Mathematical Reasoning

An Overview of Mathematical Reasoning

Do you remember, how as a middle-school grader you were often given worksheets to solve, that had questions like, "all odd numbers are prime numbers: true or false"? These statements could be true or false. These questions formed the basis of what is today known as mathematical reasoning. It is the branch of mathematics that deals with the truth in a given mathematical statement.

Math and reasoning go hand in hand, forming a very crucial part of the syllabus for JEE and other competitive exams. Let us take a look at what reasoning in math means and how we can solve important questions.

What are Mathematically Acceptable Statements?

Suppose you are given a statement that says:

“The square root of 1 is always 1.”

Thus this statement can either be true or be false, but cannot be both. This is the basic rule to follow when you solve problems in mathematics and reasoning. The statement above states that the square root of 1 is always 1. Therefore it is a true statement. Such a statement is mathematically acceptable. Other ambiguous statements, such as "the sum of prime numbers is even" makes no sense because we cannot be sure about the outcome. Therefore the rules governing a mathematically acceptable statement are:

  1. A statement can be mathematically acceptable only if it is either true or false. It cannot be both.

  2. Statements in mathematical logical reasoning can be none of these three things: exclamatory, interrogative or imperative.

  3. A statement may have variables in it. Such a declarative statement is considered an open statement, only if it becomes a statement when these variables are replaced by some constants.

  4. A mathematical statement that is a combination of two or multiple statements is known as a compound statement.

  5. Compound statements are usually joined with the help of “and” or (^). Such a statement is denoted with p^q.

  6. A compound statement containing “and” is true, if and only if all its components are found to be true.

The Following Table Illustrates the Truth Values of p^q and q^p:

Truth Table (For p^q and q^p)

p

q

q^p

p^q

T

T

T

T

T

F

F

F

F

T

F

F

F

F

F

F


Before you proceed further into compound statements, various fallacies and laws governing reasoning math, let us learn more about the various types of reasoning in maths.

Types of Maths Logical Reasoning

Logical and mathematical reasoning is key to knowing mathematics and sailing through the world of practical math. Doing, or applying mathematical principles in real life is a creative act, and reasoning is the basis of that act. It is a very useful way to make sense of the real world and nurture mathematical thinking. 

In mathematics, two kinds of reasoning demonstrate the logical validity of any statement. These are:

a. Inductive Reasoning

This involves looking for a pattern in a given set of problem statements and generalising. For instance, a student may use inductive reasoning when looking at a set of different shapes, such as rectangles, rhombuses and parallelograms. Further, he/she might try to spot the similarities and differences, and figure what is common. Therefore between a set of squares and circles, a student can use this type of reasoning to consider the shapes that are not squares. The following figure will demonstrate a similar situation.

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You may also use inductive reasoning to spot patterns in a series of even numbers, multiplying numbers by ten, or while working with roots, indices and exponents. For instance, you may look at 20/100 and 30/120 and inductively reason that common multiples in a fraction can always be cancelled. Similarly, if all these fractions form part of a series, you should remember to continue evaluating it this way, before generalising. Also, remember that inductive reasoning is always non-rigorous and statements are usually generalised. 

Rack Your Brains: Work out the following statement and find out if it uses inductive reasoning.

Statement: The cost of materials used to make a bar of Dairy Milk is Rs. 20, and the cost of labour required to manufacture it is Rs.15. The selling price of the chocolate is Rs.60.

Reasoning: It is clear from the above statement, that stores and shops will make a hefty profit selling this bar of chocolate.

Before you generalise a statement and look for the truth in it, you must practise care to prove it through deductive reasoning including mathematical reasoning.

b. Deductive Reasoning

This involves drawing logical conclusions, stating a logical argument, and then generalising a specific situation. For instance, if you know what a parallelogram or a square is, you will be able to apply this generalisation to a new pattern of figures. You will be able to conclude if each of these shapes is a square or not.

In fact, deductive reasoning is the opposite of inductive reasoning. In deductive reasoning, we apply the general rules to a given statement and see if we can make the subsequent statements true.

This was all about the different types of reasoning in math. Check out our expertly-curated collection of mathematical reasoning NCERT questions, notes on different types of statements, and reference notes, available on the site. Watch free demo classes on the Vedantu app and make math fun.

FAQ (Frequently Asked Questions)

1. What are Mathematical Statements?

Ans. A mathematical statement is the basis of all mathematics quantitative reasoning. Further, reasoning can be either inductive (also called mathematical induction) or be deductive. An assertive statement that can either be true or false is said to be an acceptable statement in mathematics. It cannot be both. Ambiguous statements are invalid in math and reasoning.

2. What are Compound Statements?

Ans. A mathematical statement that is a combination of two or multiple statements is known as a compound statement. Compound statements are usually joined with the help of “and” or (^). Such a statement is denoted with p^q. A compound statement containing “and” is true, if and only if all its components are found to be true.

3. What are the Two Main Types of Mathematical Reasoning?

Ans. Two fundamental types of reasoning govern logic and mathematics. These are:

  1. Inductive Reasoning

  2. Deductive Reasoning

4. Why is Reasoning Critical in Maths?

Ans. Reasoning helps develop our critical thinking processes, our powers of analysis and drawing logical conclusions. Doing, or applying mathematical principles in real life is a creative act, and reasoning is the basis of that act. It is a very useful way to make sense of the real world and nurture mathematical thinking.