Real Numbers

Number systems describe real numbers as combinations of rational and irrational numbers. They can be represented by a number line and can be 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. In this article, we are going to discuss the definition of the real number, the properties of real numbers and the examples of real numbers. Pi () is also a real number.

Real Numbers Definition

Real Numbers Definition: 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

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, g. Then the above properties may be defined by the usage of e, f, and g as proven below:

Commutative Property

If e and f are the numbers, then 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 might 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 3 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 sides will yield 36.

Identity Property

There 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 of the positive counting integers starting from 1 up to infinity, are part of the number line and are called the Natural numbers. Natural numbers also are referred to as counting numbers due to the fact they do not consist of 0 or negative numbers. They are part of real numbers together with only the positive integers, however not 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 zero. In contrast, zero 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 zero.

Integers

The set of positive and negative numbers, including zero, 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 zero. Example: 1, 2, 3 . . .

Negative Integers: They are the integers which are less than zero. Example: -1, -2, -3 . . .

Zero is referred to as neither negative nor positive integer. Zero 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 zero, are called rational numbers.

Types of rational numbers

• Integers such as -4, 0, 8, 100 etc

• Fractions like 6/9, -3/7, etc., where both the numerator and the denominator 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 the real numbers that can’t be represented in the format of a ratio. In different words, the real numbers that aren’t rational numbers are called irrational numbers. Hippasus, a Pythagorean philosopher, determined irrational numbers in the fifth century BC. All numbers that cannot be written in a p/q format are called irrational numbers.

Plotting real numbers on the number line

The number line can be drawn in 4 easy steps;

Step 1- a horizontal line is drawn with arrows on both sides, indicating that the lines are non-ending and increase up to infinity on both sides.

Step 2- a proper scale is chosen for drawing the number line. If a number like 5 or 30 needs to be plotted on a number line, then a scale of multiple of 5 can help plot the number.

Step 3- The most important step is to make sure 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.

Conclusion

In this article, we learnt the real numbers definition, the properties of real numbers and the examples of real numbers. Numbers play a very important role in our daily life to count, measure, or label objects and they can be easily represented on a horizontal line – this line is a number line. The 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 the addition of positive and negative integers on a number line. Real numbers help children as well as adults to count.