Types of Numbers

Types of Numbers exercise

• Explain the different types of numbers.

  Hints

  Obviously, every male Beagle is a Beagle. But on the other hand, all Beagles are not male Beagles.

  Every Beagle is a dog. But, there also exist dogs like German Shepherds that aren't Beagles.

  Solution

  You can see the system of different types of numbers in the picture on the right.

  The numbers $6$, $21$, $11$ belong to a set called natural numbers. These numbers, along with $0$, give us the set of whole numbers. Negative numbers, $-1$, $-6$, $-23$ etc., together with numbers like $0$, $73$ and $75$ are called integers.

  There are still other numbers like $\pm\frac14$ or $\pm0.375$, $\pm0.\overline{3}$, and can be rewritten in fraction form. These numbers, along with the integers combine to form the rational numbers.

  Decimal numbers like $\pi$ and $e$, which are neither terminating nor repeating and cannot be written as fractions are called irrational numbers. These numbers, together with the rational numbers, make up all the real numbers.

• Place each number in the smallest possible set.

  Hints

  The natural numbers are the numbers we use to count things.

  The union of all the numbers is the set of real numbers.

  Note that natural numbers, whole numbers, integers, and rational numbers are NOT irrational numbers.

  An irrational number is a decimal number that does not terminate or repeat and cannot be written as a fraction.

  Solution

  $6$ and $50$ are natural numbers. We use natural numbers to count things.

  Whole numbers are natural numbers but also include $0$.

  The set of negative numbers added to the set of whole numbers gives us the set of numbers called integers. For example, $-3$ and $-14$ are integers.

  There are still other numbers like $\pm\frac35$ or $\pm0.5$, $\pm0.\overline{8}$ can all be rewritten in fraction form. These numbers along with the set of integers combine to form the rational numbers.

  Decimal numbers like $\pi$, $e$ and $\sqrt3$, which do not terminate or repeat and cannot be written as fractions are called irrational numbers. These numbers, together with the rational numbers, make up the set of real numbers.

• Identify which numbers don't belong to the given number type.

  Hints

  We use natural numbers to count and order things.

  Integers are both natural numbers, zero, and negative whole numbers.

  For example, $\frac13=0.\overline3$ is a rational number. It can be written as a fraction as well as a repeating decimal.

  Solution

  The easiest way to remember the different number types is to remember examples of numbers belonging to each type.

  With the exception of the irrational numbers, all the other number types are organized like Russian nesting dolls. For example:

  • $314$ is a real number
  • $314$ is a rational number
  • $314$ is an integer
  • $314$ is a whole number
  • $314$ is a natural number

  Each number is classified by the smallest possible number type. So we assign $314$ to the natural numbers.

  The note on the right side shows the correct solution. Here you find the mistakes already corrected:

  • $-3$ is an integer but not a natural number
  • $0$ is a whole number
  • $3.14$ is a terminating decimal number and also a rational number, but it is not an integer
  • $3.1\overline4$ is a repeating decimal number, so it's a rational number, and it is not an irrational number.

• Assign each number to its correct number type.

  Hints

  $\sqrt 2$ is an irrational number. Multiples of irrational numbers are also irrational, like $3\sqrt2$.

  Natural numbers are used to count and order things.

  Some examples are:

  • $1$, $2$, $3$, ...
  • $1\text{st}$, $2\text{nd}$, $3\text{rd}$, ...

  The whole numbers are the natural numbers plus the number zero.

  Integers are positive and negative whole numbers.

  Some examples are:

  • $... , -3, -2, -1, 0, 1, 2, 3, ...$

  Solution

  We can write every number type as a set:

  • natural numbers: $\{1, 2, 3, ...\}$,
  • whole numbers: $\{0, 1, 2, 3, ...\}$,
  • integers: $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$.

  We can see that all natural numbers are whole numbers, and all whole numbers are integers.

  The representation of the rational numbers as a set isn't as easy as the number types shown above. So we'll define those numbers as follows:

  • terminating or repeating decimal numbers such as $0.75$ and $0.\overline2$
  • fractions such as $\pm\frac{435}{23}$

  Irrational numbers are all numbers which cannot be written as a fraction. So these numbers are decimal numbers which neither end nor repeat, like $\sqrt3$ or $\pi$.

• Analyze the statements about the different number types.

  Hints

  Natural numbers are positive. So, if $a$ is a natural number, then $a>0$.

  The number $0$, together with the natural numbers form the set of whole numbers.

  $\frac13=0.\overline3$ is an example of a repeating decimal, and $7.5$ is an example of a rational number.

  Solution

  We start with the natural numbers such as $11$, $21$. $-6$ is not a natural number.

  The natural numbers, together with $0$, form the whole numbers. $-1$ is not a whole number.

  Integers are numbers like $-3$ and $-99$. $-\frac23$ is not an integer.

  The rational numbers are fractions such as $\frac14$ or decimals that terminate, like $8.1$, or repeat. $-2\sqrt 5$ is not a rational number.

  Irrational numbers are not repeating decimals nor can they be written as fractions like $\pi$. $-\frac23$ is not an irrational number.

• Decide to which number type the number belongs.

  Hints

  The real numbers include both rational and irrational numbers.

  A number is either rational or irrational.

  Every natural number is a whole number, but not vice versa.

  Every whole number is a integer, but not vice versa.

  Every integer is a rational number, but not vice versa.

  Solution

  • Every natural number is also a whole number.
  • Every whole number is also an integer.
  • Every integer is also a rational number.
  • Each number which is not rational is irrational.
  • The rational and the irrational numbers together form the real numbers.

  Let's have a look at some examples:

  1. $5.4$ is a rational number because it has a terminating decimal, and since all rational numbers are also real numbers, $5.4$ is also a real number. But $5.4$ is neither an integer nor a natural number.
  2. $\sqrt 5$ is an irrational number, and therefore not a rational number. However, $\sqrt 5$ is a real number.
  3. $\frac78$ is a fraction, and therefore it's a rational number.
  4. $-42$ is an integer, rational number, and real number. But $-42$ is not a whole number nor a natural number.