What is a Real Number?
To understand real numbers, we first have to understand what rational and irrational numbers are. Rational numbers are ones that can be written in the form of p/q, where p is the numerator and q is the denominator, and both p and q are integers. For example, 7 can be written as 7/1, so it is a rational number. Irrational numbers are numbers that cannot be written in p/q form. For example,√2 is an irrational number because √2 = 1.41421 .. and continues to infinity. Hence, it cannot be written as a fraction and is non-terminating and non-recurring decimals. Rational and irrational numbers together form real numbers. You can also find a worksheet on real numbers at the end.
Introduction
The real number represents a value of a continuous quantity in Mathematics. A real number can represent a distance along a line. On the other hand, it also represents an infinite decimal expansion. The adjective real in Mathematics was introduced by René Descartes in the 17th century. He is renowned for the real and imaginary roots of polynomials.
The real numbers represent all the rational numbers. For example, the integer −5 and the fraction 4/3. As an example of irrational numbers, we can take root 2, which means 1.41421356, and so on. The square root of 10 is an example of an irrational algebraic number. The real transcendental numbers are also Included within the irrational numbers. For example, one can take π, which means 3.14159265, and so on. Real numbers are used to measure quantities such as time, mass, velocity, energy, distance, and many more. The real numbers are denoted by the symbol R or R2.
Real numbers can represent any point on an infinitely long line. That is known as a number line or real line. In that case, the points representing integers are equally spaced. A possibly infinite decimal representation can dictate any Real number. The real line is considered a part of the complex plane. Also, the real numbers can be represented as a part of the complex numbers.
The set of all real numbers goes upto infinity. Real numbers include all the natural numbers that are also the infinity setting. It does not represent any one-to-one function from the real numbers to the natural numbers. The cardinality of the set of all real numbers is represented by {\displaystyle {\Mathfrak {c}}}C. The statement that there is no subset of the reals with cardinality strictly greater than {\displaystyle \aleph _{0}}N0 and strictly smaller than {\displaystyle {\Mathfrak {c}}}C. It is known as the continuum hypothesis (CH).
There is Zermelo–Fraenkel set theory in the real numbers. That includes the axiom of choice (ZFC)—the standard foundation of modern Mathematics. Also, some models of ZFC satisfy CH, while others infarct it.
What are Mathematical Operations?
The four basic Mathematical operations are addition (+), subtraction (-), multiplication (x), and division (/). We will now understand these operations on real numbers - both rational and irrational. The real numbers worksheet will help you understand this topic better.
Operations on Two Rational Numbers
These are some of the operations:
Addition of Two Rational Numbers
When two rational numbers are added, the result is a rational number. For example, 0.24 + 0.68 = 0.92. 0.92 can be written as 92/100, which is a ratio or the p/q form.
Subtraction of Two Rational Numbers
When two rational numbers are subtracted, the result is a rational number. For example, 0.93-0.22 = 0.71 which can be written as 71/100.
Multiplication of Two Rational Numbers
When two rational numbers are multiplied, the result is a rational number. For example, 0.5 multiplied by 185 is 92.5, which can be written as 925/10.
Division of Two Rational Numbers
When a rational number is divided by another rational number, the result is a rational number. For example, 0.352 divided by 0.6 is 0.58, which can be written as 58/100.
Operations on two Irrational Numbers
Addition of Two Irrational Numbers
When two irrational numbers are added, the result can be an irrational or a rational number. For example, √3 added to (√3) is 3.46 or 2√3 which can be written as 346/100, which is a rational number. However, when 2√5 is added to 5√3, we get a non-terminating and non-recurring decimal, an irrational number. It is written as 2√5+5√3.
Subtraction of Two Irrational Numbers
Similarly, when two irrational numbers are subtracted, the result can be an irrational or a rational number. √2 is subtracted from √2, the answer is 0. When 4√5 is subtracted from 5√3, we get 5√3-4√5.
Multiplication of Two Irrational Numbers
The product of two irrational numbers can be an irrational number or a rational number. For example, when √2 is multiplied by √2, we get 2 which is a rational number. However, when √2 is multiplied by √3, we get √6 which is an irrational number.
Division of Two Irrational Numbers
Similar to multiplication, we can get either an irrational number or a rational number as a result when an irrational number is divided by another. For example, when √2 is divided by √2, we get 1 which is a rational number. But when √2 is divided by √3, we get √2/√3, which is an irrational number.
Operations on a Rational and an Irrational Number
Addition of an Irrational and a Rational Number
The sum of a rational and an irrational number is always irrational. For example, when 2 is added to 5√3, we get 2 + 5√3, which is a rational number.
Subtraction of an Irrational and a Rational Number
The difference between a rational and an irrational number is always irrational. For example, when we subtract 5√3 from 2, we get 2 - 5√3, which is irrational.
Multiplication of an Irrational and a Rational Number
The product of a rational and an irrational number might be rational or irrational. For example, when 2 is multiplied by √2, we get 2√2 which is an irrational number, but when√12 is multiplied by √3, we get √36, or 6, which is a rational number.
Division of an Irrational Number with a Rational Number
When a rational number is divided by an irrational number or vice versa, the quotient is always an irrational number. For example, when 8 is divided by √2, we get 8/√2, which is an irrational number. The answer can be further simplified to 4√2 which is also an irrational number.
Operations on Real Numbers Worksheet
Example 1:
Solve:
(7√3) x (- 5√3)
Solution: (7√3) x (- 5√3)
= 7 x -5 x √3 x √3
= -35 x 3
= -105
Example 2:
Solve:
(3√27 / 9√3)
Solution: (3√27 / 9√3)
= 3√27
= √3x3x3
= 3 x 3√3
= 9√3
= 9√3/ 9√3
= 1
Summary
Here is some fact that student must know about
Integers, rational, and irrational numbers are all included in real numbers.
Some Numbers are Not Real (imaginary, complex).
Real numbers can do arithmetic.
Every real number has a decimal representation.
There is nothing but all the real numbers in the number line.
The real number is used to measure quantities.
All Cauchy sequences converge towards the real numbers.
To understand the topic further, operations of real numbers worksheets might be of great help to students.
FAQs on Operations on Real Numbers
1. What are the fundamental arithmetic operations on real numbers?
The four fundamental arithmetic operations on real numbers are addition (+), subtraction (-), multiplication (×), and division (÷). These operations allow us to combine or manipulate any two real numbers, which include both rational and irrational numbers, to produce another real number, with the key exception of division by zero.
2. What are the key properties that govern operations on real numbers?
For any real numbers a, b, and c, the main properties for addition and multiplication are:
- Commutative Property: The order of numbers does not affect the result (e.g., a + b = b + a; a × b = b × a).
- Associative Property: The grouping of numbers does not affect the result (e.g., (a + b) + c = a + (b + c)).
- Distributive Property: Multiplication distributes over addition (e.g., a × (b + c) = a × b + a × c).
- Identity Property: The sum with 0 or product with 1 remains the original number (a + 0 = a; a × 1 = a).
- Inverse Property: Adding the opposite gives 0 (a + (-a) = 0), and multiplying by the reciprocal gives 1 (a × 1/a = 1, for a ≠ 0).
3. How are operations like addition and multiplication performed on irrational numbers?
When performing operations on irrational numbers such as square roots, we treat them similarly to algebraic variables.
- Addition and Subtraction: You can only combine 'like' terms. For example, 2√3 + 5√3 simplifies to 7√3, but an expression like 2√3 + 5√2 cannot be simplified further.
- Multiplication and Division: You can multiply or divide the rational and irrational parts separately. For example, (4√6) × (2√5) = (4 × 2) × (√6 × √5) = 8√30.
4. Why is rationalising the denominator an important skill when working with real numbers?
Rationalising the denominator is the process of removing an irrational number, like a square root, from the denominator of a fraction. This is a crucial step for two main reasons:
- Standard Form: It converts the number into a standard, simplified format, which makes it much easier to estimate its value or compare it to other numbers.
- Simplified Calculations: It makes further arithmetic operations, like addition or subtraction, far simpler. For instance, adding 2/√3 to √3 is easier once 2/√3 is rationalised to 2√3/3.
5. Is the sum or product of two irrational numbers always irrational?
No, this is a common misconception. The result of operating on two irrational numbers can be either rational or irrational. For example:
- Sum leading to a rational number: The sum of (5 + √2) and (5 - √2) is 10, which is rational.
- Product leading to a rational number: The product of √3 and √12 is √36, which equals 6, a rational number.
6. Are all four basic arithmetic operations 'closed' for the set of real numbers?
The set of real numbers is considered 'closed' under addition, subtraction, and multiplication. This means performing these operations on any two real numbers will always result in another real number. However, the set is not closed under division. This is because division by zero is undefined. Since 0 is a real number, you cannot divide any real number by it and obtain another real number as the result, which breaks the closure property.
7. How are operations on real numbers applied in practical, real-world scenarios?
Operations on real numbers are fundamental to many everyday tasks and professional fields. For example:
- Finance and Commerce: Calculating interest on loans, profit margins, discounts, and taxes involves operations on rational numbers (decimals and fractions).
- Science and Engineering: Irrational numbers like π (pi) are essential for calculating the circumference of circles or the volume of cylinders. Square roots are used in physics to calculate distances and velocities.
- Construction: The Pythagorean theorem, which uses squares and square roots, is used to ensure right angles in buildings and structures.
8. What is the identity element for addition and multiplication in real numbers?
In the set of real numbers, an identity element is a special number that, when used in an operation with another number, leaves that number unchanged.
- The additive identity is 0, because for any real number 'a', a + 0 = a.
- The multiplicative identity is 1, because for any real number 'a', a × 1 = a.
