1.01 The real number system
Our real numbers system hasn't been around in its current state forever. It was developed slowly over time. The real number system includes rationals, irrationals, integers, whole numbers, and natural numbers.
Classification of numbers is about identifying which set, or sets, a number might belong to. It might be helpful to remember the different types of numbers as a story about filling in the numbers on a number line.
The first numbers we put on the line are the natural numbers.
The set of natural numbers are the counting numbers, starting from $1$1:
$1,2,3,4,5,6,7,\ldots$1,2,3,4,5,6,7,…
Next, we will add $0$0 to our line to show the whole numbers.
The set of whole numbers are the counting numbers, starting from $0$0:
$0,1,2,3,4,5,6,7,\ldots$0,1,2,3,4,5,6,7,…
The left side of this line looks pretty empty. If we add all the negatives we now have a set of numbers called the integers.
Whole numbers together with negative numbers make up the set of integers:
$\ldots,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,\ldots$…,−7,−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6,7,…
But are there numbers between the ones we already have marked? The answer is yes - an infinite amount of numbers between every little mark!
What sort of numbers are these? Well, rational numbers are all numbers that indicate whole numbers as well as parts of whole numbers. So fractions, decimals, and percentages are added to our number line to create the set of rational numbers.
Integers together with all fractions (including repeating or repeating decimals) make up the set of Rational Numbers.
They cannot be listed, but here are some examples:
$\ldots,-8,-7.4,-7,-6,-5.33387,-4,-2,0,\frac{1}{2},75%,1,2,3,3.5656,\ldots$…,−8,−7.4,−7,−6,−5.33387,−4,−2,0,12,75%,1,2,3,3.5656,…
But wait! Our number line is still not quite full. There are still gaps in a few places. These gaps are filled with numbers we call irrational numbers. These are numbers like $\sqrt{21}$√21 and $\pi$π:
Now we can revisit our picture of the different sets of numbers in the real number system. Looking at the image below, we can see some examples of numbers that below to each set:
Notice that some number sets are entirely contained within larger number sets. For example, all of the whole numbers like $1,2,3,17,28736,\ldots$1,2,3,17,28736,… etc. are also integers. But there are some integers, like $-1,-2,-56,-98324$−1,−2,−56,−98324, that are not whole numbers.
Similarly, rational numbers are also real numbers, but the set of real numbers includes all the rational numbers and all the irrational numbers.
Practice questions
QUestion 1
Height above sea level is expressed as a positive quantity. Which set of numbers is the most appropriate for describing the position of a submarine relative to sea level?
Integers
AWhole numbers
B
QUESTION 2
Using the diagram, complete the following statement.
A real number is either:
At the top, a large rectangle is labeled "Real numbers" encompassing the entire set. Within this rectangle, there are two large subsets: the upper subset is shaded dark green and labeled "Rational numbers," and the lower subset is shaded blue and labeled "Irrational numbers." Within the upper subset is a blue-green rectangle labeled "Integers," and within it is a light-green rectangle labeled "Whole numbers."
The diagram shows the relationship between the sets of numbers. The set of real numbers includes the sets of rational and irrational numbers. The set of integers is a subset of the set of rational numbers. The set of whole numbers is a subset of the set integers, which in turn is also a subset of the set of rational numbers.
Examples of rational numbers, which include $-\frac{112}{17}$−11217, $-3.12$−3.12, $-\frac{3}{2}$−32, $\frac{1}{2}$12, $1.3$1.3, $2.\overline{6}$2.6,$\frac{13}{3}$133, are listed within the dark-green rectangle. Examples of irrational numbers, which include $\frac{-\sqrt{102}}{5}$−√1025 , $-\sqrt[3]{2}$−3√2, $\frac{1+\sqrt{5}}{2}$1+√52, $\pi$π and $\sqrt{21}$√21, are listed within in the blue rectangle. Examples of integers, which include the whole numbers, -1, -2, -3, and -4, are listed within the blue-green rectangle. Examples of whole numbers, which include 0, 1, 2, 3, 4, are listed within the light-green rectangle.
a whole number or an irrational number.
Aan integer or an irrational number.
Ba rational number or an irrational number.
Can integer or a rational number.
D
QUESTIOn 3
Consider that we can express $\sqrt{49}$√49 as $\sqrt{7^2}$√72.
Using the diagram, classify the number $\sqrt{49}$√49.
At the top, a large rectangle is labeled "Real numbers" encompassing the entire set. Within this rectangle, there are two large subsets: the upper subset is shaded dark green and labeled "Rational numbers," and the lower subset is shaded blue and labeled "Irrational numbers." Within the upper subset is a blue-green rectangle labeled "Integers," and within it is a light-green rectangle labeled "Whole numbers."
The diagram shows the relationship between the sets of numbers. The set of real numbers includes the sets of rational and irrational numbers. The set of integers is a subset of the set of rational numbers. The set of whole numbers is a subset of the set integers, which in turn is also a subset of the set of rational numbers.
Examples of rational numbers, which include $-\frac{112}{17}$−11217, $-3.12$−3.12, $-\frac{3}{2}$−32, $\frac{1}{2}$12, $1.3$1.3, $2.\overline{6}$2.6, $\frac{13}{3}$133, are listed within the dark-green rectangle. Examples of irrational numbers, which include $\frac{-\sqrt{102}}{5}$−√1025, $-\sqrt[3]{2}$−3√2, $\frac{1+\sqrt{5}}{2}$1+√52, $\pi$π and $\sqrt{21}$√21, are listed within in the blue rectangle. Examples of integers, which include the whole numbers, -1, -2, -3, and -4, are listed within the blue-green rectangle. Examples of whole numbers, which include 0, 1, 2, 3, 4, are listed within the light-green rectangle.
Select the three options that apply.
$\sqrt{49}$√49 is an irrational number.
A$\sqrt{49}$√49 is an integer.
B$\sqrt{49}$√49 is a rational number.
C$\sqrt{49}$√49 is a whole number.
D