What is Demonstrative geometry?
Demonstrative geometry is a branch of mathematics in which theorems on geometry are proved through logical reasoning. It demonstrates the truth. of mathematical statements concerning geometric figures.
Reasoning
In mathematics, reasoning involves drawing logical conclusions based on evidence or stated assumptions. Sense making may be considered as developing understanding of a situation, context, or concept by connecting it with existing knowledge or previous experience.
Basics of Reasoning
Basics of reasoning in mathematics are:
- Basic Concepts: Some concepts are accepted true without defining them
for example point, line or plane.
- Assumptions: Some statements are accepted true without proofs.
These are called basic assumptions.
Mathematical reasoning or the principle of mathematical reasoning is a part of mathematics where we determine the truth values of the given statements.
Mathematically Acceptable Statements
Consider the following Statement:
“The sum of two prime numbers is always even.”
The given statement can either be true or false since the sum of two prime numbers can be either be an even number or an odd number. Such statements are mathematically not acceptable for reasoning as this sentence is ambiguous. Thus, a sentence is only acceptable mathematically when it is “Either true or false, but not both at the same time.” Therefore, the basic entity required for mathematical reasoning is a statement. This is the mathematical statement definition.
Types of Reasoning
In terms of mathematics, reasoning can be of two major types which are:
- Inductive Reasoning
- Deductive Reasoning
Inductive Reasoning
In the Inductive method of mathematical reasoning, the validity of the statement is checked by a certain set of rules and then it is generalized. The principle of mathematical induction uses the concept of inductive reasoning.
As inductive reasoning is generalized, it is not considered in geometrical proofs. Here, is an example which will help to understand the inductive reasoning better.
Example:
Statement: The cost of goods is Rs. 10 and the cost of labor to manufacture the item is Rs. 5. The sales price of the item is Rs. 50.
Reasoning: From the above statement, it can be said that the item will provide a good profit for the stores selling it.
Deductive Reasoning
The principal of deductive reasoning is the opposite of the principle of induction. On the contrary to inductive reasoning, in deductive reasoning, we apply the rules of a general case to a given statement and make it true for particular statements. The principle of mathematical induction uses the concept of deductive reasoning (contrary to its name). The below-given example will help to understand the concept of deductive reasoning better.
Example:
Statement: Pythagorean Theorem holds true for any right-angled triangle.
Reasoning: If triangle XYZ is a right triangle, it will follow Pythagorean Theorem