Real Numbers

Introduction

Have you ever tried to measure, for example, the perimeter of a room with a ruler? If so, how accurate were your measurements? Suppose you get 20 meters. But maybe it is not exactly 20 m? May it be 20.1 meters? Or even this number is not quite accurate, but should be 20.08 meters or 20.0846 meters? Or maybe you came across such a record 20±0.15 m? But what will be the numbers between 20-0.15 m and 20+0.15 m? How many numbers are there, what are these numbers? All these questions should be answered in this article.

Definition of whole numbers

By this time, you are familiar with natural, whole numbers, integers, rational, and irrational numbers. Recall each of these numbers as meant. 

Set of numbers	Definition
Natural numbers	Numbers that are used for counting: Examples: 1, 2, 3, 4, …
Whole numbers	Natural numbers together with zero: Examples: 0, 1, 2, 3, 4, …
Integers	Whole numbers and their opposite (additive inverse) numbers: Examples: …, -3, -2, -1, 0, 1, 2, 3, …
Rational numbers	All numbers that can be represented as a fraction $\frac{m}{n}$, where n is a natural number, m is an integer: Examples: 1, -$\frac{3}{4},\frac{11}{9}$, -18, …
Irrational numbers	Numbers of the form π, e, √2,√3, …

All these numbers together form a set of real numbers. The set of real numbers is denoted as R.

Any number we can think of, except imaginary complex numbers, is a real number.

Real numbers can be positive, negative, or zero. They are called real numbers because they are not imaginary.

EXAMPLE: Identify real numbers from the list of numbers

-2.3, √5,  7i,  $\frac{2}{5}$,  0,  3 -6i,  π,  $\sqrt{-1}$,  4

SOLUTION: All imaginary numbers must be excluded from these numbers. Among them, complex numbers are numbers 7i,  3 -6i and $\sqrt{-1}$=i.  

Subsets of real numbers

Here are two illustrative ways 

to represent the set of real numbers 

and its subsets.

Circle diagram: 

Real numbers chart: 

Most often, the set of real numbers is represented as a union of two disjoint sets: the set of rational numbers and the set of irrational numbers.

Number line

In the number line, 0 is located in the middle of the line. To the right are all positive numbers, and to the left are all negative points. Any point on the number line would be considered a real number. On the number line, we can find 

• natural numbers (for example, number 6);
• whole numbers (for example, numbers 0 and 6);
• integers (for example, number -5); 
• rational number (for example, numbers $\frac{3}{2}$ and -2.4);
• irrational numbers (for example, numbers -√2 and π).

All real numbers lie on the number line, so, they can be compared and ordered. If we can order real numbers, then we can add, subtract, multiply and divide real numbers.

There are an infinite number of numbers in all of the given subsets. Natural and whole numbers can be enumerated using a certain algorithm, so the sets of these numbers are the so-called countable sets. You can also come up with an algorithm that enumerates integers and rational numbers (look for this information in the articles on integers and rational numbers) and these sets are countable too. However, irrational numbers cannot be enumerated and the set of irrational numbers is uncountable. 

The number of irrational numbers is larger than the number of rational numbers.

We can prove this by contradiction: assume irrationals can be enumerated and then find an irrational that is not in the enumeration. Remind that an irrational number’s decimal expansion has a non-repeating infinite number of digits after the decimal point. Assume that we have enumerated all irrational numbers, then they can be written in order of enumeration:

1^st number: 5.1234…

2^nd number: 4.6782…

3^rd number: 0.0301…

4^th number: 9.9999…

Now construct another irrational number. Let this number have 0 before the decimal point. Add 1 to each digit on the ith position of the i^th number and write the sum on the i^th position of this new number (if the sum turns out to be 10, we will write 0). In this way, we get the number

0.2810…

This constructed irrational number differs from the 1^st enumerated irrational number because its first decimal digit is another, it differs from the 2^nd irrational number, because its second decimal digit is another, it differs from the 3^rd irrational number, because its third decimal digit is another and so on. Therefore, we constructed one more irrational number that was not earlier enumerated, it differs from all enumerated irrational numbers. 

So, the set of irrational numbers is uncountable as well as the set of real numbers. Moreover, between arbitrary two rational numbers, there is always an irrational number and between arbitrary irrational numbers, there is always a rational number. This property of real numbers is called “density of the rational numbers in the real numbers”.

Another “strange fact”: if you take an arbitrary interval, for example (0,1), it contains as many numbers as the set of real numbers. This is quite difficult to imagine, because it seems that this interval is limited and should contain fewer numbers, but in reality, this is not the case. If we imagine the interval as an infinite elastic band that will not break no matter how long we stretch it, then at an infinite moment this elastic band will stretch in a whole real number line.

Arithmetic operations with real numbers

ADDITION: The addition combines two quantities into a single quantity called a sum. When we add 

• two rational numbers, the sum is a rational number;
• two irrational numbers, the sum is an irrational number;
• an irrational number and a rational number, the sum is an irrational number. 

For example,

4+(-7.8) = -3.8

-2√7 + √7=-√7

(√3 – 0.5)+$\frac{1}{2}$ = √3

SUBTRACTION: Subtraction is the opposite operation of addition. The result of subtraction is called a difference. When we subtract 

• two rational numbers, the difference is a rational number;
• two irrational numbers, the difference can be either rational or irrational number;
• an irrational number and a rational number, the difference can be either rational or irrational number. 

For example,

4-(-7.8)=11.8

-2√7 – √7= -3√7   but  (5+√3) – (2+√3) =3

(√3 – 0.5) – $\frac{1}{2}$ = √3 – 1 but (√3 – 0.5) – √3= -0.5

MULTIPLICATION: Multiplication combines multiple quantities into a single quantity, called the product. When we multiply 

• two rational numbers, the product is a rational number;
• two irrational numbers, the product can be either rational or irrational number;
• an irrational number and a rational number, the product is an irrational number. 

For example,

4 × (-7.8) = -31.2

-2√7 x √7 = -14 but (5+√3) x (2+√3) = 13 + 7√3

√3×5 = 5√3

DIVISION: Division is the inverse of multiplication. The result of division is called a quotient. When we divide

• two rational numbers, the quotient is a rational number;
• two irrational numbers, the quotient can be either rational or irrational number;
• an irrational number and a rational number, the quotient is an irrational number. 

For example,

4 ÷ (-7.8) = -$\frac{20}{39}$

-2√7 ÷ √7 = -2 but (5+√3) ÷ √3 = $\frac{5√3}{3}$ + 1

$\frac{5}{√5}$ = √5

EXAMPLE: Calculate $\frac{(2-√3)^2}{14-8√3}$.

SOLUTION: Start with the numerator:

(2-√3)² = 4 – 4√3 + 3 = 7 – 4√3

Apply the distributive property to the denominator:

14 – 8√3 = 2(7-4√3)

Therefore,

$\frac{(2-√3)^2}{14-8√3} = \frac{7-4√3}{2(7-4√3)} = \frac{1}{2}$ = 0.5

Arithmetic operations help us imagine where a real number is on a line. 

EXAMPLE: Where is the number √3 located on the number line? 

SOLUTION: First, we estimate between which two integers we have the number √3. 

Since 1<3<4, then 1<√3<2. So, the number √3 is located between 1 and 2.

If we want to estimate more precisely where the number √3 is located, then we note that 2.89<3<3.24 and therefore 1.7<√3<1.8.

For more accurate estimation, we can continue this process as long as we want. Moving to the right of the decimal point, we will get each time the next digit in the decimal representation of number √3 and the exact location of the number √3 on the number line.

Properties of real numbers

Let *,  ∘ be arithmetic operations with real numbers. 

CLOSURE PROPERTY: If a and b are two real numbers, then a*b is a real number too.

• addition, subtraction, multiplication are always close operations;
• division is a close operation when we divide by non-zero real number. In general, a division is not a close operation.

COMMUTATIVE PROPERTY: If a and b are two real numbers, then a*b=b*a 

• a+b=b+a for addition; 
• a×b=b×a for multiplication.

Subtraction and division are not commutative operations. It is enough to illustrate this with concrete examples:

5-6=-1  whereas  6-5=1

12÷6=2  whereas  6÷12=0.5

ASSOCIATIVE PROPERTY: If a, b and c are three real numbers, then 

(a*b)*c=a*(b*c)

• (a+b)+c = a+(b+c) for addition; 
• (a×b)×c=a×(b×c) for multiplication.

Subtraction and division are not associative operations. Let us demonstrate this with some examples:

(6-5)-4 = 1-4 = -3  whereas  6-(5-4) = 6-1 = 5

(12÷6)÷2 = 2÷2=1  whereas 12÷(6÷2) = 12÷3 = 4

DISTRIBUTIVE PROPERTY: If a, b and c are three real numbers, then 

(a*b)∘c=(a*c)∘(b*c)

• (a+b)×c = a×c+b×c – distributive property of addition over multiplication; 
• (a-b)×c = a×c-b×c – distributive property of subtraction over multiplication;
• (a+b)÷c = a÷c+b÷c – distributive property of addition over division. This property is true when the number c is a real non-zero number;
• (a-b)÷c = a÷c-b÷c – distributive property of subtraction over division. This property is true when the number c is a real non-zero number.

In other cases, the distributive property does not hold, for example,

(12-6)+4 = 6+4=10 whereas (12+4) – (6+4) = 16-10 = 6

(12×6)÷4 = 72÷4=18  whereas (12÷4) x (6÷4) = 3×1.5 = 4.5

IDENTITY PROPERTY: If a is a real number, then there exists such real number e that

a*e = e*a = a

• e=0 for addition, a+0=0+a=a;
• e=1 for multiplication, a×1=1×a=a.

There are no identities for subtraction and multiplication.

INVERSE PROPERTY: If a is a real number, then there exists such real number a^-1 that

a*a^-1 = a^-1*a = e

• a^-1=-a for addition, a+(-a)=(-a)+a=0;
• a^-1=$\frac{1}{a}$ for multiplication, a×$\frac{1}{a}$ = $\frac{1}{a}$×a=1,where a is a non-zero real number.

There are no inverse elements for subtraction and multiplication.

ZERO PRODUCT PROPERTY: If a×b=0, then a=0 or b=0 or both.

This property is very useful when solving equations. 

EXAMPLE: Solve the equation (x²-4)(x+1)=0.

SOLUTION: If the product of two real numbers is zero, then by zero product property, either 

x²-4=0 or x+1=0.

Solve each equation separately.

If x²-4=0, then x=2 or x=-2.

If x+1=0, then x=-1. 

Therefore, the equation has three real solutions, -2, -1 and 1.

NEGOTIATION PROPERTY: If a and b are two real numbers, then

• -(-a)=a;
• (-a) x (-b) = a×b;
• (-a)×b = a×(-b) = -(a×b).

 For example,

1. -(-8) = 8;
2. (-10) x (-0.1) = 10×0.1 = 1;
3. (-7) x $\frac{5}{7}$ = -(7×$\frac{5}{7}$) = -5.

Application of real numbers

It can be paraphrased that the “application of real numbers in real life” is the “usage of mathematics in real life”. Let us demonstrate real-life math problems where we see real numbers and operations with them.

1. Bills in stores. Each of us received a check at the store, which at least reflected the addition of real numbers. If you bought more than one unit of goods, then the value of the goods was multiplied by the number of goods. If you bought goods with a discount, you can also see in the check the amount of the discount and subtraction of this amount of the discount from the value of the goods.
2. Cooking. A good cook or confectioner is a great “calculator” of real numbers. Any recipe contains quantities that are given in real numbers. If you cook several portions or part of a portion, the cook is faced with the multiplication and division of real numbers. Conversion between different units is also often used, such as the conversion from the temperature in degrees Celsius to temperature in degrees Fahrenheit or between metric and imperative units of weight and volume.
3. Trips. Have you ever wondered how often you use math, and in particular real numbers when travelling? First: the clock, the conversion from seconds to hours from hours to minutes – it’s all operations with real numbers. Second: scale. Determining the distance on the map, converting measurements, moving from one measurement system to another. Third: physical quantities: the average speed of movement, the average duration, time and finally the graph of the movement, which is usually drawn using continuous number lines on which real numbers are plotted.
4. Architecture. This position is much more serious than previous ones. Because if the chef makes a mistake, the failed cake can simply be thrown in the trash, but if the architect makes a mistake, it can have very sad consequences. Therefore, all technical specialities, whether it is architecture, mechanics or rocketry, must have a well-developed mathematical apparatus and very accurate mathematical calculations performed with real numbers.
5. Physics. All the most important constants in physics are measured in real numbers with very high accuracy: the gravitational constant, the Planck constant, the densities of bodies, electric charges, the speed of light and sound – all these constants are real numbers, without which we cannot understand any physical process.
6. Optics. This important part of physics can be surely singled out as a separate item. Whether the focal length is determined correctly affects whether a person will see with made glasses or whether the photo will be of good quality. The basic laws of optics require the values of trigonometric ratios such as sines and cosines, and these ratios are very often irrational numbers, which, as you already know, are real numbers.
7. It is probably easier to name an industry where real numbers are not used. Although no, try to do it. Language? Not, because modern linguistics is computer programming, and computer programming is based on pure mathematics. Psychology? One hundred percent no, because psychology very often works with research on a large number of objects, and dealing with collected data is a statistic that also works with real numbers. History? This is a chronology in numbers. Think about whether you will be able to find a part of life around you in which there are no numbers, real numbers.

FAQs

1. What is a real number?

Any number (natural, integer, rational or irrational), except complex, is a real number. 

2. What is the non-real number? 

Non-real numbers are imaginary and complex numbers that are, numbers that contain an imaginary unit.

3. How do you identify real numbers?

This number does not contain an imaginary unit. 

4. Why there are infinitely many real numbers?

The set of real numbers consists of two disjoint subsets: the set of rational numbers and the set of irrational numbers. Each of these subsets contains infinitely many numbers, so the union of two subsets contains infinitely many numbers too.

5. Which numbers are more: rational, irrational or real?

All these sets contain infinitely many numbers, but the set of rational numbers is countable whereas the sets of irrational and real numbers are uncountable. So, the set of rational numbers is the smallest one and the sets of irrational and real numbers have the same infinite number of numbers. 

6. Where can I meet real numbers?

Throughout. It is difficult, almost impossible, to find where real numbers are not used. This article covers this issue quite widely.

7. What can I do with real numbers?

Add, subtract, multiply and divide. We can also do all the operations that come from these four, such as exponentiation, root extraction, legitimization, and exponentiation. 

8. How can a real number be represented on the number line? 

With a predetermined accuracy, we can estimate between which two rational numbers is a real number and place it on the number line.

Quiz

1. Identify real numbers from the list of numbers

-18,  √11, $\frac{i}{5}$, $\frac{13}{7}$, -1,  i, √π ,  5√-1,  3.7

SOLUTION: All imaginary numbers must be excluded from these numbers. Among them, complex numbers are numbers $\frac{i}{5}$,  i and 5√-1=5i. Therefore, real numbers are
-18, √11, $\frac{13}{7}$, -1, √π and 3.7.

ANSWER:  -18, √11, $\frac{13}{7}$, -1, √π , 3.7

2. Which of the following real numbers is represented on the number line below?

a) -√3

b) √3

c) -2√2

d) -$\frac{√2}{2}$

SOLUTION: The number line shows a real number x that lies between -3 and -2. Consider each of the given numbers and estimate its value. 

Since 1<3<4, we have that 1<√3<2. Therefore, -2<-√3<-1. So, both options a) and b) are false.

Since 1<2<4, we have that 1<√2<2. Therefore, -4<-2√2<-2 and -1<-$\frac{√2}{2}$<-$\frac{1}{2}$. So, option c) is true and option d) is false. 

ANSWER: c)

3. Calculate $\frac{16-20√2}{4}$ x √2. Explain the reasons of your steps.

SOLUTION: First, use the distributive property of subtraction over division:

$\frac{16-20√2}{4} = \frac{16}{4} – \frac{20√2}{4}$ = 4-5√2

Now, using the distributive property of subtraction over multiplication, multiply the obtained result by √2:

(4-5√2) x √2 = 4×√2 – 5√2 x √2 = 4√2-10

ANSWER: 4√2-10 

4. In each case, what is the value of x? Explain the property you used. 

a) 2,5+x=3.8+2.5;

b) (3.4×$\frac{1}{2}$) x √5 = 3.4×(x×√5);

c) a×x=1, where a is a non-zero real number.

SOLUTION: a) By the commutative property of addition, 

2.5+3.8 = 3.8+2.5

Therefore, x=3.8.

b) By the associative property of multiplication,

(3.4×$\frac{1}{2}$) x √5 = 3.4×($\frac{1}{2}$ x √5)

Therefore, x=$\frac{1}{2}$.

c) By multiplicative inverse property, for a non-zero real number a, 

a×$\frac{1}{a}$ = 1

Therefore, x=$\frac{1}{a}$.

ANSWER: a) 3.8; b) $\frac{1}{2}$; c) $\frac{1}{a}$.

5. Which of the following statements are true? 

a) Each natural number is a real number.

b) Some irrational numbers are not real.

c) Some real numbers are not rational.

d) Between every two irrational numbers there is always a rational number. 

SOLUTION: a) This statement is true because the set of natural numbers is the subset of the set of real numbers. 

b) This statement is false. Each irrational number is a real number because the set of irrational numbers is the subset of the set of real numbers.  

c) This statement is true. Rational and irrational numbers together form the set of real numbers. Therefore, there are some numbers, irrational numbers, that are not rational real numbers.

d) This statement is true. We can always find a rational number between every two irrational numbers.  

ANSWER: a), c) and d)

Conclusions

1. All known numbers, except imaginary complex numbers, are real numbers.
2. There are infinitely many real numbers that cannot be enumerated.
3. Between arbitrary two rational numbers, there is always an irrational number and between arbitrary irrational numbers, there is always a rational number. This property is called “density of the rational numbers in the real numbers”.
4. There are four basic arithmetic operations introduced for real numbers.
5. There is a range of properties that can be applied for arithmetic operations with real numbers. These properties simplify the calculations with real numbers.
6. We can see real numbers everywhere: from everyday life to the sciences.

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