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Fields Medal Prize Winners (1998)




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Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
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Rational Exponents
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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
Fractions
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
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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
Roots
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
Powers
Rewriting Algebraic Fractions
Exponents
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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Fractions

900000 ÷ 300 = ? 30 ÷ 100 = ?

Write 900,000 ÷ 300 as . We are used to seeing the divided – often called the numerator - above the fraction line and the divider (or denominator) below the fraction line. We can write 900000 as 9 × 100000 and 300 as 3 × 100. The task then looks like .

Let us first consider , which equals 3. We should have divided something 100000 time bigger than 9, making the result 100000 times bigger. We should also have divided by something 100 times bigger than 3, making the result 100 times smaller. What results from making something 100000 times bigger and then 100 times smaller? You can think of the latter as 10 times smaller and then 10 times smaller again, each time knocking one 0 off the 100000, leaving 1000 as the net 'adjustment' to the 3.

So .

In the same way, is just because there is the same m -fold increase as the m-fold decrease. Note that this does not work with, for example, (Try taking m = 3 which gives which is 2, not 3). In the case of 30 ÷ 100, (i.e. 30 %), again we start with and then make it ten times bigger (because the divided is 30, not 3) and one hundred times smaller (because the divider is 100, not 1). The net adjustment is 10 times smaller than the 3, namely 0.3

Remember: The correct value of a fraction can also always be found by long division.