1 ABSOLUTE VALUE A ABELIAN GROUP See group. ABSCISSA Abscissa means x-coordinate. The abscissa of the point (a, b) in Cartesian coordinates is a. For contrast, see ordinate. ABSOLUTE EXTREMUM An absolute maximum or an absolute minimum. mp le file ABSOLUTE MAXIMUM The absolute maximum point for a function y f (x) is the point where y has the largest value on an interval. If the function is differentiable, the absolute maximum will either be a point where there is a horizontal tangent (so the derivative is zero), or a point at one of the ends of the interval. If you consider all values of x (∞ x ∞), the function might have a finite maximum, or it might approach infinity as x goes to infinity, minus infinity, or both. For contrast, see local maximum. For diagram, see extremum. Sa ABSOLUTE MINIMUM The absolute minimum point for a function y f (x) is the point where y has the smallest value on an interval. If the function is differentiable, then the absolute minimum will either be a point where there is a horizontal tangent (so the derivative is zero), or a point at one of the ends of the interval. If you consider all values of x ( x ), the function might have a finite minimum, or it might approach minus infinity as x goes to infinity, minus infinity, or both. For contrast, see local minimum. For diagram, see extremum. ABSOLUTE VALUE The absolute value of a real number a, written as 0a 0 , is: 0a 0 a if a 0 0a 0 a if a 0 Figure 1 illustrates the absolute value function. ACCELERATION 2 file Figure 1 Absolute value function Sa mp le Absolute values are always positive or zero. If all the real numbers are represented on a number line, you can think of the absolute value of a number as being the distance from zero to that number. You can find absolute values by leaving positive numbers alone and ignoring the sign of negative numbers. For example, 017 0 17, 0105 0 105, 00 0 0 The absolute value of a complex number a bi is 2a2 b2. ACCELERATION The acceleration of an object measures the rate of change in its velocity. For example, if a car increases its velocity from 0 to 24.6 meters per second (55 miles per hour) in 12 seconds, its acceleration was 2.05 meters per second per second, or 2.05 meters/ second-squared. If x(t) represents the position of an object moving in one dimension as a function of time, then the first derivative, dx/dt, represents the velocity of the object, and the second derivative, d 2x/dt 2, represents the acceleration. Newton found that, if F represents the force acting on an object and m represents its mass, the acceleration (a) is determined from the formula F ma. 3 ALGEBRA ACUTE ANGLE An acute angle is a positive angle smaller than a 90 angle. ACUTE TRIANGLE An acute triangle is a triangle wherein each of the three angles is smaller than a 90 angle. For contrast, see obtuse triangle. ADDITION Addition is the operation of combining two numbers to form a sum. For example, 3 4 7. Addition satisfies two important properties: the commutative property, which says that file a b b a for all a and b and the associative property, which says that (a b) c a (b c) for all a, b, and c. mp le ADDITIVE IDENTITY The number zero is the additive identity element, because it satisfies the property that the addition of zero does not change a number: a 0 a for all a. Sa ADDITIVE INVERSE The sum of a number and its additive inverse is zero. The additive inverse of a (written as a) is also called the negative or the opposite of a: a (a) 0. For example, 1 is the additive inverse of 1, and 10 is the additive inverse of 10. ADJACENT ANGLES Two angles are adjacent if they share the same vertex and have one side in common between them. ALGEBRA Algebra is the study of properties of operations carried out on sets of numbers. Algebra is a generalization of arithmetic in which symbols, usually letters, are used to stand for numbers. The structure of algebra is based upon axioms (or postulates), which are statements that are assumed to be true. Some algebraic axioms include the transitive axiom: if a b and b c, then a c ALGORITHM 4 and the associative axiom of addition: (a b) c a (b c) mp le file These axioms are then used to prove theorems about the properties of operations on numbers. A common problem in algebra involves solving conditional equations—in other words, finding the values of an unknown that make the equation true. An equation of the general form ax b 0, where x is unknown and a and b are known, is called a linear equation. An equation of the general form ax2 bx c 0 is called a quadratic equation. For equations involving higher powers of x, see polynomial. For situations involving more than one equation with more than one unknown, see simultaneous equations. This article has described elementary algebra. Higher algebra involves the extension of symbolic reasoning into other areas that are beyond the scope of this book. Sa ALGORITHM An algorithm is a sequence of instructions that tell how to accomplish a task. An algorithm must be specified exactly, so that there can be no doubt about what to do next, and it must have a finite number of steps. AL-KHWARIZMI Muhammad Ibn Musa Al-Khwarizmi (c 780 AD to c 850 AD) was a Muslim mathematician whose works introduced our modern numerals (the Hinduarabic numerals) to Europe, and the title of his book Kitab al-jabr wa al-muqabalah provided the source for the term algebra. His name is the source for the term algorithm. ALTERNATE INTERIOR ANGLES When a transversal cuts two lines, it forms two pairs of alternate interior angles. In figure 2, ⬔1 and ⬔2 are a pair of alternate interior angles, and ⬔3 and ⬔4 are another pair. A theorem in Euclidian geometry says that, when a transversal cuts two parallel lines, any two alternate interior angles will equal each other. 5 AMBIGUOUS CASE file Figure 2 Alternate interior angles mp le ALTERNATING SERIES An alternating series is a series in which every term has the opposite sign from the preceding term. For example, x x3/3! x5/5! x7/7! x9/9! . . . is an alternating series. Sa ALTERNATIVE HYPOTHESIS The alternative hypothesis is the hypothesis that states, “The null hypothesis is false.” (See hypothesis testing.) ALTITUDE The altitude of a plane figure is the distance from one side, called the base, to the farthest point. The altitude of a solid is the distance from the plane containing the base to the highest point in the solid. In figure 3, the dotted lines show the altitude of a triangle, of a parallelogram, and of a cylinder. AMBIGUOUS CASE The term “ambiguous case” refers to a situation in which you know the lengths of two sides of a triangle and you know one of the angles (other than the angle between the two sides of known lengths). If the known angle is less than 90, it may not be possible to solve for the length of the third side or for the sizes of the other two angles. In figure 4, side AB of the upper triangle is the same length as side DE of the lower triangle, side AC is the same length as side DF, and angle B is the AMPLITUDE 6 Sa mp le file Figure 3 Altitudes Figure 4 Ambiguous case same size as angle E. However, the two triangles are quite different. (See also solving triangles.) AMPLITUDE The amplitude of a periodic function is onehalf the difference between the largest possible value of the function and the smallest possible value. For example, for y sin x, the largest possible value of y is 1 and the smallest possible value is 1, so the amplitude is 1. In general, the amplitude of the function y A sin x is 0A 0 .