1.1: Number Systems

In this section we introduce the number systems that we will work with in the remainder of this text.

The Natural Numbers

We begin with a definition of the natural numbers, or the counting numbers.

Definition 1: natural numbers

The set of natural numbers is the set

\[\mathbb{N}=\{1,2,3, \ldots\} \nonumber \]

The notation in equation (2) is read “\(\mathbb{N}\) is the set whose members are 1, 2, 3, and so on.” The ellipsis (the three dots) at the end in equation (2) is a mathematician’s way of saying “et-cetera.” We list just enough numbers to establish a recognizable pattern, then write “and so on,” assuming that a pattern has been sufficiently established so that the reader can intuit the rest of the numbers in the set. Thus, the next few numbers in the set \(\mathbb{N}\) are 4, 5, 6, 7, “and so on.”

Note that there are an infinite number of natural numbers. Other examples of natural numbers are 578,736 and 55,617,778. The set \(\mathbb{N}\) of natural numbers is unbounded; i.e., there is no largest natural number. For any natural number you choose, adding one to your choice produces a larger natural number.

For any natural number n, we call m a divisor or factor of n if there is another natural number k so that \(n = mk\). For example, 4 is a divisor of 12 (because 12=4 \times 3), but 5 is not. In like manner, 6 is a divisor of 12 (because 12=6 \times 2), but 8 is not.

We next define a very special subset of the natural numbers.

Definition 3: Prime NUmbers

If the only divisors of a natural number \(p\) are 1 and itself, then \(p\) is said to be prime.

For example, because its only divisors are 1 and itself, 11 is a prime number. On the other hand, 14 is not prime (it has divisors other than 1 and itself, i.e., 2 and 7). In like manner, each of the natural numbers 2, 3, 5, 7, 11, 13, 17, and 19 is prime. Note that 2 is the only even natural number that is prime.

If a natural number other than 1 is not prime, then we say that it is composite. Note that any natural number (except 1) falls into one of two classes; it is either prime, or it is composite.

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In this textbook, definitions, equations, and other labeled parts of the text are numbered consecutively, 2 regardless of the type of information. Figures are numbered separately, as are Tables.

Although the natural number 1 has only 1 and itself as divisors, mathematicians, particularly number 3 theorists, don’t consider 1 to be prime. There are good reasons for this, but that might take us too far afield. For now, just note that 1 is not a prime number. Any number that is prime has exactly two factors, namely itself and 1.

We can factor the composite number 36 as a product of prime factors, namely

\[36=2 \times 2 \times 3 \times 3 \nonumber \]

Other than rearranging the factors, this is the only way that we can express 36 as a product of prime factors.

Definition 4: Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic says that every natural number has a unique prime factorization.

No matter how you begin the factorization process, all roads lead to the same prime factorization. For example, consider two different approaches for obtaining the prime factorization of 72.

\[\begin{array}{llllll} 72 & = & 8\times 9 & 72 & = & 4\times 18 & \\ & =& (4 \times 2) \times(3 \times 3) & & = &(2 \times 2) \times(2 \times 9) \\ &=&2 \times 2 \times 2 \times 3 \times 3 & & = & 2 \times 2 \times 2 \times 3 \times 3\end{array} \nonumber \]

In each case, the result is the same, \(72=2 \times 2 \times 2 \times 3 \times 3\)

Zero

The use of zero as a placeholder and as a number has a rich and storied history. The ancient Babylonians recorded their work on clay tablets, pressing into the soft clay with a stylus. Consequently, tablets from as early as 1700 BC exist today in museums around the world. A photo of the famous Plimpton_322 is shown in Figure \(\PageIndex{1}\), where the markings are considered by some to be Pythagorean triples, or the measures of the sides of right triangles.

Figure \(\PageIndex{1}\) Plimpton_322

The people of this ancient culture had a sexagesimal (base 60) numbering system that survived without the use of zero as a placeholder for over 1000 years. In the early Babylonian system, the numbers 216 and 2106 had identical recordings on the clay tablets of the authors. One could only tell the difference between the two numbers based upon the context in which they were used. Somewhere around the year 400 BC, the Babylonians started using two wedge symbols to denote a zero as a placeholder (some tablets show a single or a double-hook for this placeholder).

The ancient Greeks were well aware of the Babylonian positional system, but most of the emphasis of Greek mathematics was geometrical, so the use of zero as a placeholder was not as important. However, there is some evidence that the Greeks used a symbol resembling a large omicron in some of their astronomical tables.

It was not until about 650 AD that the use of zero as a number began to creep into the mathematics of India. Brahmagupta (598-670?), in his work Brahmasphutasiddhanta, was one of the first recorded mathematicians who attempted arithmetic operations with the number zero. Still, he didn’t quite know what to do with division by zero when he wrote

Positive or negative numbers when divided by zero is a fraction with zero as denominator.

Note that he states that the result of division by zero is a fraction with zero in the denominator. Not very informative. Nearly 200 years later, Mahavira (800-870) didn’t do much better when he wrote

A number remains unchanged when divided by zero.

It seems that the Indian mathematicians could not admit that division by zero was impossible.

The Mayan culture (250-900 AD) had a base 20 positional system and a symbol they used as a zero placeholder. The work of the Indian mathematicians spread into the Arabic and Islamic world and was improved upon. This work eventually made its way to the far east and also into Europe. Still, as late as the 1500s European mathematicians were still not using zero as a number on a regular basis. It was not until the 1600s that the use of zero as a number became widespread.

Of course, today we know that adding zero to a number leaves that number unchanged and that division by zero is meaningless,4 but as we struggle with these concepts, we should keep in mind how long it took humanity to come to grips with this powerful abstraction (zero as a number).

If we add the number zero to the set of natural numbers, we have a new set of numbers which are called the whole numbers

Definition 5

The set of whole numbers is the set

\[\mathbb{W}=\{0,1,2,3, \ldots\} \nonumber \]

The Integers

Today, much as we take for granted the fact that there exists a number zero, denoted by 0, such that

% It makes no sense to ask how many groups of zero are in five. Thus, 5/0 is undefined. 4

\[a+0=a \nonumber \]

for any whole number a, we similarly take for granted that for any whole number a there exists a unique number −a, called the “negative” or “opposite” of a, so that \[a+(-a)=0 \nonumber \]

In a natural way, or so it seems to modern-day mathematicians, this easily introduces the concept of a negative number. However, history teaches us that the concept of negative numbers was not embraced wholeheartedly by mathematicians until somewhere around the 17th century.

In his work Arithmetica (c. 250 AD?), the Greek mathematician Diophantus (c. 200-284 AD?), who some call the “Father of Algebra,” described the equation 4 = 4x + 20 as “absurd,” for how could one talk about an answer less than nothing? Girolamo Cardano (1501-1576), in his seminal work Ars Magna (c. 1545 AD) referred to negative numbers as “numeri ficti,” while the German mathematician Michael Stifel (1487-1567) referred to them as “numeri absurdi.” John Napier (1550-1617) (the creator of logarithms) called negative numbers “defectivi,” and Rene Descartes (1596-1650) (the creator of analytic geometry) labeled negative solutions of algebraic equations as “false roots.”

On the other hand, there were mathematicians whose treatment of negative numbers resembled somewhat our modern notions of the properties held by negative numbers. The Indian mathematician Brahmagupta, whose work with zero we’ve already mentioned, described arithmetical rules in terms of fortunes (positive number) and debts (negative numbers). Indeed, in his work Brahmasphutasiddhanta, he writes “a fortune subtracted from zero is a debt,” which in modern notation would resemble 0 − 4 = −4. Further, “a debt subtracted from zero is a fortune,” which resonates as 0 − (−4) = 4. Further, Brahmagupta describes rules for multiplication and division of positive and negative numbers:

• The product or quotient of two fortunes is one fortune.
• The product or quotient of two debts is one fortune.
• The product or quotient of a debt and a fortune is a debt.
• The product or quotient of a fortune and a debt is a debt.

In modern-day use we might say that “like signs give a positive answer,” while “unlike signs give a negative answer.” Modern examples of Brahmagupta’s first two rules are (5)(4) = 20 and (−5)(−4) = 20, while examples of the latter two are (−5)(4) = −20 and (5)(−4) = −20. The rules are similar for division.

In any event, if we begin with the set of natural numbers \(\mathbb{N} = {1, 2, 3, . . .}\), add zero, then add the negative of each natural number, we obtain the set of integers.

Definition 8

The set of integers is the set

\[\mathbb{Z}=\{\ldots,-3,-2,-1,0,1,2,3, \ldots\} \nonumber \]

The letter \(\mathbb{Z}\) comes from the word Zahl, which is a German word for “number.”

It is important to note that an integer is a “whole” number, either positive, negative, or zero. Thus, −11 456, −57, 0, 235, and 41 234 576 are integers, but the numbers −2/5, 0.125, \(\sqrt{2}\) and \(\pi\) are not. We’ll have more to say about the classification of the latter numbers in the sections that follow.

Rational Numbers

You might have noticed that every natural number is also a whole number. That is, every number in the set \(\mathbb{N}=\{1,2,3, \ldots\}\) is also a number in the set \(\mathbb{W}=\{0,1,2,3, \ldots\}\). Mathematicians say that “\(\mathbb{N}\) is a subset of \(\mathbb{W}\),” meaning that each member of the set \(\mathbb{N}\) is also a member of the set \(\mathbb{W}\). In a similar vein, each whole number is also an integer, so the set \(\mathbb{W}\) is a subset of the set \(\mathbb{Z}=\{\ldots,-2,-2,-1,0,1,2,3, \dots\}\).

We will now add fractions to our growing set of numbers. Fractions have been used since ancient times. They were well known and used by the ancient Babylonians and Egyptians.

In modern times, we use the phrase rational number to describe any number that is the ratio of two integers. We will denote the set of rational numbers with the letter \(\mathbb{Q}\).

Definition 10

The set of rational numbers is the set

\[\mathbb{Q}=\{{\frac{m}{n} : \text{m, n are integers}, n \neq 0}\} \nonumber \]

This notation is read “the set of all ratios m/n, such that m and n are integers, and n is not 0.” The restriction on n is required because division by zero is undefined.

Clearly, numbers such as −221/31, −8/9, and 447/119, being the ratios of two integers, are rational numbers (fractions). However, if we think of the integer 6 as the ratio 6/1 (or alternately, as 24/4, −48/ − 8, etc.), then we note that 6 is also a rational number. In this way, any integer can be thought of as a rational number (e.g., 12 = 12/1, −13 = −13/1, etc.). Therefore, the set \(\mathbb{Z}\) of integers is a subset of the set \(\mathbb{Q}\) of rational numbers.

But wait, there is more. Any decimal that terminates is also a rational number. For example,

\[0.25=\frac{25}{100}, \quad 0.125=\frac{125}{1000}, \quad and -7.6642=-\frac{76642}{10000} \nonumber \]

The process for converting a terminating decimal to a fraction is clear; count the number of decimal places, then write 1 followed by that number of zeros for the denominator.

For example, in 7.638 there are three decimal places, so place the number over 1000, as in

\[\frac{7638}{1000} \nonumber \]

But wait, there is still more, for any decimal that repeats can also be expressed as the ratio of two integers. Consider, for example, the repeating decimal

\[0.0 \overline{21}=0.0212121 \ldots \nonumber \]

Note that the sequence of integers under the “repeating bar” are repeated over and over indefinitely. Further, in the case of \(0.0\overline{21}\), there are precisely two digits under the repeating bar. Thus, if we let \(x=0.0\overline{21}\), then

\[x=0.0212121 \ldots \nonumber \]

and multiplying by 100 moves the decimal two places to the right.

\[100 x=2.12121 \ldots \nonumber \]

If we align these two results

\[\begin{aligned} 100 x &=2.12121 \ldots \\-x &=0.02121 \ldots \end{aligned} \nonumber \]

and subtract, then the result is

\[\begin{aligned} 99 x &=2.1 \\ x &=\frac{2.1}{99} \end{aligned} \nonumber \]

However, this last result is not a ratio of two integers. This is easily rectified by multiplying both numerator and denominator by 10.

\[x=\frac{21}{990} \nonumber \]

We can reduce this last result by dividing both numerator and denominator by 3. Thus, \(0.0 \overline{21}=7 / 330\), being the ratio of two integers, is a rational number.

Let’s look at another example.

Example \(\PageIndex{1}\)

Show that \(0 . \overline{621}\) is a rational number.

Solution

In this case, there are three digits under the repeating bar. If we let x = 0.621, then multiply by 1000 (three zeros), this will move the decimal three places to the right.

\[\begin{aligned} 1000 x &=621.621621 \ldots \\ x &=0.621621 \ldots \end{aligned} \nonumber \]

Subtracting,

\[\begin{aligned} 999 x &=621 \\ x &=\frac{621}{999} \end{aligned} \nonumber \]

Dividing numerator and denominator by 27 (or first by 9 then by 3), we find that \(0 . \overline{621}=23 / 37\). Thus, 0.621, being the ratio of two integers, is a rational number.

At this point, it is natural to wonder, “Are all numbers rational?” Or, “Are there other types of numbers we haven’t discussed as yet?” Let’s investigate further.

The Irrational Numbers

If a number is not rational, mathematicians say that it is irrational

Definition 13

Any number that cannot be expressed as a ratio of two integers is called an irrational number.

Mathematicians have struggled with the concept of irrational numbers throughout history. Pictured in Figure \(\PageIndex{2}\) is an ancient Babylonian artifact called The Square Root of Two Tablet.

Figure \(\PageIndex{2}\) The Square Root of Two Tablet.

There is an ancient fable that tells of a disciple of Pythagoras who provided a geometrical proof of the irrationality of \(\sqrt{2}\). However, the Pythagoreans believed in the absoluteness of numbers, and could not abide the thought of numbers that were not rational. As a punishment, Pythagoras sentenced his disciple to death by drowning, or so the story goes.

But what about \(\sqrt{2}\)? Is it rational or not? A classic proof, known in the time of Euclid (the “Father of Geometry,” c. 300 BC), uses proof by contradiction. Let us assume that \(\sqrt{2}\) is indeed rational, which means that \(\sqrt{2}\) can be expressed as the ratio of two integers p and q as follows.

\[\sqrt{2}=\frac{p}{q} \nonumber \]

Square both sides,

\[2=\frac{p^{2}}{q^{2}} \nonumber \]

then clear the equation of fractions by multiplying both sides by \(q^{2}\).

\[p^{2}=2 q^{2} \nonumber \]

Now p and q each have their own unique prime factorizations. Both \(p^{2}\) and \(q^{2}\) have an even number of factors in their prime factorizations.6 But this contradicts equation 14, because the left side would have an even number of factors in its prime factorization, while the right side would have an odd number of factors in its prime factorization (there’s one extra 2 on the right side).

Therefore, our assumption that \(\sqrt{2}\) was rational is false. Thus, \(\sqrt{2}\) is irrational.

There are many other examples of irrational numbers. For example, \(\pi\) is an irrational number, as is the number \(e\), which we will encounter when we study exponential functions. Decimals that neither repeat nor terminate, such as

\[0.1411411141114 \ldots \nonumber \]

are also irrational. Proofs of the irrationality of such numbers are beyond the scope of this course, but if you decide on a career in mathematics, you will someday look closely at these proofs. Suffice it to say, there are a lot of irrational numbers out there. Indeed, there are many more irrational numbers than there are rational numbers.

The Real Numbers

If we take all of the numbers that we have discussed in this section, the natural numbers, the whole numbers, the integers, the rational numbers, and the irrational numbers, and lump them all into one giant set of numbers, then we have what is known as the set of real numbers. We will use the letter R to denote the set of all real numbers.

Definition 15

\[\mathbb{R}=\{x : \quad\text{ x is a real number} \} \nonumber \]

This notation is read “the set of all x such that x is a real number.” The set of real numbers \(\mathbb{R}\) encompasses all of the numbers that we will encounter in this course.