# Foundations

## Decimals


### Round Decimals

Decimals are another way of writing fractions whose denominators are powers of ten.
Just as in whole numbers, each digit of a decimal corresponds to the place value based on the powers of ten.  shows the names of the place values to the left and right of the decimal point.
When we work with decimals, it is often necessary to round the number to the nearest required place value. We summarize the steps for rounding a decimal here.

### Add and Subtract Decimals

To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. We then add or subtract the numbers as if they were whole numbers and then place the decimal point in the sum.

### Multiply and Divide Decimals

When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. We multiply the numbers temporarily ignoring the decimal point and then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product. Finally, we write the product with the appropriate sign.
Often, especially in the sciences, you will multiply decimals by powers of 10 (10, 100, 1000, etc). If you multiply a few products on paper, you may notice a pattern relating the number of zeros in the power of 10 to number of decimal places we move the decimal point to the right to get the product.
Just as with multiplication, division of signed decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed and the sign of the quotient. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.
We review the notation and vocabulary for division:
We’ll write the steps to take when dividing decimals for easy reference.

### Convert Decimals, Fractions, and Percents

In our work, it is often necessary to change the form of a number. We may have to change fractions to decimals or decimals to percent.
We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal  the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.
The steps to take to convert a decimal to a fraction are summarized in the procedure box.
A percent is a ratio whose denominator is 100. Percent means per hundred. We use the percent symbol, %, to show percent. Since a percent is a ratio, it can easily be expressed as a fraction. Percent means per 100, so the denominator of the fraction is 100. We then change the fraction to a decimal by dividing the numerator by the denominator. After doing this many times, you may see the pattern.
To convert a percent number to a decimal number, we move the decimal point two places to the left.
To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is 100, it is easy to change that fraction to a percent. After many conversions, you may recognize the pattern.
To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.

### Simplify Expressions with Square Roots

Remember that when a number  is multiplied by itself, we write  and read it “ squared.” The result is called the square of a number n. For example,  is read “8 squared” and 64 is called the square of 8. Similarly, 121 is the square of 11 because  is 121. It will be helpful to learn to recognize the perfect square numbers.
What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.
Because  we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is m is called a square root of a number m.
Notice  also, so  is also a square root of 100. Therefore, both 10 and  are square roots of 100. So, every positive number has two square roots—one positive and one negative. The radical sign, , denotes the positive square root. The positive square root is called the principal square root. When we use the radical sign that always means we want the principal square root.
We know that every positive number has two square roots and the radical sign indicates the positive one. We write  If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example,  We read  as “the opposite of the principal square root of 10.”

### Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

We have already described numbers as counting numbers, whole numbers, and integers. What is the difference between these types of numbers? Difference could be confused with subtraction. How about asking how we distinguish between these types of numbers?
What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.
In general, any decimal that ends after a number of digits (such as 7.3 or ) is a rational number. We can use the reciprocal (or multiplicative inverse) of the place value of the last digit as the denominator when writing the decimal as a fraction. The decimal for  is the number  The bar over the 3 indicates that the number 3 repeats infinitely. Continuously has an important meaning in calculus. The number(s) under the bar is called the repeating block and it repeats continuously.
Since all integers can be written as a fraction whose denominator is 1, the integers (and so also the counting and whole numbers. are rational numbers.
Every rational number can be written both as a ratio of integers  where p and q are integers and  and as a decimal that stops or repeats.
Are there any decimals that do not stop or repeat? Yes! The number  (the Greek letter pi, pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat. We use three dots (…) to indicate the decimal does not stop or repeat.
The square root of a number that is not a perfect square is a decimal that does not stop or repeat.
A numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call this an irrational number.
Let’s summarize a method we can use to determine whether a number is rational or irrational.
We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.
Later in this course we will introduce numbers beyond the real numbers.  illustrates how the number sets we’ve used so far fit together.
Does the term “real numbers” seem strange to you? Are there any numbers that are not “real,” and, if so, what could they be? Can we simplify  Is there a number whose square is 
None of the numbers that we have dealt with so far has a square that is  Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to  The square root of a negative number is not a real number.

### Locate Fractions and Decimals on the Number Line

We now want to include fractions and decimals on the number line. Let’s start with fractions and locate  and  on the number line.
We’ll start with the whole numbers 3 and  because they are the easiest to plot. See .
The proper fractions listed are  and  We know the proper fraction  has value less than one and so would be located between 0 and 1. The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts  We plot 
Similarly,  is between 0 and  After dividing the unit into 5 equal parts we plot 
Finally, look at the improper fractions  Locating these points may be easier if you change each of them to a mixed number.
 shows the number line with all the points plotted.
Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

### Key Concepts

1. How to round decimals.


2. How to add or subtract decimals.


3. How to multiply decimals.


4. How to multiply a decimal by a power of ten.


5. How to divide decimals.


6. How to convert a decimal to a proper fraction and a fraction to a decimal.


7. How to convert a percent to a decimal and a decimal to a percent.


8. Square Root Notation
 is read “the square root of m.”
If  then  for 
The square root of m,  is the positive number whose square is m.
9. Rational or Irrational
If the decimal form of a number


10. Real Numbers


### Practice Makes Perfect

Round Decimals
In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.
Add and Subtract Decimals
In the following exercises, add or subtract.
Multiply and Divide Decimals
In the following exercises, multiply.
In the following exercises, divide. Round money monetary answers to the nearest cent.
Convert Decimals, Fractions and Percents
In the following exercises, write each decimal as a fraction.
In the following exercises, convert each fraction to a decimal.
In the following exercises, convert each percent to a decimal.
In the following exercises, convert each decimal to a percent.
Simplify Expressions with Square Roots
In the following exercises, simplify.
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.
Locate Fractions and Decimals on the Number Line
In the following exercises, locate the numbers on a number line.

### Writing Exercises


### Self Check

ⓐ Use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?