The Irrational Numbers and the Real Number System 1.4 The Irrational Numbers and the Real Number System

Pythagorean Theorem Pythagoras, a Greek mathematician, is credited with proving that in any right triangle, the square of the length of one side (a2) added to the square of the length of the other side (b2) equals the square of the length of the hypotenuse (c2) . a2 + b2 = c2

Irrational Numbers An irrational number is a real number whose decimal representation is a nonterminating, nonrepeating decimal number. Examples of irrational numbers:

Radicals are all irrational numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand.

Principal Square Root The principal (or positive) square root of a number n, written is the positive number that when multiplied by itself, gives n. For example,

Perfect Square Any number that is the square of a natural number is said to be a perfect square. The numbers 1, 4, 9, 16, 25, 36, and 49 are the first few perfect squares.

Product Rule for Radicals Simplify: a) b)

Addition and Subtraction of Irrational Numbers To add or subtract two or more square roots with the same radicand, add or subtract their coefficients. The answer is the sum or difference of the coefficients multiplied by the common radical.

Example: Adding or Subtracting Irrational Numbers Simplify: Simplify:

Multiplication of Irrational Numbers Simplify:

Quotient Rule for Radicals

Example: Division Divide: Solution: Divide: Solution:

Rationalizing the Denominator A denominator is rationalized when it contains no radical expressions. To rationalize the denominator, multiply BOTH the numerator and the denominator by a number that will result in the radicand in the denominator becoming a perfect square. Then simplify the result.

Example: Rationalize Rationalize the denominator of Solution:

Real Numbers and their Properties 1.5 Real Numbers and their Properties

Real Numbers The set of real numbers is formed by the union of the rational and irrational numbers. The symbol for the set of real numbers is

Relationships Among Sets Irrational numbers Rational numbers Integers Whole numbers Natural numbers Real numbers

Properties of the Real Number System Closure If an operation is performed on any two elements of a set and the result is an element of the set, we say that the set is closed under that given operation.

Commutative Property Addition a + b = b + a for any real numbers a and b. Multiplication a • b = b • a for any real numbers a and b.

Example 8 + 12 = 12 + 8 is a true statement. Note: The commutative property does not hold true for subtraction or division.

Associative Property Addition (a + b) + c = a + (b + c), for any real numbers a, b, and c. Multiplication (a • b) • c = a • (b • c), for any real numbers a, b, and c.

Example (3 + 5) + 6 = 3 + (5 + 6) is true. Note: The associative property does not hold true for subtraction or division.

Distributive Property Distributive property of multiplication over addition a • (b + c) = a • b + a • c for any real numbers a, b, and c. Example: 6 • (r + 12) = 6 • r + 6 • 12 = 6r + 72

Rules of Exponents and Scientific Notation 1.6 Rules of Exponents and Scientific Notation

Exponents When a number is written with an exponent, there are two parts to the expression: baseexponent The exponent tells how many times the base should be multiplied together.

Product Rule Simplify: 34 • 39 34 • 39 = 34 + 9 = 313 64 • 65 = 64 + 5 = 69

Quotient Rule Simplify: Simplify:

Zero Exponent Rule Simplify: (3y)0 (3y)0 = 1 Simplify: 3y0 = 3(1) = 3

Negative Exponent Rule Simplify: 64

Power Rule Simplify: (32)3 (32)3 = 32•3 = 36 Simplify: (23)5 (23)5 = 23•5 = 215

Scientific Notation Many scientific problems deal with very large or very small numbers. 93,000,000,000,000 is a very large number. 0.000000000482 is a very small number.

Scientific Notation continued Scientific notation is a shorthand method used to write these numbers. 9.3  1013 and 4.82  10–10 are two examples of numbers in scientific notation.

To Write a Number in Scientific Notation 1. Move the decimal point in the original number to the right or left until you obtain a number greater than or equal to 1 and less than 10. 2. Count the number of places you have moved the decimal point to obtain the number in step 1. If the decimal point was moved to the left, the count is to be considered positive. If the decimal point was moved to the right, the count is to be considered negative. 3. Multiply the number obtained in step 1 by 10 raised to the count found in step 2. (The count found in step 2 is the exponent on the base 10.)

Example Write each number in scientific notation. a) 1,265,000,000. 1.265  109 b) 0.000000000432 4.32  1010

To Change a Number in Scientific Notation to Decimal Notation 1. Observe the exponent on the 10. 2. a) If the exponent is positive, move the decimal point in the number to the right the same number of places as the exponent. Adding zeros to the number might be necessary. b) If the exponent is negative, move the decimal point in the number to the left the same number of places as the exponent. Adding zeros might be necessary.

Example Write each number in decimal notation. a) 4.67  105 467,000 0.000000145