Exponential Functions
Linera Equations
Simple Trinomials as Products of Binomials
Laws of Exponents and Dividing Monomials
Solving Equations
Multiplying Polynomials
Multiplying and Dividing Rational Expressions
Solving Systems of Linear Inequalities
Mixed-Number Notation
Linear Equations and Inequalities in One Variable
The Quadratic Formula
Fractions and Decimals
Graphing Logarithmic Functions
Multiplication by 111
Solving Systems of Equations - Two Lines
Solving Nonlinear Equations by Factoring
Solving Linear Systems of Equations by Elimination
Rationalizing the Denominator
Simplifying Complex Fractions
Factoring Trinomials
Linear Relations and Functions
Axis of Symmetry and Vertices
Equations Quadratic in Form
The Appearance of a Polynomial Equation
Subtracting Reverses
Non-Linear Equations
Exponents and Order of Operations
Factoring Trinomials by Grouping
Factoring Trinomials of the Type ax 2 + bx + c
The Distance Formula
Invariants Under Rotation
Multiplying and Dividing Monomials
Solving a System of Three Linear Equations by Elimination
Multiplication by 25
Powers of i
Solving Quadratic and Polynomial Equations
Slope-intercept Form for the Equation of a Line
Equations of Lines
Square Roots
Integral Exponents
Product Rule for Radicals
Solving Compound Linear Inequalities
Axis of Symmetry and Vertices
Multiplying Rational Expressions
Reducing Rational Expressions
Properties of Negative Exponents
Numbers, Factors, and Reducing Fractions to Lowest Terms
Solving Quadratic Equations
Factoring Completely General Quadratic Trinomials
Solving a Formula for a Given Variable
Factoring Polynomials
Decimal Numbers and Fractions
Multiplication Properties of Exponents
Multiplying Fractions
Multiplication by 50

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Decimal Numbers

Decimal numbers make it easy to work with fractional amounts. We show fractions of dollars using the decimal system. We can make distance calculations easier by using the metric system instead of miles-feet-inches.

Decimal numbers or decimal fractions are a proper fraction based on the number 10. They use a decimal point separating the fraction from the whole number. Decimal numbers are written without showing denominators. Decimal numbers have three parts: an integer, a decimal point, and a decimal number.

We use the same place value system for decimals that we use for whole numbers. Anywhere you look in a whole number, the value of the place to the right of a digit is always ten times smaller. The tens’ place is ten times smaller than the hundreds’ place, and the ones’ place is ten times smaller than the tens’ place.

To find the value of the place to the right of the ones’ place, divide one by ten: 1 ÷ 10 = 1/10 = one-tenth. The first place to the right of the decimal point is the tenths’ place. The value to the right of the tenths’ place is the 1/10 ÷ 10, and is called the hundredths’ place. This chart shows some common place values.

Writing Decimal Numbers

Between each whole number, this number line is divided into tenths. Decimals with values less than 1 are written with a 0 in the ones’ place.

To write five tenths, write a 0 in the ones’ place and a 5 in the tenths’ place, with a decimal point in between: 0.5.

Whole numbers can also be written with one or more zeros after the decimal point:

3 = 3.0,

1 = 1.000,

0 = 0.00000

The number line below shows what is in the area between 0 and 0.5 from the number line above. Between each of the tenths’ values, the number line is divided into hundredths. Notice the denominator, hundredths, names the last place on the right that is holding a digit.

To write five hundredths, write a 0 in the tenths place and a 5 in the hundredths’ place: 0.05. The 0 is needed as a placeholder between the decimal point and the hundredths’ digit.


How would you write three and nine hundredths as a decimal?


The and tells you where to put the decimal point: after the 3. The hundredths tells you there are two decimal places in the number and that a 9 goes in the hundredths’ place: 3.09

Changing Decimals to Fractions

It is easy to convert decimals to fractions. Look at the number 0.27, or twenty-seven hundredths. The hundredths tells you the denominator is 100. The numerator will be the numeral to the right of the decimal point, which is 27 in this case. Do not write the decimal point in the fraction.

You must look for fractions that can be reduced. For example, look at the number 0.32, or thirty-two hundredths. Write the fraction and then reduce it to lowest terms.

Changing Fractions to Decimals

Decimal numbers are really just a shortcut method of writing common fractions, where the denominator is 10 or 100 or 1000 and so forth. You can convert fractions into decimal numbers by making the denominator a multiple of 10. The denominator will indicate the smallest place value that should appear in the decimal.

You can also change a fraction to a decimal if the fraction has a denominator other than 10, but you need to know about decimal division to do this. Or, use a calculator.

Note: When a decimal number is less than one, we always write the zero to the left of the decimal point. Without the leading zero, the reader might not notice that tiny little mark for the decimal point.

Repeating Decimals

Some fractions when written as a decimal number will repeat forever. To write a repeating decimal, put a bar over the rightmost digit (or digits) that repeat. For example, or simply .

Why do some fractions repeat and others don't? Notice that decimal numbers represent the addition of and and and so forth. Now take a closer look at the factors in these fractions: and . In fact, every position in the decimal number represents some multiple of 2's and 5's. This works well for fractions containing any multiples of halves and fifths. But it is impossible to exactly represent thirds, sixths or anything with factors other than 2 or 5!