Real number

While it may seem like we only have and use one type of number in our daily lives, there are actually many different types of numbers. Take a look at some of the numbers in the image below.

 

real_number_line

 

Notice any difference between these numbers? While this group has many different types of numbers, they are all real numbers. Real numbers are any of the following:

  • Whole numbers
  • Integers
  • Rational numbers
  • Irrational numbers

 

To understand what these different types of numbers are, take a look at the table below.

 

Examples Definition
Whole 0, 5, 65, 100, 788 Positive numbers that are not fractions or decimals
Integer -5, -75, -340, -800, 55, 90 Positive and negative numbers that aren’t fractions or decimals
Rational \frac{1}{2}, 0.67 A number that is made from the division of two integers
Irrational \pi, \sqrt{2} A number that can’t be written as a fraction or decimal

 

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Number Line

In order to better understand the differences between these different types of numbers, it can be helpful to use a number line. A number line can help you identify what types of numbers you’re dealing with. Take the following set of numbers as an example, which give 5 numbers. Let’s see if we can figure out what kind of number they are.

 

Number Type
1 3.3 ?
2 5 ?
3 -6 ?
4 -5.3 ?
5 1.4142135623….. ?

 

A number line is a plot of numbers on a straight line. We can use a number line to help us decide what kind of numbers we have. Take the following as an example.

 

number_line

 

As you can see, number lines are pretty easy to understand. You can easily plot positive and negative numbers, rational and irrational numbers and more. Let’s plot the numbers given in the table above.

 

real_number_line

 

Now, it is clear what type of number each of the 5 numbers are. 

Number Type Why
1 3.3 Rational A number that is made by dividing 33 by 10
2 5 Whole A positive number that’s not a decimal
3 -6 Integer A negative number that’s not a decimal
4 -5.3 Rational A number made by dividing -53 by 10
5 1.4142135623….. Irrational A number that can’t be written as a fraction

 

Irrational Number

As stated in the first section, an irrational number is different from a rational number. While a rational number is any number that can be written as a decimal or a fraction, an irrational number can’t be expressed as a ratio of two integers. Find some examples in the table below.

 

In Decimal or Fraction Form Description
5 \div 10 \frac{5}{10} A ratio of 5 to 10
.0066 \frac{66}{10000} A ratio of 66 to 10 000
\sqrt{3} 1.7320508075688772... Cannot be expressed as a ratio

 

Generally, there are many roots that are irrational numbers. Recall that roots can be squares, cube and more. However, keep in mind that not all roots are irrational numbers. Take the following examples:

 

Result Type
\sqrt{4} 2 Rational
\sqrt[^3]{27} 3 Rational
\sqrt[^3]{81} 4.3267487109... Irrational

 

Examples

In this section, we’ll go over some examples of irrational numbers. These irrational numbers are the ones you are likely to encounter in other classes, as they are very famous.

 

Pi

Pi is the most common example of an irrational number. It is represented by the Greek letter pi:

 

pi_maths

 

Pi cannot be written as a decimal or a fraction. In fact, some people have tried to calculate pi to hundreds of decimal places without finding any sort of pattern. Take a look at pi below, which is cut off at the 15th number.

 

number_pi
Of course, it is possible to get close to pie using some fractions. Take a look at the table below, which gives some examples.

 

Fraction Decimal
22/5 4.4
22/6 3.666667
22/7 3.142857
22/8 2.75

 

Euler’s Number

Euler’s number is the second most common irrational number that you’ll encounter in math. Euler’s number is similar to pie in that it can be calculated to many decimal places but still does not show any pattern, or any sign of stopping.

 

Euler’s number is written as an e, which can be seen below.

 

euler's_number
Euler’s number is also known as the natural base. Take a look at the table below, which describes the difference between when you should use the two terms.

 

Description Example
Euler’s Using only e 5 * e, e^6
Natural Base Using it as a base for a logarithm \log _{e} x

 

When we use Euler’s number as the natural base, we can call the logarithm a natural logarithm, which is written as ln.

 

natural_base

 

As you can see, having a logarithm with the natural base is the same as having the natural logarithm of x. This means that the natural base is the inverse of the natural logarithm. Take a look at the table below, which has some of the rules for natural logarithms. 

Example
\log _{e} x = \ln x \log _{e} 5 = \ln 5
\ln _{e} e^{x} = x \ln _{e} e^{11} = 11

 

Golden Ratio

The golden ratio is represented by the Greek letter phi and can be found in many different aspects of life. It can often be found in art and architecture. To find the golden ratio, let’s start with a rectangle.

 

golden_ratio

 

As you can see, the golden ratio is present when the long part of some line, a, divided by the smaller part, b, has the same result as adding the small and long part, a + b, and dividing it by the larger part, a. 

This sounds a bit complicated, but we can try with an example.

 

golden_ratio_example

 

Result
\frac{4181}{2584} 1.6180341
\frac{4181}{2584} 1.6180341

 

As you can see, this approximates the golden ratio.

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.