

bly complex) roots if the roots are counted according to

their “multiplicity.” For instance, the equation 2x3–

9x2+ 12x– 4 = 0 has three roots, if the solution x= 2 is

counted twice, as is appropriate.

Numerical methods such as the

BISECTION METHOD

and N

EWTON

’

S METHOD

can be used to find roots of

equations to any desired degree of accuracy.

The nth root of a number a, denoted n

√

–

a, is any

value xthat satisfies the polynomial equation xn= a.

For example, the number 2 is a 10th root of 1,024

(since 210 = 1,024), and –1 is a sixth root of 1, since

(–1)6= 1. If n= 2, then an nth root is called a

SQUARE

ROOT

. If n= 3, then it is called a

CUBE ROOT

.

root test See

CONVERGENT SERIES

.

rose A planar curve shaped like a collection of petals

with a common origin is called a rose. In polar coordi-

nates, such a curve has an equation of the form r=

asin(nθ) or r= acos(nθ) for some constant aand posi-

tive integer n. The angle θtakes values between zero

and 360°, and may consequently yield a negative value

for r. In this case the corresponding point on the curve

is drawn a distance |r| from the origin in a direction

opposite to that indicated by angle θ.

If nis odd, the number of petals that appear

around the origin is n. If nis even, then 2nloops

appear. In 1728 Italian mathematician Guido Grandi

called these curves “rhodonea.”

rotation See

GEOMETRIC TRANSFORMATION

;

LINEAR

TRANSFORMATION

.

rounding If a number has more digits than can be

conveniently handled or stored, then it is often conve-

nient to replace the figure by the number closest to it

with the desired number of digits. This process is called

rounding. For example, the closest two-digit number to

68.7 is 69. Thus 68.7 “rounded to two-digits” is 69.

Rounding 5.237 to one decimal place yields 5.2.

Rounding 453 to the nearest 10 yields 450, whereas

rounding it the nearest 100 yields 500.

It is conventional to “round up” if the number

under consideration is equally distant from two approx-

imations. For instance, 9 is deemed to be the closest

integer to 8.5.

Errors may be produced if a number is rounded

more than once. For example, rounding 78.347 to two

decimal places yields 78.35. If we later decide to round

to one decimal place, working with the 78.35 yields

78.4, whereas the original number rounded to one dec-

imal place is 78.3. It is essential, then, that all rounding

processes be executed in just one step.