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# Real Number|Definition & Meaning

## Definition

All **non-imaginary numbers** are called **real numbers**. Therefore, the set of real numbers includes all other sets except imaginary numbers, i.e., rational, irrational, integer, whole, and natural numbers are all **subsets** of real numbers.

Figure 1 – Real Numbers

Figure 1 illustrates the graphical representation of real numbers.

The union of rational and irrational numbers is called real numbers. They are represented by the letter $\textbf{R}$ and can be either **positive or negative**. This category includes all-natural integers, decimals, and fractions.

**Integers** that can be expressed as **fractions** p/q **are** **called** rational **numbers.** The fraction has a numerator of “p” and a denominator of “q,” where “q” is not equal to zero. Natural numbers, whole numbers, decimals, and integers can all be **rational numbers. **

The group of real numbers known as **irrational numbers** is a set of numbers that cannot be represented as a fraction of the form p/q. In this form, p and q must be integers, and q must be non-zero (q ≠ 0).

The **foregoing informal explanations** of real numbers are insufficient to **guarantee the accuracy** of the **theorem** proofs that include real numbers.

**Real analysis,** the study of real functions, and real-valued sequences are all built on the awareness that a better definition was required and the invention of one. This was a significant mathematical development of the **19th century.**

The unique Dedekind-complete ordered field of real numbers is now defined axiomatically. Cauchy sequences (of rational numbers), Dedekind **intersections,** and **equivalence** **classes** **of** **infinitesimal** representations are **other** **common** definitions of real numbers. **All** these definitions are **the** **same** **because** **they** all **conform** **to** **the** **axiomatic** **definition.**

**Properties of Real Number**

The **closure, associative, commutative, and distributive** properties of the set of real numbers are satisfied, much as those of the sets of natural and integer numbers.

**Closure Property**

According to the closure property, the result of **adding and multiplying** two real numbers is always a real number. The following is how R’s closure property is expressed: If a, b ∈ R, a + b ∈ R and ab ∈ R.

**Associative Property**

Even when the order of the numbers is altered, the **sum or product** of any three real numbers stays the same. Following is a description of R’s associative property: If a,b,c ∈ R, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c

**Commutative Property**

Even after switching the numbers’ order, the **total and product** of two real numbers stay the same. The following sentence expresses R’s commutative property: If a, b ∈ R, a + b = b + a and a × b = b × a.

**Distributive Property**

**Real numbers** are distributive. Hence, they are real numbers. a (b + c) = (a$\times$b) + (a$\times$c) is the distributive property of multiplication over addition, whereas a (b – c) = (a$\times$b) – (a$\times$c) is the distributive property of **multiplication over subtraction.**

**Differences Between Real Numbers and Integers**

**Rational, irrational, whole, and natural numbers** are all examples of real numbers. $\textbf{R}$ is the symbol used to represent real numbers. A **distinct actual number** is displayed at each place on the number line. Fractions and the decimal system are regarded as real numbers.

**Negative, positive, and zero numbers** are all examples of integers. Integers are represented by the letter $\textbf{Z}$. On a number line, integers are represented only by **whole numbers and negative** values. Fractions and decimals are excluded from integers.

**Real Line**

The points on an infinitely long line known as **the number line** or the **real line**, where the points corresponding to integers (…, 2, 1, 0, 1, 2,…) are evenly spaced, can be thought of as **real numbers.**

Figure 2 illustrates the real numbers on a real line from -8 to 8.

**Applications of Real Numbers**

Most **physical variables**, including position, mass, speed, and electric charge, as well as physical constants, such as the universal gravitational constant, are represented in the physical sciences by **real numbers**.

**Real** measurements of physical quantities are of finite **precision**, **but** **fundamental** physical **theories** such as classical mechanics, electromagnetism, quantum mechanics, general relativity, and the standard **model** are **actually** mathematical structures, **usually** smooth **is** **described** **in** **a** **simple** **manifold** or Hilbert **space b****ased** on real numbers.

Since **real numbers** have several **topological qualities** that are a technical annoyance, the Baire space is employed as a substitute for them in **set theory**, more especially descriptive set theory. “Reals” are the components of Baire space.

The Zermelo-Fraenkel axiomatization of **set theory** is most frequently used to formalize real numbers. However, other mathematicians also **investigate real numbers** using different logical mathematical foundations. In particular, constructive mathematics and reverse mathematics both study **real numbers.**

The **hyper-real numbers**, created by Edwin Hewitt, Abraham Robinson, and others, add infinitesimal and infinite numbers to the set of **real numbers**, allowing for the construction of infinitesimal calculus in a manner that is more in line with the original intuitions of Leibniz, Euler, Cauchy, and others.

Since **finite computers** are unable to directly store infinite amounts of digits or other infinite representations, electronic calculators and computers cannot work on arbitrary real numbers. Furthermore, they typically don’t even work with random, **difficult-to-manipulate actual numbers.**

Instead, computers commonly use **floating-point numbers**, which approximate objects with finite accuracy and resemble **scientific notation**. The amount of data storage space allotted for each number, whether it be represented as **a fixed-point**, **floating-point, arbitrary-precision number**, or in another way, determines the possible accuracy.

Most scientific calculations employ binary floating-point arithmetic, which frequently has a 64-bit representation and a **precision** of about 16 **decimal digits.**

**Some Examples of Real Numbers**

**Example 1 **

Illustrate the real numbers from one to seven on real line.

**Solution**

Draw a line from one to seven with each digit to be equally spaced as illustrated in figure 3. The line is known as real line as real numbers are illustrated on the line.

**Example 2**

What should be multiplied by 3.25 so this term will be equal to one.

**Solution**

We can write 3.25 as:

\[ \dfrac{325}{100} \]

If we multiply it with the reciprocal, then the answer will be equal to one.

\[ \frac{325}{100} \times \frac{100}{325} = 1 \]

All the terms will cancel out the answer will be one.

*All images were created using GeoGebra.*