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Closed Captioned Video: Rational Numbers: Variable Expressions with Rational Numbers

Closed Captioned Video: Rational Numbers: Variable Expressions with Rational Numbers

This is part of a collection of video tutorials on the topic of Rational Numbers. This includes defining rational numbers, rational number operations, comparing and ordering rational numbers, and applications of rational numbers.

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To see the complete collection of these videos on fractions, click on this link.

The following section includes background information on rational numbers. Refer to this section as you view the videos, or as review material afterward.


A Brief Review of Integers

Before defining rational numbers, we first need to review integers. The definition of a rational number involves integers, so let's make sure you're comfortable with the definition of integers.

Integers include the whole numbers, zero, and positive numbers 1, 2, 3, etc., but also includes a different class of numbers, negative numbers.

 

The definition of integers.

You can use a number line to represent the integers.

A number line showing integers.

Notice that every integer and its opposite is the same distance from 0 on the number line. Also, the arrow heads on the number line mean that the integers extend to infinity.

Notice that every integer and its opposite is the same distance from 0 on the number line. Also, the arrow heads on the number line mean that the integers extend to infinity.

Representing distance on a number line is shown below.

Integers

Distance is always a positive number, so use the absolute value symbol to ensure the result is positive. Subtract one value from another and find the absolute value of the difference.

Every integer has its opposite. Notice on the number line that 1 and -1 are opposites, as are 2 and -2, and so on. Every integer and its opposite is the same distance from zero on the number line, as shown below.

Integers

As you can see both 4 and -4 are four units from zero. This same pattern applies to all integers and their opposites. Also, the sum of any integer and its opposite is zero.

1 + (-1) = 0

2 + (-2) = 0

This pattern continues for all integers and their opposites. 

The technique of finding the absolute value of a difference applies to any pair of integers, as shown below.

Integers

 

Defining Rational Numbers

All rational numbers can be written as shown below. A rational number looks like a fraction, but  it is a ratio of two numbers. As with fractions, the denominator cannot be zero.

The definition of a rational number.

Rational numbers can be written in different ways.

Examples of rational numbers.

Integers are examples of rational numbers.

Examples of rational numbers.

If a number cannot be written as the ratio of two integers, then it is an irrational number.

Examples of irrational numbers.

Rational numbers can be graphed on a number line. Make a note of the rational numbers in decimal form graphed on the number line below. However, keep in mind that the integer values are also rational numbers. 

Examples of graphs of rational numbers.

Rational numbers and their opposites, as with integers, add up to zero. Every rational number has its opposite.

Adding and Subtracting Rational Numbers

Add and subtract rational numbers the way you would add fractions. Here is an example of combining a positive and negative rational number.

Rational Numbers

Here is another example. In this example the result is a negative number.

Rational Numbers

Adding and subtracting rational numbers in fraction form is a lot like adding and subtraction fractions. The main difference is that these fractions can be negative.

When expressed as decimals, combine rational numbers the way you would decimals.

Rational numbers

Adding and subtracting rational numbers in decimal form is a lot like adding and subtraction decimals. The main difference is that these decimals can be negative.

Multiplying Rational Numbers

Multiply rational numbers the way you would multiply fractions. Here are examples of multiplying rational numbers. The second example shows how to divide out common factors before multiplying.

Rational Numbers

Multiplying rational numbers in fraction form is a lot like multiplying fractions. The main difference is that these fractions can be negative.

Rational Expressions

Rational numbers are ratios of two integers. Similarly, a rational expression is the ratio of two algebraic expressions. The algebraic expression can be a number or a polynomial. Here are three examples of rational expressions.

Rational Expressions

Like rational numbers, the denominator can't be zero. Let's see what that looks like for the expressions above.

Rational Expressions

Like rational numbers, rational expressions can be combined. Use fraction operations to combine rational expressions.

Rational Expressions

Simplify rational expressions by finding common factors in the numerator and denominator. In this example there is a common term, x + 1, that divides out of both the numerator and denominator.

Rational Expressions

Rational Equations

When a rational expression equals a number, this forms a rational equation, which can be solved for the variable. Here is an example of a rational equation and its solution.

Rational Equations

This is part of a collection of video tutorials on the topic of Rational Numbers.

To see the complete collection of these videos, click on this link.

Note: The download is an MP4 file.

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Math Video Library

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Common Core Standards CCSS.MATH.CONTENT.7.NS.A.2.A, CCSS.MATH.CONTENT.8.NS.A.1, CCSS.MATH.CONTENT.6.NS.C.5, CCSS.MATH.CONTENT.6.NS.C.6, CCSS.Math.CONTENT.7.NS.A.1, CCSS.MATH.CONTENT.7.NS.A.1.C, CCSS.MATH.CONTENT.7.NS.A.1.D, CCSS.MATH.CONTENT.7.NS.A.3
Duration 4.67 minutes
Grade Range 6 - 8
Curriculum Nodes Algebra
    • Rational Expressions and Functions
        • Rational Expressions
Copyright Year 2020
Keywords rational numbers, rational numbers video tutorials, video tutorials, algebraic expressions, video tutorial, Closed Captioned Video