2.1: Integers and Rational Numbers
Integers and rational numbers are important in daily life. The price per square yard of carpet is a rational number. The number of frogs in a pond is expressed using an integer. The organization of real numbers can be drawn as a hierarchy. Look at the hierarchy below.
The most generic number is the real number; it can be a combination of negative, positive, decimal, fractional, or non-repeating decimal values. Real numbers have two major categories: rational numbers and irrational numbers. Irrational numbers are non-repeating, non-terminating decimals such as \begin{align*}\pi\end{align*} or \begin{align*}\sqrt{2}\end{align*}. The discussion in this lesson revolves around rational numbers.
Definition: A rational number is a number that can be written in the form \begin{align*}\frac{a}{b}\end{align*}, where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are integers and \begin{align*}b \ne 0\end{align*}.
A Review of Fractions
You can think of a rational number as a fraction of a cake. If you cut the cake into \begin{align*}b\end{align*} slices, your share is \begin{align*}a\end{align*} of those slices. For example, when we see the rational number \begin{align*}\frac{1}{2}\end{align*}, we imagine cutting the cake into two parts. Our share is one of those parts. Visually, the rational number \begin{align*}\frac{1}{2}\end{align*} looks like this.
There are three main types of fractions:
- Proper fractions are rational numbers where the numerator is less than the denominator. A proper fraction represents a number less than one. With a proper fraction you always end up with less than a whole cake!
- Improper fractions are rational numbers where the numerator is greater than the denominator. Improper fractions can be rewritten as a mixed number – an integer plus a proper fraction. An improper fraction represents a number greater than one.
- Equivalent fractions are two fractions that give the same numerical value when evaluated. For example, look at a visual representation of the rational number \begin{align*}\frac{2}{4}\end{align*}.
The visual of \begin{align*}\frac{1}{2}\end{align*} is equivalent to the visual of \begin{align*}\frac{2}{4}\end{align*}. We can write out the prime factors of both the numerator and the denominator and cancel matching factors that appear in both the numerator and denominator.
\begin{align*}\left (\frac{2}{4}\right ) = \left (\frac{\cancel{2}\cdot 1}{\cancel{2}\cdot 2 \cdot 1}\right )\end{align*} We then re-multiply the remaining factors. \begin{align*}\left (\frac{2}{4}\right ) = \left (\frac{1}{2}\right )\end{align*}
Therefore, \begin{align*}\frac{1}{2} = \frac{2}{4}\end{align*}. This process is called reducing the fraction, or writing the fraction in lowest terms. Reducing a fraction does not change the value of the fraction; it simplifies the way we write it. When we have canceled all common factors, we have a fraction in its simplest form.
Example 1: Classify and simplify the following rational numbers.
a) \begin{align*}\left (\frac{3}{7}\right )\end{align*}
b) \begin{align*}\left (\frac{9}{3}\right )\end{align*}
c) \begin{align*}\left (\frac{50}{60}\right )\end{align*}
Solution:
a) Because both 3 and 7 are prime numbers, \begin{align*}\frac{3}{7}\end{align*} is a proper fraction written in its simplest form.
b) The numerator is larger than the denominator; therefore, this is an improper fraction.
\begin{align*}\frac{9}{3}= \frac{3 \times 3}{3}= \frac{3}{1}=3\end{align*}
c) This is a proper fraction; \begin{align*}\frac{50}{60}= \frac{5 \times 2 \times 5}{6 \times 2 \times 5}= \frac{5}{6}\end{align*}
Ordering Rational Numbers
To order rational numbers is to arrange them according to a set of directions, such as ascending (lowest to highest) or descending (highest to lowest). Ordering rational numbers is useful when determining which unit cost is the cheapest.
Example 2: Cans of tomato sauce come in three sizes: 8 ounces, 16 ounces, and 32 ounces. The costs for each size are $0.59, $0.99, and $1.29, respectively. Find the unit cost and order the rational numbers in ascending order.
Solution: Use proportions to find the cost per ounce. \begin{align*}\frac{\$0.59}{8} = \frac{\$0.07375}{ounce}; \ \frac{\$0.99}{16} = \frac{\$0.061875}{ounce}; \ \frac{\$1.29}{32} = \frac{\$0.0403125}{ounce}\end{align*}. Arranging the rational numbers in ascending order: 0.0403125, 0.061875, 0.07375
Example 3: Which is greater \begin{align*}\frac{3}{7}\end{align*} or \begin{align*}\frac{4}{9}\end{align*}?
Solution: Begin by creating a common denominator for these two fractions. Which number is evenly divisible by 7 and 9? \begin{align*}7 \times 9 = 63\end{align*}, therefore the common denominator is 63.
\begin{align*}\frac{3 \times 9}{7 \times 9} = \frac{27}{63} && \frac{4 \times 7}{9 \times 7} = \frac{28}{63}\end{align*}
Because \begin{align*}28 > 27, \ \frac{4}{9} > \frac{3}{7}\end{align*}
For more information regarding how to order fractions, watch this YouTube video.
Khan Academy: Ordering Fractions
Graph and Compare Integers
More specific than the rational numbers are the integers. Integers are whole numbers and their negatives. When comparing integers, you will use the math verbs such as less than, greater than, approximately equal to, and equal to. To graph an integer on a number line, place a dot above the number you want to represent.
Example 4: Compare the numbers 2 and –5.
Solution: First, we will plot the two numbers on a number line.
We can compare integers by noting which is the greatest and which is the least. The greatest number is farthest to the right, and the least is farthest to the left.
In the diagram above, we can see that 2 is farther to the right on the number line than –5, so we say that 2 is greater than –5. We use the symbol > to mean “greater than.”
Therefore, \begin{align*}2 > -5\end{align*}.
Numbers and Their Opposites
Every number has an opposite, which represents the same distance from zero but in the other direction.
A special situation arises when adding a number to its opposite. The sum is zero. This is summarized in the following property.
The Additive Inverse Property: For any real number \begin{align*}a, \ a + (- a) = 0\end{align*}.
Absolute Value
Absolute value represents the distance from zero when graphed on a number line. For example, the number 7 is 7 units away from zero. The number –7 is also 7 units away from zero. The absolute value of a number is the distance it is from zero, so the absolute value of 7 and the absolute value of –7 are both 7.
We write the absolute value of –7 like this \begin{align*}|-7|\end{align*}
We read the expression \begin{align*}|x|\end{align*} like this: “the absolute value of \begin{align*}x\end{align*}.”
- Treat absolute value expressions like parentheses. If there is an operation inside the absolute value symbols, evaluate that operation first.
- The absolute value of a number or an expression is always positive or zero. It cannot be negative. With absolute value, we are only interested in how far a number is from zero, not the direction.
Example 5: Evaluate the following absolute value expressions.
a) \begin{align*}|5 + 4|\end{align*}
b) \begin{align*}3 - |4 - 9|\end{align*}
c) \begin{align*}|-5 - 11|\end{align*}
d) \begin{align*}-|7-22|\end{align*}
Solution:
a) \begin{align*}|5 + 4| &= |9|\\ &= 9\end{align*}
b) \begin{align*}3 - |4 - 9| &= 3 - |-5|\\ &= 3 - 5\\ &= -2\end{align*}
c) \begin{align*}|-5-11|&=|-16|\\ &= 16\end{align*}
d) \begin{align*}-|7 - 22| &= -|-15|\\ &= -(15)\\ &= -15\end{align*}
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Integers and Rational Numbers (13:00)
- Define absolute value.
- What are the three types of fractions?
- Give an example of a real number that is not an integer.
- What standards separate a rational number from an irrational number?
- The tick-marks on the number line represent evenly spaced integers. Find the values of \begin{align*}a, b, c, d,\end{align*} and \begin{align*}e\end{align*}.
In 6 – 8, determine what fraction of the whole each shaded region represents.
In 9 – 12, place the following sets of rational numbers in order from least to greatest.
- \begin{align*}\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\end{align*}
- \begin{align*}\frac{11}{12}, \frac{12}{11}, \frac{13}{10}\end{align*}
- \begin{align*}\frac{39}{60}, \frac{49}{80}, \frac{59}{100}\end{align*}
- \begin{align*}\frac{7}{11}, \frac{8}{13}, \frac{12}{19}\end{align*}
In 13 – 18, find the simplest form of the following rational numbers.
- \begin{align*}\frac{22}{44}\end{align*}
- \begin{align*}\frac{9}{27}\end{align*}
- \begin{align*}\frac{12}{18}\end{align*}
- \begin{align*}\frac{315}{420}\end{align*}
- \begin{align*}\frac{19}{101}\end{align*}
- \begin{align*}\frac{99}{11}\end{align*}
In 19 – 24, find the opposite of each of the following.
- 1.001
- –9.345
- (16 – 45)
- (5 – 11)
- \begin{align*}(x + y)\end{align*}
- \begin{align*}(x - y)\end{align*}
In 25 – 34, simplify.
- \begin{align*}|-98.4|\end{align*}
- \begin{align*}|123.567|\end{align*}
- \begin{align*}-|16-98|\end{align*}
- \begin{align*}11 - |-4|\end{align*}
- \begin{align*}|4 - 9|-|-5|\end{align*}
- \begin{align*}|-5-11|\end{align*}
- \begin{align*}7-|22-15-19|\end{align*}
- \begin{align*}-|-7|\end{align*}
- \begin{align*}|-2-88| - |88 + 2|\end{align*}
- \begin{align*}|-5-99| + -|16-7|\end{align*}
In 35 – 38, compare the two real numbers.
- 8 and 7.99999
- –4.25 and \begin{align*}\frac{-17}{4}\end{align*}
- 65 and –1
- 10 units left of zero and 9 units right of zero
- A frog is sitting perfectly on top of number 7 on a number line. The frog jumps randomly to the left or right, but always jumps a distance of exactly 2. Describe the set of numbers that the frog may land on, and list all the possibilities for the frog’s position after exactly 5 jumps.
- Will a real number always have an additive identity? Explain your reasoning.
Mixed Review
- Evaluate the following expression: \begin{align*}\frac{5}{6} d + 7a^2\end{align*}; use \begin{align*}a=(-1), \ d=24\end{align*}.
- The length of a rectangle is one more inch than its width. If the perimeter is 22 inches, what are the dimensions of the rectangle?
- Determine if \begin{align*}x=-2\end{align*} is a solution to \begin{align*}4x+7 \le 15\end{align*}.
- Simplify: \begin{align*}\frac{(7+3) \div 2 \times 3^2-5}{(58-8)}\end{align*}.