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In mathematics, real numbers are described as combinations of rational and irrational numbers. They can be represented by a number line and used for all arithmetic operations. A complex number can, however, also be expressed as imaginary numbers because they cannot be expressed numerically. 57, -92, 512.45, 5/9 and so on are some examples of real numbers. Pi () is also a real number. In this article, we are going to discuss the definition of real numbers, the properties of real numbers and why are real numbers important. Moreover, we will examine the real-life applications of real numbers.
Definition of Real Numbers
Rational and irrational numbers can be defined together as real numbers. They can be both positive and negative. This category includes natural numbers, decimals and fractions.
Properties of Real Numbers
The four crucial properties of real numbers are as follows:
Commutative Property
Associative Property
Distributive Property
Identity Property
Let us assume three distinct real numbers – e, f and g. Then, the above properties may be defined by the usage of e, f and g as shown below.
Commutative Property
If e and f are the numbers, the overall form might be e + f = f + e for addition and e.f = f.e for multiplication.
Addition: e + f = f + e. For instance, 7 + 2 = 2 + 7, 8 + 2 = 2 + 8.
Multiplication: e × f = f × e. For instance, 7 × 4 = 4 × 7, 5 × 9 = 9 × 5.
Associative Property
If e, f and g are the numbers, the general form will be e + (f + g) = (e + f) + g for addition and (e f) g = e (f g) for multiplication.
Addition: The general form might be e + (f + g) = (e + f) + g. An instance of additive associative property is 23 + (4 + 3) = (23 + 4) + 3.
Multiplication: (e f) g = e (f g). An instance of a multiplicative associative property is (6 × 7) 3 = 6 (7 × 3).
Distributive Property
For three numbers (e, f, and g) that are real in nature, the distributive property is represented as
e (f + g) = e f + e g and (e + f) g = e g + f g.
Example of distributive property is: 6 (4 + 2) = 6 × 4 + 6 × 2. Here, each side will yield 36.
Identity Property
These are additive and multiplicative identities.
For addition: e + 0 = e. (0 is the additive identity)
For multiplication: e × 1 = 1 × e = e (1 is the multiplicative identity).
Natural Numbers
All positive counting integers starting from 1 up to infinity are part of the number line and are called natural numbers. Natural numbers are also referred to as counting numbers, as they do not consist of 0 or negative numbers. They are part of real numbers with only the positive integers; however, they are not a part of 0, fractions, decimals and negative numbers.
Whole Numbers
In general, natural numbers are a set of counting numbers starting from 1, and they are also known as whole numbers when they are combined with 0. In contrast, 0 is an undefined identity, representing a null result or no result at all. Simply put, whole numbers are numbers that do not include fractions, decimals or even negative numbers. A whole number consists of positive integers plus zero. You can also define whole numbers as non-negative integers. Whole numbers are distinguished from natural numbers by the presence of 0.
Integers
The set of positive and negative numbers, including 0, contains integers with no decimal or fractional parts. Examples of integers are: -7, 0, 9, 52, 86, 89 and 4,573. A set of integers, which is denoted as Z, includes
Positive Integers: They are the integers which are greater than 0. Example: 1, 2, 3 . . .
Negative Integers: They are the integers which are less than 0. Example: -1, -2, -3 . . .
Zero is referred to as neither a negative nor positive integer. 0 is a whole number.
Rational Numbers
All numbers that can be written in a p/q format, where the denominator q is not equal to 0, are called rational numbers.
Types of rational numbers
Integers such as -4, 0, 8, 100, etc
Fractions such as 6/9, -3/7, etc., whose numerators and denominators are integers
Terminating decimals such as 0.97, 0.3416, 0.4737, etc
Non-terminating decimals such as 0.7777…, 0.121212…, etc., with some repeating patterns after the decimal point. They are mostly called non-terminating repeating decimals
Irrational Numbers
Irrational numbers are real numbers that cannot be represented in a ratio format. In other words, real numbers that are not rational numbers are called irrational numbers. Hippasus, a Pythagorean philosopher, determined irrational numbers in the 5th century BC. All numbers that cannot be written in a p/q format are called irrational numbers.
Plotting real numbers on a number line
The number line can be drawn in 4 easy steps as described below.
Step 1- A horizontal line is drawn with arrows on both sides, indicating that the lines are non-ending and increase to infinity on both sides.
Step 2- A proper scale is chosen for drawing the number line. If a number like 50 or 20 needs to be plotted on a number line, then a multiple scale of 5 can help plot the number.
Step 3 – The most important step is to ensure that the points are marked at equal intervals.
Step 4 – The last step is to locate the desired number on the number line and then highlight or circle it for reference.
What are the real life applications of real numbers?
For measuring airspeed, rainfall, wind speed and distance
In insurance policies
In Medical instruments and for checking heartbeat rate
To check fuel amount, car driving instrument and rpm
In ticket number and at a train driver’s desk
For measuring speed, velocity, and acceleration
In graphs indicating stock market prices
Conclusion
In this article, we understood the definition of real numbers, properties of real numbers and examples of real numbers. Numbers play a very vital role in our daily life for counting, measuring and labelling objects, and they can be easily represented on a horizontal line; ‘-’ this line is a number line. A natural number, whole number, integer, rational number and irrational number are all parts of real numbers. The number line consists of both positive and negative integers. There is a different process for adding positive and negative integers on a number line. Real numbers help children as well as adults to count.
