QED The abbreviation QED is short for quod erat
demonstrandum, Latin for “which was to be demonstrated.” EUCLID of the third century B.C.E. wrote the
Greek equivalent of this phrase at the end of each of his
proofs to indicate that its conclusion had been reached.
Many mathematicians still follow this tradition.
The initials QEF, for quod erat faciendum (which
was to be done), are sometimes added after the completion of a geometrical construction, and QEI, for quod
erat inveniendum (which was to be found), after the
completion of a calculation.
quadrant In a system of CARTESIAN COORDINATES,
the two coordinate axes divide the plane into four
regions called quadrants. By convention, the quadrants
are numbered, with the first quadrant being the one
above the x-axis and to the right of the y-axis (this is
the region in which both x and y are positive), and the
second, third, and fourth quadrants arranged counterclockwise about the origin.
Many popular science fiction texts describe threedimensional space as being divided into four quadrants. This is incorrect usage of the term.
One-quarter of the circumference of a circle is sometimes called a quadrant, as is one quarter of the interior
of a circle (bounded by two perpendicular radii).
See also OCTANT.
quadratic (quadric) Any expression, function, or
equation containing variables raised to the second
power, but no higher power, is described as quadratic.
Thus a quadratic polynomial, for instance, is a POLYNOMIAL of second degree, and a quadratic equation is
an equation formed by setting a quadratic polynomial
equal to zero. Such an equation has the form ax
2 + bx
+ c = 0 with a, b, and c either REAL or COMPLEX NUMBERS and a nonzero.
Quadratic equations often arise when computing
the dimensions of rectangles. (This explains the use of
the prefix “quad” in their name, derived from the Latin
name quattuor for “four.”) For instance, told that a
quadrangle of area 55 ft
2 has length 6 units longer than
its width leads to the equation (given in terms of the
unknown width x) x(x + 6) = 55 which is the quadratic
equation x
2 + 6x – 55 = 0. Ancient Egyptian and Babylonian scholars of ca. 2000 B.C.E. were the first to contend with problems of this type.
The technique of COMPLETING THE SQUARE provides a simple method for solving all quadratic equations. It also leads to the famous quadratic formula:
A quadratic equation of the form ax
2 + bx + c
= 0 with a ≠ 0 has solutions given by:
For example, the quadratic formula shows that x
2 + 6x
– 55 = 0 has solutions x = 5 and x = –11. (The positive
solution provides an acceptable answer to our quadrangle problem.)
x
b
b
ac
a
=
− ±
−
2
4
2
428
Q
demonstrandum, Latin for “which was to be demonstrated.” EUCLID of the third century B.C.E. wrote the
Greek equivalent of this phrase at the end of each of his
proofs to indicate that its conclusion had been reached.
Many mathematicians still follow this tradition.
The initials QEF, for quod erat faciendum (which
was to be done), are sometimes added after the completion of a geometrical construction, and QEI, for quod
erat inveniendum (which was to be found), after the
completion of a calculation.
quadrant In a system of CARTESIAN COORDINATES,
the two coordinate axes divide the plane into four
regions called quadrants. By convention, the quadrants
are numbered, with the first quadrant being the one
above the x-axis and to the right of the y-axis (this is
the region in which both x and y are positive), and the
second, third, and fourth quadrants arranged counterclockwise about the origin.
Many popular science fiction texts describe threedimensional space as being divided into four quadrants. This is incorrect usage of the term.
One-quarter of the circumference of a circle is sometimes called a quadrant, as is one quarter of the interior
of a circle (bounded by two perpendicular radii).
See also OCTANT.
quadratic (quadric) Any expression, function, or
equation containing variables raised to the second
power, but no higher power, is described as quadratic.
Thus a quadratic polynomial, for instance, is a POLYNOMIAL of second degree, and a quadratic equation is
an equation formed by setting a quadratic polynomial
equal to zero. Such an equation has the form ax
2 + bx
+ c = 0 with a, b, and c either REAL or COMPLEX NUMBERS and a nonzero.
Quadratic equations often arise when computing
the dimensions of rectangles. (This explains the use of
the prefix “quad” in their name, derived from the Latin
name quattuor for “four.”) For instance, told that a
quadrangle of area 55 ft
2 has length 6 units longer than
its width leads to the equation (given in terms of the
unknown width x) x(x + 6) = 55 which is the quadratic
equation x
2 + 6x – 55 = 0. Ancient Egyptian and Babylonian scholars of ca. 2000 B.C.E. were the first to contend with problems of this type.
The technique of COMPLETING THE SQUARE provides a simple method for solving all quadratic equations. It also leads to the famous quadratic formula:
A quadratic equation of the form ax
2 + bx + c
= 0 with a ≠ 0 has solutions given by:
For example, the quadratic formula shows that x
2 + 6x
– 55 = 0 has solutions x = 5 and x = –11. (The positive
solution provides an acceptable answer to our quadrangle problem.)
x
b
b
ac
a
=
− ±
−
2
4
2
428
Q