Rational and Real Numbers

 

Learning Objective(s)

·         Identify the subset(s) of the real numbers that a given number belongs to.

·         Locate points on a number line.

·         Compare rational numbers.

·         Identify rational and irrational numbers.

 

You’ve worked with fractions and decimals, like 3.8 and . These numbers can be found between the integer numbers on a number line. There are other numbers that can be found on a number line, too. When you include all the numbers that can be put on a number line, you have the real number line. Let's dig deeper into the number line and see what those numbers look like. Let’s take a closer look to see where these numbers fall on the number line.

 

 

The fraction , mixed number , and decimal 5.33… (or ) all represent the same number. This number belongs to a set of numbers that mathematicians call rational numbers. Rational numbers are numbers that can be written as a ratio of two integers. Regardless of the form used,  is rational because this number can be written as the ratio of 16 over 3, or .

Examples of rational numbers include the following.

0.5, as it can be written as

 

, as it can be written as

 

−1.6, as it can be written as

 

4, as it can be written as

 

-10, as it can be written as

 

All of these numbers can be written as the ratio of two integers.

 

You can locate these points on the number line.

 

In the following illustration, points are shown for 0.5 or , and for 2.75 or .

As you have seen, rational numbers can be negative. Each positive rational number has an opposite. The opposite of  is , for example.

 

Be careful when placing negative numbers on a number line. The negative sign means the number is to the left of 0, and the absolute value of the number is the distance from 0. So to place −1.6 on a number line, you would find a point that is |−1.6| or 1.6 units to the left of 0. This is more than 1 unit away, but less than 2.

 

 

 

 

Which of the following points represents ?   Show/Hide Answer A) Incorrect. This point is just over 2 units to the left of 0. The point should be 1.25 units to the left of 0. The correct answer is point B.   B) Correct. Negative numbers are to the left of 0, and  should be 1.25 units to the left. Point B is the only point that’s more than 1 unit and less than 2 units to the left of 0.   C) Incorrect. Notice that this point is between 0 and the first unit mark to the left of 0, so it represents a number between −1 and 0. The point for  should be 1.25 units to the left of 0. You may have correctly found 1 unit to the left, but instead of continuing to the left another 0.25 unit, you moved right. The correct answer is point B.   D) Incorrect. Negative numbers are to the left of 0, not to the right. The point for  should be 1.25 units to the left of 0. The correct answer is point B.   E) Incorrect. This point is 1.25 units to right of 0, so it has the correct distance but in the wrong direction. Negative numbers are to the left of 0. The correct answer is point B.

 

 

 

When two whole numbers are graphed on a number line, the number to the right on the number line is always greater than the number on the left.

 

The same is true when comparing two integers or rational numbers. The number to the right on the number line is always greater than the one on the left.

 

Here are some examples.

 

 

 

 

Which of the following are true?   i. −4.1 > 3.2 ii. −3.2 > −4.1 iii. 3.2 > 4.1 iv. −4.6 < −4.1   A) i and iv B) i and ii C) ii and iii D) ii and iv E) i, ii, and iii   Show/Hide Answer A) i and iv Incorrect. −4.6 is to the left of −4.1, so −4.6 < −4.1. However, positive numbers such as 3.2 are always to the right of negative numbers such as −4.1, so 3.2 > −4.1 or −4.1 < 3.2. The correct answer is ii and iv, −3.2 > −4.1 and −4.6 < −4.1.   B) i and ii Incorrect. −3.2 is to the right of −4.1, so −3.2 > −4.1. However, positive numbers such as 3.2 are always to the right of negative numbers such as −4.1, so 3.2 > −4.1 or −4.1 < 3.2. The correct answer is ii and iv, −3.2 > −4.1 and −4.6 < −4.1.   C) ii and iii Incorrect. −3.2 is to the right of −4.1, so −3.2 > −4.1. However, 3.2 is to the left of 4.1, so 3.2 < 4.1. The correct answer is ii and iv, −3.2 > −4.1 and −4.6 < −4.1.   D) ii and iv Correct. −3.2 is to the right of −4.1, so −3.2 > −4.1. Also, −4.6 is to the left of −4.1, so −4.6 < −4.1.   E) i, ii, and iii Incorrect. −3.2 is to the right of −4.1, so −3.2 > −4.1. However, positive numbers such as 3.2 are always to the right of negative numbers such as −4.1, so 3.2 > −4.1 or −4.1 < 3.2. Also, 3.2 is to the left of 4.1, so 3.2 < 4.1. The correct answer is ii and iv, −3.2 > −4.1 and −4.6 < −4.1.

 

 

 

There are also numbers that are not rational. Irrational numbers cannot be written as the ratio of two integers.

 

Any square root of a number that is not a perfect square, for example , is irrational. Irrational numbers are most commonly written in one of three ways: as a root (such as a square root), using a special symbol (such as ), or as a nonrepeating, nonterminating decimal.

 

Numbers with a decimal part can either be terminating decimals or nonterminating decimals. Terminating means the digits stop eventually (although you can always write 0s at the end). For example, 1.3 is terminating, because there’s a last digit. The decimal form of  is 0.25. Terminating decimals are always rational.

 

Nonterminating decimals have digits (other than 0) that continue forever. For example, consider the decimal form of , which is 0.3333…. The 3s continue indefinitely. Or the decimal form of  , which is 0.090909…: the sequence “09” continues forever.

 

In addition to being nonterminating, these two numbers are also repeating decimals. Their decimal parts are made of a number or sequence of numbers that repeats again and again. A nonrepeating decimal has digits that never form a repeating pattern. The value of, for example, is 1.414213562…. No matter how far you carry out the numbers, the digits will never repeat a previous sequence.

 

If a number is terminating or repeating, it must be rational; if it is both nonterminating and nonrepeating, the number is irrational.

 

 

 

 

 

 

 

The set of real numbers is made by combining the set of rational numbers and the set of irrational numbers. The real numbers include natural numbers or counting numbers, whole numbers, integers, rational numbers (fractions and repeating or terminating decimals), and irrational numbers. The set of real numbers is all the numbers that have a location on the number line.

 

Sets of Numbers   Natural numbers                      1, 2, 3, …   Whole numbers                       0, 1, 2, 3, …   Integers                                    …, −3, −2, −1, 0, 1, 2, 3, …   Rational numbers                    numbers that can be written as a ratio of two integers—rational numbers are terminating or repeating when written in decimal form   Irrational numbers                    numbers than cannot be written as a ratio of two integers—irrational numbers are nonterminating and nonrepeating when written in decimal form   Real numbers                          any number that is rational or irrational

 

 

 

 

 

 

 

 

Which of the following sets does  belong to?   whole numbers integers rational numbers irrational numbers real numbers   A) rational numbers only B) irrational numbers only C) rational and real numbers D) irrational and real numbers E) integers, rational numbers, and real numbers F) whole numbers, integers, rational numbers, and real numbers   Show/Hide Answer A) rational numbers only Incorrect. The number is rational (it's written as a ratio of two integers) but it's also real. All rational numbers are also real numbers. The correct answer is rational and real numbers, because all rational numbers are also real.   B) irrational numbers only Incorrect. Irrational numbers can't be written as a ratio of two integers. The correct answer is rational and real numbers, because all rational numbers are also real.   C) rational and real numbers Correct. The number is between integers, so it can't be an integer or a whole number. It's written as a ratio of two integers, so it's a rational number and not irrational. All rational numbers are real numbers, so this number is rational and real.   D) irrational and real numbers Incorrect. Irrational numbers can't be written as a ratio of two integers. The correct answer is rational and real numbers, because all rational numbers are also real.   E) integers, rational numbers, and real numbers Incorrect. The number is between integers, not an integer itself. The correct answer is rational and real numbers.   F) whole numbers, integers, rational numbers, and real numbers Incorrect. The number is between integers, so it can't be an integer or a whole number. The correct answer is rational and real numbers.

 

The set of real numbers is all numbers that can be shown on a number line. This includes natural or counting numbers, whole numbers, and integers. It also includes rational numbers, which are numbers that can be written as a ratio of two integers, and irrational numbers, which cannot be written as a the ratio of two integers. When comparing two numbers, the one with the greater value would appear on the number line to the right of the other one.