corollary

What is a Corollary in Math? Definition

A corollary in math is a statement that follows directly from a previously proven theorem or proposition. It is a result that can be easily deduced from the main theorem without requiring any additional proof. Corollaries are often used to provide further clarification or to extend the implications of a theorem.

History of Corollary

The concept of corollary has been used in mathematics for centuries. The term "corollary" itself comes from the Latin word "corollarium," which means "a deduction." The use of corollaries can be traced back to ancient Greek mathematicians, such as Euclid, who employed them extensively in his famous work "Elements."

What Grade Level is Corollary For?

Corollaries are typically introduced in advanced mathematics courses, usually in high school or college. They are commonly encountered in subjects like geometry, algebra, calculus, and number theory. Due to their reliance on previously proven theorems, corollaries are more suitable for students who have a solid understanding of mathematical concepts and proofs.

Knowledge Points in Corollary and Detailed Explanation Step by Step

Corollaries are derived from theorems and propositions, so a thorough understanding of these concepts is necessary to comprehend corollaries fully. Here is a step-by-step explanation of how a corollary is derived:

1. Start with a previously proven theorem or proposition.
2. Identify a statement that can be directly deduced from the main theorem without requiring any additional proof.
3. State the derived statement as a corollary, making sure to reference the original theorem.
4. Provide a brief explanation of why the corollary follows logically from the main theorem.

Types of Corollary

Corollaries can take various forms depending on the nature of the original theorem. Some common types of corollaries include:

1. Geometric Corollaries: These corollaries are derived from geometric theorems and often involve properties of angles, lines, or shapes.
2. Algebraic Corollaries: These corollaries are derived from algebraic theorems and involve equations, inequalities, or algebraic manipulations.
3. Calculus Corollaries: These corollaries are derived from calculus theorems and often involve derivatives, integrals, or limits.
4. Number Theory Corollaries: These corollaries are derived from number theory theorems and involve properties of integers, primes, or divisibility.

Properties of Corollary

Corollaries inherit the properties of the theorems from which they are derived. Some common properties of corollaries include:

1. Logical Consequence: Corollaries are logical consequences of the main theorem and can be deduced directly from it.
2. Simplification: Corollaries often simplify the application of the main theorem by providing a more specific or specialized result.
3. Extension: Corollaries can extend the implications of the main theorem by providing additional insights or consequences.

How to Find or Calculate Corollary?

Corollaries are not found or calculated independently. Instead, they are derived from previously proven theorems or propositions. To find a corollary, one must carefully analyze the main theorem and identify statements that can be directly deduced from it without requiring any additional proof.

Formula or Equation for Corollary

Corollaries do not have specific formulas or equations. They are statements that follow logically from the main theorem. However, the main theorem may have associated formulas or equations that can be used to derive the corollary.

How to Apply the Corollary Formula or Equation?

As mentioned earlier, corollaries do not have specific formulas or equations. Instead, they rely on the formulas or equations associated with the main theorem. To apply a corollary, one must understand the conditions and assumptions of the main theorem and use the corollary to simplify or extend the implications of the theorem.

Symbol or Abbreviation for Corollary

There is no specific symbol or abbreviation for corollary. It is commonly denoted as "Cor." followed by a number or letter to distinguish it from other corollaries.

Methods for Corollary

The methods for deriving corollaries involve logical deduction and analysis of the main theorem. Some common methods include:

1. Direct Deduction: Identify statements that can be directly deduced from the main theorem without requiring any additional proof.
2. Logical Reasoning: Use logical reasoning to establish the connection between the main theorem and the derived corollary.
3. Mathematical Manipulation: Apply mathematical manipulations, such as algebraic simplifications or geometric transformations, to derive corollaries.

Solved Examples on Corollary

Example 1: Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Corollary: If the square of the hypotenuse is greater than the sum of the squares of the other two sides, then the triangle is acute-angled.

Explanation: The corollary follows directly from the theorem. If the square of the hypotenuse is greater than the sum of the squares of the other two sides, it implies that the angle opposite the hypotenuse is acute, as in an acute-angled triangle.

Example 2: Theorem: The sum of the angles in a triangle is 180 degrees. Corollary: If one angle in a triangle is a right angle (90 degrees), then the other two angles are acute angles.

Explanation: The corollary is a direct consequence of the theorem. If one angle in a triangle is a right angle, the sum of the other two angles must be 90 degrees. Since the sum of the angles in a triangle is 180 degrees, the other two angles must be acute.

Example 3: Theorem: If a number is divisible by 6, it is also divisible by both 2 and 3. Corollary: If a number is not divisible by either 2 or 3, it is not divisible by 6.

Explanation: The corollary can be derived from the theorem by negating the statement. If a number is not divisible by either 2 or 3, it implies that it cannot be divisible by 6, as 6 is a multiple of both 2 and 3.

Practice Problems on Corollary

1. Given a triangle with angles measuring 30 degrees, 60 degrees, and 90 degrees, find the length of the hypotenuse.
2. If a number is divisible by 4, is it also divisible by 2? Explain using a corollary.
3. Prove that the sum of the exterior angles of any polygon is 360 degrees.

FAQ on Corollary

Question: What is a corollary? A corollary is a statement that follows directly from a previously proven theorem or proposition without requiring any additional proof.

Question: How are corollaries derived? Corollaries are derived by analyzing the main theorem and identifying statements that can be directly deduced from it.

Question: Are corollaries always true? Corollaries are true as long as the main theorem from which they are derived is true.

Question: Can corollaries be used to prove other theorems? Corollaries are often used to provide further clarification or to extend the implications of a theorem but cannot be used to prove other theorems independently.

Question: Can corollaries be proven independently? Corollaries cannot be proven independently as they rely on the proof of the main theorem from which they are derived.