The Rational Numbers

Section 9.2 The Real Numbers

Recall the definition of the set of rational numbers.

Since every rational number can be represented as a fraction or ratio of two integers ,
every rational number either has a terminating or repeating decimal representation .

However, numbers exist whose decimal representations neither terminate nor repeat. In
other words, there are numbers with infinite nonrepeating decimal representations. These
numbers form the set of irrational numbers and completely fill in all of the gaps in the
rational number line .

When we combine the set of rational numbers with the set of irrational numbers the set of
real numbers is created. The following diagram illustrates how all of the number sets we
have studied so far are related to one another.

Real Numbers

Note: Q∪I = R and Q∩I = { }.

Properties of Real Numbers
• Closure – The sum , difference, product , or quotient of two real numbers is another
real number.

• Commutativity –

o Addition: a + b = b + a
o Multiplication: ab = ba

• Associativity –

o Addition: (a + b)+ c = a + (b + c)
o Multiplication: (ab)c = a(bc)

• Identity –

o Addition: a + 0 = 0 + a = a
o Multiplication: a ∙1 =1 ∙ a = a

• Inverse –

o Addition: a + (− a) = 0
o Multiplication:

• Distributivity –

o Multiplication Over Addition: a(b + c) = ab + ac
o Multiplication Over Subtraction: a(b − c) = ab − ac
o Division Over Addition:
o Division Over Subtraction:

• Multiplication Property of –1 : a(−1) = (−1)a = −a

• Multiplication Property of 0: a ∙0 = 0 ∙a = 0

• Opposite of a Sum: − (a + b) = −a + (− b)

Properties of Ordering Real Numbers
• Transitivity – If a < b and b < c , then a < c .

• Addition – If a < b , then a + c < b + c

• Subtraction – If a < b , then a − c < b − c

• Multiplication

o By a positive number p: If a < b , then ap < bp .
o By a negative number n: If a < b , then an > bn .

• Division –

o By a positive number p: If a < b , then a ÷ p < b ÷ p .
o By a negative number n: If a < b , then a ÷ n > b ÷ n .

• Density: If a < b , there exists a real number c such that a < c < b .

Squares and Square Roots
A perfect square is defined to be the result of squaring a rational number.

If we need to reverse this operation – that is, go from the square of a number back to the
number itself – we take the square root of that number .

All positive real numbers except zero have two square roots: one is positive and the other
is negative. We call the positive number the positive root or principal root and the
negative number the negative root.

Note:
• The square root of zero is zero.
• The square root of a negative number is not defined among the real numbers.

Radical Expressions and Evaluating the nth Root
A radical expression is an expression that can be written in the form , where n is the
index (which is always an integer greater than one) , is called the radical symbol, and
a is called the radicand.

We call radicals with even indices even roots.
Radicals with odd indices are called odd roots.

The square root is an even root of index two. When using square roots, it is convention
not to write the index in the expression or equation .

To evaluate a radical expression :
• If n is an integer > 1 and a = 0, then the result is zero.
• If n is an even integer > 1 and a < 0, then the result is not a real number.
• If n is an even integer > 1 and a > 0, then
• If n is an odd integer > 1,

To solve a radical expression using your TI-83 calculator :
• First, enter the index n in the home screen.
• Press MATH , 5.
• Open a parentheses, enter the radicand a, then close the parentheses.
• Press ENTER.
• If you suspect the result should be a fraction, press MATH, 1, ENTER.

Rational Exponents
A radical expression can be written using rational exponents.

In these equations, a is a real number and n is a positive integer > 1.

To evaluate an expression with a rational exponent having a numerator of one , write an
equivalent radical expression. Then, evaluate as directed previously.

Because we can write radical expressions using rational exponents we can apply the rules
of exponents to them.

Recall that Well, where a is a real number,
is defined, m and n have no common factors other than one, m is an integer not equal
to zero, and n is a positive integer > 1.