1.4 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: 5.12639573...

6.1011011101111...

0.525225222...

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

Radicals

2, 17, 53

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,

16 = 4 since 44 =16

49 = 7 since 77 = 49

n

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)

ab a b, a 0, b 0

40 410 4 10 2 10 2 10

125 255 25 5 5 5 5 5

40

125

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: 4 7 3 7

4 7 3 7

(4 3) 7

7 7

8 5 125

8 5 125

8 5 25 5

8 5 5 5

(8 5) 5

3 5

Multiplication of Irrational Numbers

• Simplify:

6 54

6 54 654 324 18

Quotient Rule for Radicals

a

b

a

b, a 0, b 0

Example: Division

• Divide:

• Solution:

• Divide:

• Solution:

16

4

144

2

16

4

16

4 4 2

144

2

144

2 72

362 36 2

6 2

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:

8

12.

8

12

8

12

2

3

2

3

2

3

3

3

6

3

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 numbersNatural numbers

Real numbers

Properties of the Real Number System

• ClosureIf 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.• 5 9 = 9 5 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.

• (4 6) 2 = 4 (6 2) is true.

• Note: The associative property does not hold true for subtraction or division.

Distributive Property

• Distributive property of multiplication over additiona • (b + c) = a • b + a • cfor any real numbers a, b, and c.

• Example: 6 • (r + 12) = 6 • r + 6 • 12 = 6r + 72

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.

45 44444

Product Rule

• Simplify: 34 • 39

34 • 39 = 34 + 9 = 313

• Simplify: 64 • 65

64 • 65 = 64 + 5 = 69

amg an amn

Quotient Rule

• Simplify: • Simplify:

am

anam n, a 0

75

72

915

98

75

7275 2 73

915

98915 8 97

Zero Exponent Rule

• Simplify: (3y)0

(3y)0 = 1

• Simplify: 3y0

3y0 = 3 (y0) = 3(1) = 3

a0 1, a 0

Negative Exponent Rule

• Simplify: 64

a m

1

am, a 0

6 4

1

64

1

1296

Power Rule

• Simplify: (32)3

(32)3 = 32•3 = 36

• Simplify: (23)5

(23)5 = 23•5 = 215

(am )n amgn

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 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.

Scientific Notation continued

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) 0.0000000004324.32 1010

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

b) 1.45 10–7

0.000000145