A Glossary of Mathematical Terms and Concepts for ESL Students

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Q - Z

QUADRANT
The plane is divided into four parts by the x and y axes. They are called the first, second, third and fourth quadrant, respectively.

a plane with x and y axes and sections labeled clockwise from upper right: first quadrant, fourth duadrant, third quadrant, second quadrant
QUADRILATERAL
A quadrilateral is a polygon of four sides. a polygon of four sides

Some special quadrilaterals:
Square
a square: a quadrilateral of four equal length sides		           Rectangle
a rectangle: a quadrilateral all of whose angles are right angles; with adjacent sides of unequal length
Parallelogram
a parallelogram: a quadrilateral with opposite sides parallel and equal length		           Trapezoid
a trapezoid: a quadrilateral having only two sides parallel
QUOTIENT
A quotient is the quantity resulting from the division of one quantity by another. (See also division and ratio.)

Examples:
6
2		 ,    2x + 1
x		 ,    7
3
These may be reduced to 3, 2 + 1
x		 ,  2 1
3		 (We may regard 2 as quotient, 1 as remainder.)
RADICAL
A radical is the positive part of the n-th root of a quantity.

Examples:

square root of 16, cube root of 27, square root of (50x^3)
These may be simplified as 4, 3, 5xsquare root of 2x

A fraction with radicals in the denominator may be changed to an equivalent fraction without radicals in the denominator by rationalizing the denominator.

1 / (square root of 3) = 1 / (square root of 3) multiplied by [(square root of 3) / (square root of 3)] = (square root of 3) / 3

1 / (3 + square root of 2) = 1 / (3 + square root of 2) multiplied by [(3 - square root of 2) / (3 - square root of 2)] = (3 - square root of 2) / 9 -2 = (3 - square root of 2) / 7
RATIO
A ratio is the quotient of two numbers. (See also quotient and proportion.)

Example:
The ratio of 4 to 3 is expressed as 4 : 3 or 4.
3
RECIPROCAL
The reciprocal of a non-zero quantity a is a quantity b such that ab = 1. b is also known as the inverse of a.

Examples:
The reciprocal of 2   is   1
2
The reciprocal of 1
3		   is   3
The reciprocal of 4
7		   is   7
4
The reciprocal of   2  
x + 1		   is   x + 1
2
ROOT
1. root of an equation: (See also equation.)
4x + 2 = 6     x = 1 is a root
x2 - 4 = 0     x = ± 2 are roots.

2. root of a quantity: (See also power and radical.)
square root square root of 25 = 5  
cube root cube root of 8 = 2  
n-th root n-th root of a (the n-th root of a is a number b such that bn = a)
SIGNED NUMBER
A signed number is a number with a plus '+' or a minus '-' sign.
Positive Number: +3, +1.6, 4
Negative Number: -5, -7.2, -99

addition of signed numbers:
(+3) + (+2) = +5
(+3) + (-2) = 1
(-3) + (-2) = -5
(-3) + (+2) = -1

subtraction of signed numbers:
(+3) - (+2) = 3 - 2 = 1
(+3) - (-2) = 3 + 2 = 5
(-3) - (-2) = -3 + 2 = -1
(-3) - (+2) = -3 -2 = 5

multiplication of signed numbers:
(+3) (+2) = 6
(+3) (-2) = -6
(-3) (-2) = 6
(-3) (+2) = -6

division of signed numbers:
+3
+2		 = 1 1
2		              +3
-2		 = -1 1
2
 
-3
-2		 = 1 1
2		        -3
+2		 = -1 1
2
SIMPLIFY
To simplify is to change a mathematical expression to its simplest form. Also "perform the indicated operations" has the same purpose. (See also evaluate.)

Examples:
3 + 4(x - 2) = 3 + 4x -8
  = 4x - 5
(-2x2y)3(5xy3)2 = (-2)3x6y352x2y6
  = -200x8y9
(x^2 - 3xy) / x = x - 3y
SLOPE OF A STRAIGHT LINE
slope: m = tan A            a straight, diagonal line from first quadrant to third quadrant, passing through second quadrant; two points lying on the line and in the first quadrant are labeled: (x sub 1, y sub 1), and (x sub 2, y sub 2); a vertical line drawn through (x sub 2, y sub 2) intersects a horizontal line drawn through (x sub 1, y sub 1); the resulting triangle has an angle labeld 'A'
  m = (y sub 2 - y sub 1) / (x sub 2 - x sub 1)
(See also equation of a straight line.)
SOLVE
To solve an equation is to find the root(s) of an equation. To solve a problem is to find the answer(s) of the problem. (See also equation and root.)
SPEED
speed = distance
time
Examples:
When an object travels 10 miles in two hours, then its speed is
10 miles
 2 hours		 = 5 miles per hour (or 5 mph).

When an object moves 20 feet in 5 seconds, then its speed is
20 ft
 5 sec		 = 4ft/sec.
SUBTRACTION -
subtract x from 5 5 - x
the difference between 6 and 4 6 - 4
y decreased by 2 y - 2
7 minus n 7 - n
x less than 8 8 - x
SYSTEMS OF LINEAR EQUATIONS
 x + y = 3     (1)
2x - y = 0     (2)
(1) and (2) together are called a system of linear equations with two unknowns.

To solve by addition:
(1) + (2) 3x = 3 so x = 1
Hence 1 + y = 3 so y = 2
x = 1
y = 2		 } solution
To solve by substitution:
From (1) y = 3 - x  
substitute in (2) 2x - (3 - x) = 0  
3x - 3 = 0 so x = 1 and y = 2
To solve by graph:

a graph of x + y = 3 and 2x - y =0 with intersecting point labeled '(1, 2) solution'
TRIANGLES
acute: all interior angles less than 90°
obtuse: one interior angle greater than 90°
right: one interior angle equals 90°
scalene: no two sides equal
isosceles: two sides equal
equilateral:    three sides equal

Examples:
acute
a polygon with three sides and all acute interior angles		            obtuse
a polygon with three sides and one interior angle greater than 90 degrees
right
a polygon with three sides and one interior angle equal 90 degrees		 scalene
a polygon with three sides, of which no two are of equal length
isosceles
a polygon with three sides, two of which are of equal length		 equilateral
a polygon with three equal length sides

Some properties of a triangle:
a triangle; angle A + angle B + angle C = 180            a triangle, angle DCB = angle A + angle B; angle DCB is an exterior angle
TRIGONOMETRIC FUNCTIONS
sin A =  a 
c		 =   opposite  
hypotenuse		       a triangle with angle and opposite leg labeled: 'A, a; B, b; C, c'
cos A =  b 
c		 =   adjacent  
hypotenuse
tan A =  a 
b		 =  opposite 
adjacent
cot A =    1   
tan A		 =  b 
a
sec A =    1   
cos A		 =  c 
b
csc A =    1   
sin A		 =  c 
a
sin mathematical symbol theta = y                graph of a straight line from first quadrant to third quadrant, passing through the origin
cos mathematical symbol theta = x    
tan mathematical symbol theta =  y 
x		    
cot mathematical symbol theta =    1   
tan mathematical symbol theta		 =  y 
x
sec mathematical symbol theta =    1   
cos mathematical symbol theta		 =  1 
y
csc mathematical symbol theta =    1   
sin mathematical symbol theta		 =  1 
x
VARIABLE
A variable is a symbol in a mathematical expression that can assume different values. Usually the symbols are the alphabet: a, b, c, …, x, y, z.

Examples: x + 2y - 3xy + 4
In this expression x and y are variables while 4 is a constant.

y = 2x + 3
In this expression y is considered as a function of x. Here x is called an independent variable and y is called the dependent variable. (See also function and polynomial.)
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