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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Variables and Expressions

Objective Learn the notion of variable and algebraic expression.

The most important concept in algebra is that of a variable. This is a concept that you may have seen before, but it is important to begin by reviewing and discussing this concept. Remember that a variable is simply a letter or sometimes a symbol used to represent a quantity. For example, suppose b represents the number of boys in the class and g represents the number of girls. A verbal expression like "There are 14 girls in the class." can be written as an algebraic expression or equation, which involves a variable.

g = 14

We can also write a verbal expression from an algebraic one. Because we know that the variable b represents the number of boys in the class, the algebraic equation b = 15 can be translated to the verbal sentence "There are 15 boys in the class."

What is an algebraic expression?

Have a look at these definitions:

  • A variable is a letter or symbol used to represent a quantity.
  • An algebraic expression consists of one or more numbers and variables along with one or more arithmetic operations.

Some examples of algebraic expressions are shown below.

You might write a table like the following to show the relation between verbal and algebraic expressions.

Verbal Expression Algebraic Expression
2 more than a number x x + 2
The square of a number x divided by three times a number y.
The number of eyes in the classroom (Let s represent the number of students in the class.) 2s + 2 or 2(s + 1)

(Each student has two eyes and the teacher has two eyes)

Remember the terms that describe expressions like x 5. These expressions are called powers. The x is the base of the power and the number 5 is called the exponent. The expression x 5 is described verbally as "x raised to the fifth power" or simply "x to the fifth."

Translating between algebraic and verbal expressions.

Example 1

Write an algebraic expression for the sum of 3 and the number x divided by 16.

Solution (3 + x ) ÷ 16 or .

Example 2

Write an algebraic expression for six subtracted from the number y, all raised to the fourth power.

Solution ( y - 6) 4

Example 3

Write a verbal expression for 5m - n 2 .


The product of 5 and the number m minus the number n raised to the second power (or n squared)

Example 4

Write a verbal expression for ( x + 2) y.


The sum of the number x and 2, all raised to the exponent y (or the y th power)

There is one final term that you should describe to your students in this lesson. To evaluate an expression means to find its values. Use the following examples to illustrate this definition.


Example 5

Evaluate 4 3 .


4 3 = 4 · 4 · 4 = 64


Example 6

Evaluate the expression ( x + 3) 2 when x = 3.


Substitute 3 for x in the expression. ( x + 3) 2 = (3 + 3) 2 = 6 2 = 36

The processes of assigning variables to quantities and of translating between verbal and algebraic expressions are fundamental in algebra.