Math 80 Lecture - Chapter 4 Merced College; Don Power

4.1 Introduction to Decimals

Place Value

Name the place value of a given digit

Write a fraction as a decimal

Divide numerator by denominator

The decimal form must either terminate or repeat

You may be asked to round the result

Do not use the remainder form of the answer for decimal division.

Write a decimal as a fraction

Read the number with the appropriate place value ("tenths," "hundredths," etc.)

Write what you just said as a fraction (the place value indicated the denominator)

Caution: where is the decimal point in an integer?

Read or write a number in words

Whole number part - as though there were no decimal

Decimal point: "and"

For the decimal part:

Read as though it were the complete number

Specify the place value of the final digit

Given a number in words, write the number in standard form

Proper or improper fraction:

Numerator: Ignore decimal point and write the entire number

Denominator: corresponds to the place value

Mixed number:

Whole number part is the number to the left of the decimal point

Numerator: Everything to the right of the decimal point

Denominator: Corresponds to the place value

Order Relations

Place the correct symbol, < or >, between the numbers

Write the numbers in order from smallest to largest

Rounding

Find the place to which the rounding is to be done

"Round to the nearest ____ place" (our text) or "...to _____ significant digits" (science classes)

Cut-off line after (to right) of that point

If next digit is

4 or below ... round down (just drop all from cutoff line on)

5 or above ... round up (add 1 to last digit you keep; carry if necessary; drop all after cutoff)

You can use "wavy =" to show that the answer is approximate

Applications

Rounding

Comparing numbers in a table: which is smallest or greatest?

4.2 Operations

Addition and Subtraction: Line up the decimal points; fill in 0's on the decimal side

Mult: Don't line up the decimals; multiply as if there are no decimals; then count decimal places

Division:

Case 1: No decimals in divisor: decimal point moves straight up from its location in the dividend

Then divide as if there is no decimal point

Do not use remainders in decimal division: they are meaningless

The decimal portion of the answer will either terminate or repeat

Case 2: Divisor contains a decimal point:

Move the decimal point the same distance and direction in both the divisor and dividend

Fractions and Decimals: A fraction is a division problem

Be able to convert fractions -> decimals

Decimals -> fractions (if they are terminating fractions)

Applications and formulas

3.14 as an estimate of π

4.3 Solving Equations with Decimals

Solving equations

The usual steps:

Clear grouping symbols

Clear fractions (we can also clear decimals here) - optional step

Move terms (by adding or subtracting)

Collect like terms

Divide

Applications:

These applications involve substituting into a formula and solving

4.4 Radical Expressions

Square roots of perfect squares

What are perfect squares? List up to 20²

What are the square roots of the perfect squares?

List the positive and negative square roots

Radical symbol (Square root symbol) refers only to the positive root

To refer to the negative square root, put a negative sign before the radical symbol

Know the ones up to sqrt(400)

Be able to use a calculator

For squaring

For taking square roots

Ex: −sqrt(49);

sqrt(16/25);

sqrt(81)+2*sqrt(25);

Evaluate 2*sqrt(a−b) when a=211, b=42

Square roots of whole numbers

Calculator task: Find the square root of any positive real number

Expect ugly decimals except for perfect squares

Ex: 2*sqrt(11)

Ex: Between what whole numbers is sqrt(131) (can be done w/o calc)

Non-calculator task: Simplify without a calculator by pulling out a perfect square

Principle: sqrt(a*b) = sqrt(a) * sqrt(b)

If you can factor out a perfect square, you can simplify the expression.

Ex: simplify sqrt(150)

4.5 Real Numbers

Real numbers are any numbers that can be located on a number line. They include

A. Rational Numbers:

They can be written as fractions of integers

Ex: −4 = −4/1

Decimal forms either:

Terminate: 3/8 = 0.125; or

Repeat: 4/11 = 0.363636...

Rational numbers include:

1. Integers

a. Whole numbers

(1) Zero

(2) Counting numbers (natural numbers, positive integers)

b. Negative integers

2. Fractions that cannot be reduced to integers

B. Irrational numbers.

(Like π or square root of 2)

They cannot be written as fractions of two integers.

Decimal forms do not terminate or repeat.

Tasks:

Graph real numbers on a number line: Put a dot in the appropriate location

Graph intervals of real numbers on a number line, defined by:

Statements like " the real numbers...

less than 2" Use a close parenthesis ) at 2 on the graph

greater than or equal to −5" Use [ at −5

between −3 and 7" Use ( and ) at −3 and 7 respectively

between −3 and 7 inclusive" Use [ and ] at −3 and 7 respectively

Inequalities: i.e. statements written with inequality symbols.

The following inequalities correspond to the statements above:

x<2

x≥−5

−3<x<7

−3≤x ≤7

Return to:  Merced College; Don Power               Updated 10/05/06 by Don Power