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5: Decimals

Last updated

Aug 13, 2020
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- 4.S: Fractions (Summary)
- 5.1: Decimals (Part 1)

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Gasoline price changes all the time. They might go down for a period of time, but then they usually rise again. One thing that stays the same is that the price is not usually a whole number. Instead, it is shown using a decimal point to describe the cost in dollars and cents. We use decimal numbers all the time, especially when dealing with money. In this chapter, we will explore decimal numbers and how to perform operations using them.

Topic hierarchy

5.1: Decimals (Part 1)
Decimals are another way of writing fractions whose denominators are powers of ten. To convert a decimal number to a fraction or mixed number, look at the number to the left of the decimal. If it is zero, the decimal converts to a proper fraction. If not, the decimal converts to a mixed number. The numbers to right of the decimal point become the numerator while the place value corresponding to the final digit represent to the denominator. Finally, simplify the fraction if possible.
5.2: Decimals (Part 2)
Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line. To round a decimal, locate the given place value and mark it with an arrow. Underline the digit to the right of the place value and determine if it is greater than or equal to 5. If it is, add one to the digit in the given place value. If not, don't change the digit. Finally, rewrite the number, removing all digits to the right of the given place value.
5.3: Decimal Operations (Part 1)
To add or subtract decimals, write the numbers vertically so the decimal points line up. Use zeros for place holders, as needed. Then, add or subtract the numbers as if they were whole numbers. Lastly, place the decimal in the answer under the decimal points in the given numbers. Multiplying decimals is like multiplying whole numbers—we just have to determine where to place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.
5.4: Decimal Operations (Part 2)
Just as with multiplication, division of decimals is very much like dividing whole numbers. To divide a decimal by a whole number, we place the decimal point in the quotient above the decimal point in the dividend and then divide as usual with long division. Sometimes we need to use extra zeros at the end of the dividend to keep dividing until there is no remainder. To divide decimals, we multiply both the numerator and denominator by the same power of 10 to make the denominator a whole number.
5.5: Decimals and Fractions (Part 1)
To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction. To add a fraction and a decimal, we would need to either convert the fraction to a decimal or the decimal to a fraction. To compare a decimal to a fraction, we will first convert the fraction to a decimal and then compare the decimals. A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly.
5.6: Decimals and Fractions (Part 2)
All circles have exactly the same shape, but their sizes are affected by the length of the radius. A line segment that passes through a circle’s center connecting two points on the circle is called a diameter. The diameter is twice as long as the radius. The size of a circle can be measured in two ways. The distance around a circle is called its circumference. Archimedes discovered that for circles of all different sizes, dividing the circumference by the diameter always gives the same number.
5.7: Rational and Irrational Numbers
A rational number is a number that can be written in the form p/q, where p and q are integers and q ≠ 0. Rational numbers consist of many decimals and all fractions and integers, both positive and negative. An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form neither stops nor repeats. Some irrational numbers include pi and the square roots of numbers that are not perfect squares. Real numbers are numbers that are either rational or irrational.
5.8: Averages and Probability (Part 1)
The mean is often called the arithmetic average. It is computed by dividing the sum of the values by the number of values. The median of a set of data values is the middle value. So, half of the data values are less than or equal to the median while the other half of the data values are greater than or equal to the median. The mode of a set of numbers is the number with the highest frequency. The frequency is the number of times a number occurs.
5.9: Averages and Probability (Part 2)
The probability of an event is the number of favorable outcomes divided by the total number of outcomes possible. The probability of an event tells us how likely that event is to occur. We usually write probabilities as fractions or decimals. The basic definition of probability assumes that all the outcomes are equally likely to occur.
5.10: Ratios and Rate (Part 1)
A ratio compares two numbers or two quantities that are measured with the same unit. When a ratio is written in fraction form, the fraction should be simplified. To find the ratio of two measurements, we must make sure the quantities have been measured with the same unit. If the measurements are not in the same units, we must first convert them to the same units. A rate compares two quantities of different units. A rate is usually written as a fraction.
5.11: Ratios and Rate (Part 2)
A unit rate is a rate with denominator of 1 unit. Unit rates are very common in our lives. For example, when we say that we are driving at a speed of 68 miles per hour we mean that we travel 68 miles in 1 hour. A unit price is a unit rate that gives the price of one item. Unit prices are very useful if you comparison shop. The better buy is the item with the lower unit price. Most grocery stores list the unit price of each item on the shelves.
5.12: Simplify and Use Square Roots (Part 1)
If m is the square of n, then n is the square root of m and m is the product of a number n multiplied by itself. If n is a whole number, then m is a perfect square. Any positive number squared is positive, and any negative number squared is also positive. When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. The radical sign stands for the positive square root, which is also called the principal square root.
5.13: Simplify and Use Square Roots (Part 2)
There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator’s display. When we use a variable in a square root expression, for our work, we will assume that the variable represents a nonnegative number.
5.14: Commutative and Associative Properties (Part 1)
The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same. When adding or multiplying three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Addition and Multiplication, respectively. So, addition and multiplication are commutative and associative. But, subtraction and division are neither commutative nor associative.
5.15: Commutative and Associative Properties (Part 2)
When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first instead of automatically following the order of operations. No matter what you are doing, it is always a good idea to think ahead. When simplifying an expression, think about what your steps will be. For example, when adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.
5.16: Distributive Property
The Distributive Property states that if a, b, c are real numbers, then a(b + c) = ab + ac. In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. When you distribute a negative number, you need to be extra careful to get the signs correct. Sometimes we need to use the Distributive Property as part of the order of operations.
5.17: Properties of Identity, Inverses, and Zero
Adding zero to any number doesn’t change the value. For this reason, we call 0 the additive identity. The opposite of a number is its additive inverse. The reciprocal of a number is its multiplicative inverse. A number and its reciprocal multiply to 1, which is the multiplicative identity. The product of any real number and 0 is 0. Zero divided by any real number except zero is zero. But division by zero is undefined.
5.18: Solve Equations with Decimals
Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family’s budget, or planning for the future of your business, you’ll be solving equations with decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal.
5.E: Decimals (Exercises)
5.E: The Properties of Real Numbers (Exercises)
5.S: Decimals (Summary)
5.S: The Properties of Real Numbers (Summary)

Figure 5.1 - The price of a gallon of gasoline is written as a decimal number. (credit: Mark Turnauckus, Flickr)

Contributors

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."

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