Number Sets

 

Learning Objective(s)

·         Identify and define counting, natural, whole, integer, rational, irrational, and real numbers.

 

 

Mathematicians recognize several sets of numbers that share certain characteristics. These categories are helpful to know when only certain kinds of numbers are valid for values and variables. Our understanding and classification of the different sets of numbers has developed over thousands of years.

 

 

Early civilizations found different ways to write numbers, but they all started with the same set of numbers preschool children learn today: the natural numbers (also called the counting numbers). These are the numbers 1, 2, 3, and so on—the numbers we use when counting. They are natural because our natural understanding of numbers begins with noticing multiple copies of things, like how many fingers we have, or the size of sets, like how many toys we have.

 

Although the earliest civilizations understood “nothing”—they knew when they had no cows, or no children, of course—the number zero has an interesting history. The first use of a symbol to represent “nothing” wasn't until 300 BC. The Babylonian number system used the symbols only as a placeholder in a place value system, much as we use 0 in the number 702 to represent no 10s. The earliest recognition of zero as a number, in the same way that 1 and 23 are numbers, is uncertain, but may date to the 9^th century in India. When 0 is added to the set of 1, 2, 3, and so on, it forms the whole numbers. These are called “whole” because they have no fractional parts.

 

The integers are the whole numbers plus their negative counterparts: …, -3, -2, -1, 0, 1, 2, 3, …. Negative numbers first appeared in China as early as the first century BC. (That’s as much as 1000 years before zero was recognized as a number itself!) However, despite their usefulness in representing concepts such as debt, it was only since the 18^th century—less than 300 years ago—that they gained widespread acceptance as actual numbers.

 

Which of the following sets does the number 0 belong to?   natural numbers whole numbers integers   A) Natural numbers only B) Whole numbers only C) Natural and whole numbers D) Integers only E) Whole numbers and integers                   Show/Hide Answer A) Incorrect. The natural numbers are 1, 2, 3, and so on. They don’t include 0. The correct answer is whole numbers and integers.   B) Incorrect. While the whole numbers include 0, so do the integers. The correct answer is whole numbers and integers.   C) Incorrect. The natural numbers are 1, 2, 3, and so on. They don’t include 0. The correct answer is whole numbers and integers.   D) Incorrect. While the integers include 0, so do the whole numbers. The correct answer is whole numbers and integers.   E) Correct. Both the whole numbers and integers include 0, but the natural numbers do not.

 

 

 

Fractional numbers have been around for much longer than zero or negative numbers. Ancient Egyptians (from as early as the 21^st century BC) studied fractions. Today, fractional numbers are included in the set of rational numbers, which are numbers that can be written in the form  where p and q are both integers. Rational numbers can be written in many forms. For example,  can also be written as , 5.66…, or . Regardless of the form used, because this number can be written as the ratio of two integers, the number is rational.

 

Note that all integers (and that means all whole numbers and natural numbers) are rational numbers because they can be written using 1 as the denominator q. For example, -3 can be written as , so it is rational as well.

 

So far, the types of numbers we have described create a nesting series of sets. We began with the natural numbers, then expanded that set with 0 to form the whole numbers. Then we included negative numbers with the whole numbers to create the integers. Now we have the rational numbers, which includes all the integers plus many other numbers. This diagram shows how these number sets are “nested”:

 

 

 

 

The Greek mathematician Pythagoras, for whom the Pythagorean Theorem is named, was the leader of a group known as the Pythagoreans. They believed that all quantities could be expressed as a natural number or a ratio of natural numbers. Legend tells that they believed this so fervently that when one of their members used the Pythagorean Theorem to prove that the hypotenuse of the triangle below cannot be expressed as a ratio of natural numbers, they exiled him. (Or worse—some stories say they threw him overboard while at sea and watched him drown!)

 

Today we know that the Pythagoreans' thinking was wrong, and there are indeed quantities that are not rational. These irrational numbers cannot be expressed as a ratio of 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, using a special symbol (such as ), or as a nonrepeating, nonterminating decimal.

 

Numbers with a decimal part can either be terminating or nonterminating.  Terminating means the digits stop eventually (although we can always write 0s at the end). A nonterminating decimal has 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 decimal is nonrepeating if the digits never form a repeating pattern. The value of, for example, is 1.414213562…. No matter how far we 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.

 

Consider a terminating decimal, like 3.529. Since this number has a terminating decimal, it is a rational number. We can write 3.529 as a ratio of integers: the numerator is the number without the decimal point (in this case, 3,529) and the denominator is a power of 10 corresponding to the place value of the last digit. Since the 9 is in the thousandths place, the denominator is 1,000. (Another way to do this is to count the decimal places to the right of the decimal point in the original number. The denominator is 1 followed by that many 0s. There are three digits to the right of the decimal place in 3.529, so the denominator is 1 followed by three 0s.)

 

 

Here's one more example:

 

 

 

What about a nonterminating, repeating decimal? Here’s one method for expressing a repeating decimal as a ratio of integers. It works with all repeating decimals:

 

1.      Write an equation, x = the number with a bar over the repeating portion.

2.      Count how many digits are in the repeating portion of the decimal, and multiply both sides of the equation in step 1 by 10 raised to that power to create a second equation. (For example, if there are 3 digits that repeat, multiply by 10³ or 1000.) Keep the repeating portion the same, even if you could move the repetition bar over.

3.      Subtract the equation in step 1 from the equation in step 2. In doing this, the repeating decimal parts will line up. All the digits from that point on will be 0s.

4.      Divide both sides of the new equation in step 3 by the coefficient of x.

5.      Multiply both numerator and denominator by 10 until any decimals are removed, or rewrite the number in lowest terms.

 

Let’s try it to see if it makes sense:

 

 

That did work! We were able to write a nonterminating but repeating decimal as the ratio of two integers. That proves that the number is rational.

 

Is  rational or irrational?   A) Rational B) Irrational   Show/Hide Answer A) Correct. The decimal is nonterminating, but it is repeating, so it can be written as a ratio of integers.   B) Incorrect. Although the decimal in nonterminating, it is repeating. Repeating decimals can be written as a ratio of integers. The number is rational.

 

 

The set of real numbers is made by combining the set of rational numbers and the set of 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                    any number that can be expressed in the form , where p and q are integers, rational numbers are terminating or repeating when written in decimal form   Irrational numbers                    any number that cannot be written as  (where p and q are integers), irrational numbers are nonterminating and nonrepeating when written in decimal form   Real numbers                          any number that is rational or irrational.

 

The diagram below shows how all the real numbers are related to each other. Notice there is no overlap between the rational numbers and the irrational numbers, and the two sets together make up the set of real numbers.

 

 

 

 

The numbers that can be represented on a number line are called real numbers. These numbers can be separated into two sets that have no numbers in common: the irrational numbers and the rational numbers. Irrational numbers have decimal forms that are nonterminating and nonrepeating. Rational numbers have decimal forms that are terminating or repeating. Inside the set of rational numbers are several smaller, nested sets: integers, whole numbers, and natural numbers.

 

The sets of numbers are defined as follows:

 

Real numbers	any number that is rational or irrational
Rational numbers	any number that can be written as the ratio of two integers and that is terminating or repeating in decimal form
Integers	…, -3, -2, -1, 0, 1, 2, 3, …
Whole numbers	0, 1, 2, 3, …
Natural numbers	1, 2, 3, …
Irrational numbers	any number that can't be written as the ratio of two integers and that is nonterminating and nonrepeating in decimal form