Any number that can be found in the real world is a real number. We find numbers everywhere around us. Natural numbers are used for counting objects, rational numbers are used for representing fractions, irrational numbers are used for calculating the square root of a number, integers for measuring temperature, and so on.
These different types of numbers make a collection of real numbers. In this lesson, we will learn all about real numbers and their important properties.
Any number that we can think of, except complex numbers, is a real number. The set of real numbers, which is denoted by R, is the union of the set of rational numbers (Q) and the set of irrational numbers (Q). So, we can write the set of real numbers as, R = Q ∪ Q. This indicates that real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
For example, 3, 0, 1.5, 3/2, √5, and so on are real numbers.
We know that real numbers include rational numbers and irrational numbers. Thus, there does not exist any real number that is neither rational nor irrational. It simply means that if we pick up any number from R, it is either rational or irrational.
Any number which can be defined in the form of a fraction p/q is called a rational number. The numerator in the fraction is represented as ‘p’ and the denominator as ‘q’, where ‘q’ is not equal to zero. A rational number can be a natural number, a whole number, a decimal, or an integer. For example, 1/2, -2/3, 0.5, 0.333 are rational numbers.
Irrational numbers are the set of real numbers that cannot be expressed in the form of a fraction p/q where ‘p’ and ‘q’ are integers and the denominator ‘q’ is not equal to zero (q≠0.). For example, π (pi) is an irrational number. π = 3.14159265…In this case, the decimal value never ends at any point. Therefore, numbers like √2, -√7, and so on are irrational numbers.
Real numbers are represented by the symbol R. Here is a list of the symbols of the other types of numbers.
N - Natural numbers W - Whole numbers Z - Integers Q - Rational numbers ¯¯¯¯ Q - Irrational numbers
All numbers except complex numbers are real numbers. Therefore, real numbers have the following five subsets:
Number Sets (Notation) The font that the symbols are written in (i.e. \(\mathbb{N}\), \(\mathbb{R}\)) is known as blackboard font.
\(\mathbb{N}\) Natural Numbers (\(0,1,2,3\)) (Not used in the CIS102 Syllabus)
\(\mathbb{Z}\) Integers (\(-3,-2,-1,0,1,2,3, \ldots\))
[\(\ast\)] \(\mathbb{Z}^{+}\) Positive Integers
[\(\ast\)] \(\mathbb{Z}^{-}\) Negative Integers
\(\mathbb{Q}\) Rational Numbers
\(\mathbb{R}\) Real Numbers
\(\mathbb{Z}\) Set of all integers
\(\mathbb{Q}\) Set of all rational numbers
\(\mathbb{R}\) Set of all real numbers
\(\mathbb{Z}^{+}\) Set of all positive integers
\(\mathbb{Z}^{-}\) Set of all negative integers
\(\mathbb{R}^{+}\) Set of all positive real numbers
\(\mathbb{R}^{-}\) Set of all negative real numbers
Irrational numbers
Inequality symbols
Floating-point notation
Number Sets: Natural Numbers, Integers, Rational Numbers and Real Numbers.
mantissa - the part of a floating-point number which represents the significant digits of that number.
radix point - a radix point or radix character is a symbol used in the display of numbers to separate the integer part of the value from its fractional part. A decimal point is a well-known radix point.
\(\mathbb{N}\) : natural numbers (or positive integers) \(\{1,2,3,\ldots\}\)
\(\mathbb{Z}\) : integers \(\{-3,-2,-1,0,1,2,3,\ldots\}\) \begin{itemize}
(The letter \(\mathbb{Z}\) comes from the word which means ``numbers” in German.)
\(\mathbb{Q}\) : rational numbers
\(\mathbb{R}\) : real numbers
\(\mathbb{N} \subseteq \mathbb{Z } \subseteq \mathbb{Q} \subseteq \mathbb{R}\)
(All natural numbers are integers. All integers are rational numbers. All rational numbers are real numbers.)
In computing, floating point describes a method of representing an approximation of a real number in a way that can support a wide range of values.
The numbers are, in general, represented approximately to a fixed number of significant digits (the mantissa) and scaled using an exponent.
In essence, computers are integer machines and are capable of representing real numbers only by using complex codes. The most popular code for representing real numbers is called the IEEE Floating-Point Standard . The term floating point is derived from the fact that there is no fixed number of digits before and after the decimal point; that is, the decimal point can float.
There are also representations in which the number of digits before and after the decimal point is set, called fixed-point representations.
In general, floating-point representations are slower and less accurate than fixed-point representations, but they can handle a larger range of numbers.
In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat.
An irrational number cannot be expressed as a ratio between two numbers and it cannot be written as a simple fraction because there is not a finite number of numbers when written as a decimal. Instead, the numbers in the decimal would go on forever, without repeating.
You can tell if it is Rational or Irrational by trying to write the number as a simple fraction.
The fractional numbers are figured out by two whole numbers (the fraction terms) that are separated by a horizontal line (the fraction line).
The number above the line (the numerator) can be every whole number and the number below the line (the denominator) should be different from zero. Here are some examples of some kinds of fractions:
3/4 ;9/2 ;2 1/3 ;5/2 = 10/4 ;4/3 ;8/100 ;
It may seem weird, but the answer is No. First of all because
the definition tells us that fractional numbers are numbers that
represent one or more parts of the whole. So, for instance, the number
10/2, which is written as a fraction is not a fractional
number, since it figures out number 5 and this one is not
part of a whole.
sqrt 2 / 3 is also written as a fraction, but it
is not a fractional number since the numerator is not a whole number.
A fractional number, commonly referred to as a fraction, represents a part of a whole or a ratio between two quantities. It consists of two components: a numerator and a denominator.
Fractions are essential in various fields, including mathematics, science, engineering, and everyday life, for representing and manipulating quantities that are not whole numbers.
Floating point notation is a method used to represent real numbers that can accommodate a wide range of values. This notation is particularly useful in scientific and engineering calculations where extremely large or small numbers are common. Unlike fixed-point notation, which has a fixed number of digits before and after the decimal point, floating point notation can “float” the decimal point, allowing for greater flexibility and precision.
At its core, floating point notation expresses numbers in the form of:
\[ N = M \times B^E \]
where: - \(N\) is the number being represented. - \(M\) is the mantissa (or significand), representing the significant digits of the number. - \(B\) is the base (or radix), typically 2 for binary systems used in computers. - \(E\) is the exponent, indicating the power to which the base is raised.
For example, the decimal number 123.45 can be represented in floating point notation as:
\[ 1.2345 \times 10^2 \]
Here, 1.2345 is the mantissa, 10 is the base, and 2 is the exponent.
The IEEE 754 standard is the most widely used standard for floating point computation. It defines the representation and behavior of floating point numbers in binary and decimal formats.
Consider the decimal number -42.75. To represent this in IEEE 754 single precision:
The IEEE 754 representation of -42.75 in single precision is:
\[ 1\ 10000100\ 01010110000000000000000 \]
Floating point notation is crucial in various fields:
Floating point notation provides a flexible and efficient way to represent real numbers in computing. By understanding its structure and limitations, one can effectively utilize this notation in various applications, ensuring accurate and reliable numerical computations. The IEEE 754 standard has standardized this representation, making it an essential tool in modern computing and digital systems.
Scientific notation is a way of expressing very large or very small numbers in a compact and manageable form. It is widely used in science, engineering, and mathematics to make calculations easier and to clearly communicate precise values. Scientific notation represents numbers as a product of a coefficient and a power of 10.
A number in scientific notation is written as: \[ N = M \times 10^E \] where: - \(N\) is the number being represented. - \(M\) is the mantissa (or coefficient), a decimal number greater than or equal to 1 and less than 10. - \(10\) is the base. - \(E\) is the exponent, an integer that indicates how many times the base (10) should be multiplied or divided.
Scientific notation and floating point notation are both methods for representing real numbers, especially very large or very small ones. While they are related concepts, they have different applications and structures.
While both scientific and floating point notations serve to represent real numbers efficiently, scientific notation is more suitable for manual calculations and general use, whereas floating point notation is designed for computational efficiency and precision in digital systems.
By understanding and utilizing scientific notation, one can effectively manage and communicate precise numerical values in various fields of study and practice.
A repeating decimal (or recurring decimal) is a decimal number that has a sequence of digits that repeats infinitely. The repeating part of the decimal can be a single digit or a group of digits. In mathematical notation, repeating decimals are often represented by placing a bar (vinculum) over the repeating digits.
To convert a repeating decimal to a fraction, algebraic methods are often used. Here’s a step-by-step example for converting \(0.\overline{6}\) to a fraction:
So, \(0.\overline{6}\) is equivalent to the fraction \(\frac{2}{3}\).
Repeating decimals are crucial in understanding the relationship between rational numbers and their decimal representations. They also appear frequently in calculations and real-world applications, making their understanding essential for mathematical literacy.
A repeating decimal, also called a recurring decimal, is a number whose decimal representation eventually becomes periodic (i.e., the same sequence of digits repeats indefinitely). The repeating portion of a decimal expansion is conventionally denoted with a vinculum so, for example,
\[ 1⁄7 = 0.\overline{142857} = 0.1428571428571428571...\]
The minimum number of digits that repeats in such a number is known as the decimal period.