Welcome to the exciting world of proofs in mathematics! Proofs are among the most powerful tools mathematicians use to establish the truth. They are the reasons why mathematics is often described as the most exact of all sciences. Let us explore how this concept fits into our daily lives and learning.
While the concept of proof can be introduced as early as grade 3, it becomes more formal and significant around grades 6 to 9. Students start engaging with simple proofs at these levels, particularly in geometry and number theory. As students get older and their mathematical skills develop, the complexity and importance of proofs also increase.
Proof exists across all domains of mathematics. However, they are most commonly found in Geometry, Algebra, and Number Theory at the school level. Students will encounter proofs in Calculus, Abstract Algebra, and Real Analysis as they move on to more advanced topics.
Applicable Common Core Standards
The concept of proving satisfies various Common Core Standards, especially in Geometry and Expressions & Equations. For example:
CCSS.MATH.CONTENT.HSG.CO.C.9: Prove theorems about lines and angles.
CCSS.MATH.CONTENT.HSG.SRT.B.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Definition of the Topic
Proof is a logical argument that uses rules and definitions to show that a mathematical statement is true. It consists of a series of statements, each following logically from the previous ones. A proof concludes with the statement being proved, ensuring that the conclusion is a logical consequence of the given information.
Theorem: A mathematical statement that we can prove to be true.
Axiom or Postulate: A statement assumed to be true without proof.
Proof: A logical argument showing that a theorem is true.
Proposition: A proved and accepted statement, but less central than a theorem.
Lemma: A minor result whose main purpose is to help prove a theorem.
Corollary: A result which follows directly from a theorem that has already been proven.
Discussion with Illustrative Examples
What is proof? A mathematical concept can be found true by supporting proof or a series of logical proofs. It is a logical argument that stabilizes the truth of a statement.
One of the standard proof formats is the two-column proof. It consists of two columns, wherein the first column contains numbered logical arguments called statements. The second column has the justification that supports the statements in column one, called reasons.
Two-Column Proof Format
In proving, we use geometric concepts and properties to solve problems. Here are a few fundamental ideas and congruence-related properties.
Line segments that are the same length are congruent to each other. From the figure above, both line segments CJ and YD measure 6 inches. Therefore, we can conclude that line segment CJ ≅ line segment YD.
Two angles that have the exact measurement are congruent to each other. Angles JYH and TKV both measure 45॰. Therefore, we can conclude that ∠JYH ≅ ∠TKV.
If two triangles are the same size and shape, they are congruent. From the figure above, triangles SKY and HCV have the same sides and equal angle measurements. Therefore, triangles SKY and HCV are congruent with each other.
Properties of Congruence
Shapes, line segments, and angles are always congruent to themselves.
Symmetric Property of Congruence
If a shape is congruent to a second shape, then the second shape is also congruent to the first shape.”
Transitive Property of Congruence
If an object is congruent to a second object and the second object is congruent to a third one, then the first object is congruent to the third object.
SAS (Side-Angle-Side) Postulate
If one triangle’s two sides and included angle correspond to another triangle’s two sides and included angle, the two triangles are congruent.
ASA (Angle-Side-Angle) Postulate
The two triangles are congruent if one triangle’s two angles and included side correspond to the other triangle’s corresponding two angles and included side.
SSS (Side-Side-Side) Theorem
The two triangles are congruent if the three sides of one triangle correspond to the three sides of the other triangle.
Corresponding Parts of Congruent Triangles are Congruent
If two triangles are congruent, the six pairs of their corresponding parts are congruent.
AAS (Angle-Angle-Side) Theorem
If the corresponding two angles and the non-included side of one triangle match those of the other’s corresponding two angles and the non-included side, the two triangles are congruent.
Steps in Writing an Indirect Proof
Step 1: Know the statement to be proved. Assume that the statement is false by negating it or making it’s opposite a true statement.
Step 2: State logical proofs until a contradiction (two statements cannot be both true) is made.
Step 3: The assumption (negating the desired conclusion) has been proven false by contradiction. Therefore, the original conclusion must be true.
You can write an indirect proof in this format:
Case 1: Write the assumption (Suppose… We assume that…).
Case 2: State the logical reasons.
Case 3: Write the conclusion.
Examples with Solution
Given: J is not the midpoint of YH.
Prove: YJ ≇ JH
We assume that YJ ≅ JH.
Then YJ and JH are congruent by the Definition of Midpoint (Midpoint bisects a line segment into two congruent parts).
This contradicts the given information that J is not the midpoint of YH. Therefore, the assumption that YJ ≅ JH must be false, and the original conclusion, YJ ≇ JH, must be true.
Prove that a triangle’s total angles equal 180 degrees.
Draw a line that passes through the other vertex of the triangle while remaining parallel to the triangle’s base.
By the definition of alternate angles, ∠3 = ∠4 and ∠2 = ∠5. So, ∠4 + ∠5 = ∠2 + ∠3.
Adding ∠1 on both sides, ∠1 + ∠4 + ∠5 = ∠1 + ∠2 + ∠3
The adjacent angles along the line sum up to 180 degrees (a straight line).
The triangle’s angles correspond to these angles.
As a result, a triangle’s angles add up to 180 degrees.
Consider the Pythagorean theorem, a fundamental theorem in geometry. According to this theorem, the square of the side (hypotenuse) opposite the right angle equals the sum or total of the squares of the other two sides. Show that the equation is a² + b² = c².
To prove this theorem, we often use a geometrical proof where we draw squares on each side of the right triangle and show that the two smaller squares’ combined area is equal to the area of the larger square.
Can you tell us which pair of woods are congruent to each other?
Y ≅ J
C ≅ S
H ≅ T
Real-life Application with Solution
The understanding of proof is not just limited to mathematics but also extends to everyday life. For instance, a detective might need to prove that a particular person committed a crime.
Suppose we have three clues:
The criminal has a red car.
The criminal has a pet parrot.
The criminal lives in a blue house.
Let us say we have three suspects, and based on the clues, we have the following information:
Suspect A has a red car and a blue house but has a pet dog.
Suspect B has a green car, a pet parrot, and a blue house.
Suspect C has a red car, a pet parrot, and a blue house.
We apply the logical process similar to a mathematical proof:
We first observe that Suspect A cannot be the criminal because even though he has a red car and a blue house, he doesn’t have a pet parrot.
Similarly, Suspect B can’t be the criminal because his car is not red, even though he has a pet parrot and a blue house.
Therefore, by the process of elimination, we deduce that Suspect C must be the criminal because he satisfies all the given conditions.
This is an application of how proof works in real-life situations.
Now, let’s practice a bit! Try solving these problems:
1. If you know that a number is even, prove that its square will also be even.
2. Prove that the sum of two odd numbers is always even.
3. Prove that the diagonals of a rectangle are equal.
1. A number is an even number if it is divisible by 2.
If we have 2n as an even number, squaring it shall give us (2n)(2n) = 4n2 = 2(2n2). Therefore, the obtained number is divisible by 2.
2. Let us say that 2m and 2n are even numbers because they can be divided by 2, then 2m + 1 and 2n +1 are odd numbers. If we add the odd numbers ( 2m + 1 ) + ( 2n + 1) equals 2m + 2n + 2. The sum can be written as 2 (m + n + 1 ). Since it has 2 as its factor, two odd integers add up to an even number.
3. Say we have a rectangle ABCD, then AC and BD are diagonals.
In ∆ABC and ∆BCD, ∠ABC and ∠BCD are angles of rectangles which are both 90°.
BC is a common side.
AB = CD since opposite sides of a parallelogram are equal.
By Side-Angle-Side (SAS) Congruency, ∆ABC ≅ ∆BCD.
By corresponding parts of congruent triangles, AC = DC.
Hence, the diagonals are equal.
Frequently Asked Questions (FAQs)
What is mathematical proof?
Mathematical proof is a logical argument that uses rules and definitions to show a mathematical statement is true.
Why are proofs important in mathematics?
Proofs are important because they ensure that mathematical theorems are universally and undeniably true, given that the axioms and definitions they are based on are true.
Can a mathematical proof be wrong?
A proof could be incorrect if it contains a logical fallacy, a mistake in the reasoning, or if it is based on false axioms or definitions. However, it is considered universally true once a proof has been established and verified.
How do theorems and proofs differ from one another?
A proof is a process or logical argument showing a theorem
is true, while a theorem is a mathematical statement or assertion that has been proven true.
Are proofs necessary in all areas of mathematics?
Proofs are a fundamental part of all areas of mathematics. While some areas, like geometry, may use more visual proofs, every branch of mathematics relies on proofs to establish the truth of mathematical statements.