Fields Medal Prize Winners (1998)


Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Rewriting Algebraic Fractions
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Order and Inequalities

One important property of real numbers is that they can be ordered. If a and b are real numbers, a is less than b if b - a is positive. This order is denoted by the inequality a < b.

The statement “b is greater than a” is equivalent to saying that a is less than b. When three real numbers a, b, and c are ordered such that a < b and b < c we say that b is between a and c and a < b < c.

Geometrically, a < b if and only if a lies to the left of b on the real line (see the figure below).

For example, 1 < 2 because 1 lies to the left of 2 on the real line.

The following properties are used in working with inequalities. Similar properties are obtained if < is replaced by and > is replaced by ≥. (The symbols and mean less than or equal to and greater than or equal to, respectively.)

Properties of Inequalities

Let a, b, c, d, and k be real numbers.

1. If a < b and b < c, then a < c. Transitive Property.
2. If a < b and c < d, then a + c < b + d. Add inequalities.
3. If a < b, then a + k < b + k. Add a constant.
4. If a < b and k > 0, then ak < bk. Multiply by a positive constant.
4. If a < b and k < 0, then ak > bk. Multiply by a negative constant.

NOTE Note that you reverse the inequality when you multiply by a negative number. For example, if x < 3 then -4x > -12. This also applies to division by a negative number. Thus, if -2x > 4, then x < -2.

A set is a collection of elements. Two common sets are the set of real numbers and the set of points on the real line. Many problems in calculus involve subsets of one of these two sets. In such cases it is convenient to use set notation of the form {x: condition on x}, which is read as follows.

For example, you can describe the set of positive real numbers as

{x: x > 0} Set of positive real numbers

Similarly, you can describe the set of nonnegative real numbers as

{x: x 0} Set of nonnegative real numbers

The union of two sets A and B, denoted by is the set of elements that are members of A or B or both. The intersection of two sets A and B, denoted by is the set of elements that are members of A and B. Two sets are disjoint if they have no elements in common.

The most commonly used subsets are intervals on the real line. For example, the open interval

(a, b) = {x: a < x < b} Open interval

is the set of all real numbers greater than a and less than b, where a and b are the endpoints of the interval. Note that the endpoints are not included in an open interval. Intervals that include their endpoints are closed and are denoted by

[a, b] = {x: a x b} Closed interval

The nine basic types of intervals on the real line are shown in the table below. The first four are bounded intervals and the remaining five are unbounded intervals. Unbounded intervals are also classified as open or closed. The intervals (-∞, b) and (a, ) are open, the intervals (-∞, b] and [a, ) are closed, and the interval (-∞, ∞) is considered to be both open and closed.