Definition Examples And Key Differences Between Theorem Lemma And Corollary
In this article, we are going to talk about theorems, lemmas, and corollary; theorems are those mathematical statements which are true and have logical proof. For example, De Moivre’s Theorem, Alternate Segment Theorem, etc. In this article, we are going to talk about how to prove different theorems and also solve some questions based on them. A corollary is a statement that follows naturally from some other statement that has either been proven or is generally accepted as true. For example, The sum of the interior angles of any triangle is always 180 degrees. A lemma is like a mini-theorem that helps you prove a bigger theorem or statement. For example, Euclid’s Division Lemma.
Table of Contents
An Overview of Theorems, Lemma, and Corollary
History of the Mathematician
Definition of Theorem
Statement of De Moivre’s Theorem
Proof of De Moivre’s Theorem
Definition of Lemma
Definition of Corollary
Solved Examples
Important Points to Remember
History of Euclid
Euclid
Image Credit: Wikimedia
Name: Euclid
Born: Mid-4th century BC
Field: Mathematics
Contribution: Euclid was the very first person to start discovering these theorems and Lemma.
Nationality: Greek
Definition of Theorem
A theorem is a mathematical statement that is true and has very logical proof. It can either be for algebra or geometry, but the result of a theorem can always be proved.
Let us take the example of De Moivre’s Theorem to explain how a theorem can be proved.
Statement of De Moivre’s Theorem
According to the De Moivre Theorem, if we raise the power of a polar complex number by n, then it is equivalent to increasing the modulus to the same power and multiplying it by the argument raised to the same power, which means:
${{(\cos x+\sin x)}^{n}}=\cos (nx)+\sin (nx)$.
Proof of De Moivre’s Theorem
Let us take a complex number $z=x+iy$. This complex number can be written in polar form as $z=r(\cos \theta +i\sin \theta )$.
Here, $r=\sqrt{{{x}^{2}}+{{y}^{2}}}$ ($r$ is called the modulus/absolute value of the complex number).
When we plot the complex number on the argand plane:
Argand Plane
$\cos \theta =\dfrac{x}{r}$
$\sin \theta =\dfrac{y}{r}$
Now, let’s raise the complex number $z$ to the power $n$.
$z=r(\cos \theta +i\sin \theta )$
${{z}^{n}}={{(r(\cos \theta +i\sin \theta ))}^{n}}$
${{z}^{n}}={{r}^{n}}{{(\cos \theta +i\sin \theta )}^{n}}$
Solving using the principle of mathematical induction:
For $n=1$
${{(\cos \theta +i\sin \theta )}^{1}}=\cos (1\theta )+\sin (1\theta )$
Assuming this to be true for $n=k$,
${{(\cos \theta +i\sin \theta )}^{k}}=\cos (k\theta )+\sin (k\theta )$
Proving this to be true for $n=k+1$,
${{(\cos \theta +i\sin \theta )}^{k+1}}={{(\cos \theta +i\sin \theta )}^{k}}.{{(\cos \theta +i\sin \theta )}^{1}}$
${{(\cos \theta +i\sin \theta )}^{k+1}}=(\cos (k\theta )+i\sin (k\theta )).(\cos \theta +i\sin \theta )$
${{(\cos \theta +i\sin \theta )}^{k+1}}=\cos (k\theta ).\cos (\theta )+i\cos (k\theta ).\sin \theta +i\sin (k\theta )\cos (\theta )-\sin (k\theta )\sin (\theta )$
${{(\cos \theta +i\sin \theta )}^{k+1}}=\cos (k\theta ).\cos (\theta )-\sin (k\theta )\sin (\theta )+i(\cos (k\theta ).\sin \theta +\sin (k\theta )\cos (\theta ))$
Using trigonometry formulae:
${{(\cos \theta +i\sin \theta )}^{k+1}}=\cos (k+1)\theta +i(\sin (k+1)\theta )$
Hence Proved.
Definition of Lemma
A lemma is nothing but a proven statement that is used to prove other statements. Lemma is like a mini-theorem that helps you prove a bigger theorem or statement.
A few examples of a lemma are as follows:
Euclid’s Division Lemma - According to Euclid's division lemma, we will always get a unique integer as the quotient and a unique integer as the remainder when we divide one integer by another non-zero integer. The theorem states that, for given two positive integers, $a$ and $b$, there exist unique integers, $q$ and $r$, such that $a=bq+r$, where $0\le r<b$. The integer $q$ is the quotient and the integer $r$ is the remainder. The quotient and the remainder are unique.
Definition of Corollary
The corollary is a statement that follows with little or no proof required from an already proven statement. For example, there is a theorem in geometry that the angles opposite to two congruent sides of a triangle are also congruent. A corollary to this statement is that an equilateral triangle is equiangular.
So for a corollary, the proof relies heavily on a given certain theorem. A corollary can be proved but it depends on some theorem heavily. There can't be any assumptions in the proof.
Solved Examples
1. Find \[X{}^\circ +Y{}^\circ \].
A Circle with Two Tangents
Ans. Here, \[\angle Y\] is the angle between the chord and the tangent and \[\angle X\] is the angle subtended at the circumference by the chord in the alternate segment.
Hence, \[\angle X=\angle Y\].
\[\angle X\] and \[30{}^\circ \] are the opposite angles of a cyclic quadrilateral:
\[\angle X+30{}^\circ =180{}^\circ \]
\[\angle X=150{}^\circ \]
\[\angle Y=150{}^\circ \]
2. Calculate the value of angles x and y in the given diagram.
A Circle
Ans. The tangent to the circle has the point of contact at \[C\], the angle formed between the tangent and the chord \[AC\] is \[\angle x\] and the chord \[AC\] is subtending an \[\angle y\] in the alternate segment of the circle.
\[\angle x=\angle y\] (Alternate Segment Theorem)
\[\angle ABC+\angle BAC+\angle ACB=180{}^\circ \](Angle sum property of Triangle)
\[\angle y+90+30=180\]
\[\angle y+120=180\]
\[\angle y=60{}^\circ \]
3. Find DE
Ans. According to the basic proportionality theorem:
\[\frac{AE}{DE}=\frac{BE}{CE}\]
\[\frac{4}{DE}=\frac{6}{8.5}\]
\[\frac{4*8.5}{6}=DE\]
\[DE=5.66\]
Important Points to Remember
Lemma is nothing but just a follow-up of a theorem, it helps us in proving bigger theorems.
The corollary is an idea developed from a theorem that is already proved; here, the word idea means understanding the same theorem differently to try to prove another theorem.
List of Related Articles
FAQs on Understanding Theorem Lemma And Corollary In Mathematics
1. What is the difference between a theorem, lemma, and corollary?
The difference between a theorem, lemma, and corollary lies in their role in mathematical reasoning: a theorem is a main proven result, a lemma is a supporting result, and a corollary is a direct consequence of a theorem.
- Theorem: A major or central mathematical statement that has been proven true.
- Lemma: A helper result used to prove a theorem.
- Corollary: A result that follows immediately from a theorem with little or no additional proof.
2. What is a theorem in mathematics?
A theorem is a mathematical statement that has been logically proven true using axioms, definitions, and previously established results.
- It is usually an important or central result in a topic.
- It requires a formal proof.
- Example: The Pythagorean Theorem states that in a right triangle, a² + b² = c².
3. What is a lemma in mathematics?
A lemma is a proven statement used primarily to help prove a larger theorem.
- It is sometimes called a “helper theorem.”
- It may not seem important on its own.
- Example: Euclid’s Lemma states that if a prime number divides a product of two integers, it divides at least one of them.
4. What is a corollary in mathematics?
A corollary is a statement that follows directly and easily from a previously proven theorem.
- It requires little or no additional proof.
- It is a natural consequence of a theorem.
- Example: From the Pythagorean Theorem, a corollary is that a triangle with sides satisfying a² + b² = c² is a right triangle.
5. Can a lemma ever be more important than a theorem?
Yes, a lemma can become more famous or important than the theorem it helps prove.
- Some lemmas gain independent significance.
- Example: Zorn’s Lemma is widely used in algebra and set theory.
- Its importance extends far beyond its original use.
6. How are theorem, lemma, and corollary used in proofs?
In mathematical proofs, a lemma supports a theorem, the theorem establishes the main result, and a corollary follows from that theorem.
- Step 1: Prove one or more lemmas.
- Step 2: Use those lemmas to prove the main theorem.
- Step 3: Deduce corollaries directly from the theorem.
7. Do theorem, lemma, and corollary all require proof?
Yes, a theorem, lemma, and corollary all require proof, but the amount of proof may differ.
- Theorem: Usually requires a detailed proof.
- Lemma: Also requires proof before being used.
- Corollary: Often needs only a short explanation since it follows directly from a theorem.
8. What is an example showing the difference between theorem, lemma, and corollary?
An example of the difference is seen in number theory using prime numbers.
- Lemma: If a prime number divides a product, it divides at least one factor (Euclid’s Lemma).
- Theorem: Every integer greater than 1 can be written uniquely as a product of primes (Fundamental Theorem of Arithmetic).
- Corollary: The prime factorization of a number is unique.
9. What is the difference between a corollary and a theorem?
The difference between a corollary and a theorem is that a theorem is a main proven result, while a corollary is a direct consequence of that theorem.
- Theorem: Requires independent and often detailed proof.
- Corollary: Follows quickly from a theorem with minimal extra reasoning.
10. Why do mathematicians use different terms like theorem, lemma, and corollary?
Mathematicians use terms like theorem, lemma, and corollary to clarify the role each statement plays in logical reasoning.
- They organize mathematical arguments clearly.
- They show which results are central and which are supporting.
- They help readers understand the structure of proofs.
