Ready to unlock the mysteries of pi, e and beyond? Welcome to our transcendental number quiz, a free math trivia challenge designed for curious minds and budding mathematicians. Whether you're brushing up via a number system quiz or diving into a real numbers algebraic expressions quiz , this number theory quiz will put your skills to the test on algebraic vs transcendental definitions and offer fascinating transcendental numbers examples. Think you can conquer these college-level puzzles? Grab your pencil, click "Start Quiz" and see if you can ace our free transcendental number quiz now!
Which of the following best describes a transcendental number?
A real or complex number not a root of any non-zero polynomial equation with integer coefficients.
A number that can be expressed as a ratio of two integers.
A number that satisfies at least one polynomial equation with leading coefficient 1.
An integer greater than 1 with no positive divisors other than 1 and itself.
A transcendental number is defined by not satisfying any algebraic equation with integer or equivalently rational coefficients. All algebraic numbers are solutions to such polynomial equations, while transcendental numbers lie beyond this set. This is the standard definition found in number theory. Learn more.
Which of the following numbers is known to be transcendental?
? (pi)
?2
? (the golden ratio)
0.101001000100001… (Champernowne’s constant)
The transcendence of ? was proved by Ferdinand von Lindemann in 1882, showing it cannot satisfy any polynomial with rational coefficients. ?2 and the golden ratio are algebraic since they solve simple polynomial equations. Champernowne’s constant is normal but algebraic status is not transcendental. Read more.
The set of all transcendental numbers is:
Uncountable
Countably infinite
Finite
Empty
Algebraic numbers are countable, so their complement among the reals—the transcendental numbers—is uncountable. This means almost all real numbers are transcendental in the sense of Lebesgue measure. The cardinality of transcendental numbers equals the continuum. Source.
Which statement is true regarding transcendental numbers?
Every transcendental number is irrational.
Every irrational number is transcendental.
Every algebraic number is transcendental.
Every rational number is transcendental.
Transcendental numbers cannot be roots of any polynomial equation with integer coefficients, so they cannot be rational or even algebraic irrational. Thus all transcendental numbers are irrational, but not all irrationals are transcendental. Many irrational numbers are algebraic (e.g., ?2). More details.
Which theorem is central to proving that the number e is transcendental?
Lindemann–Weierstrass theorem
Fundamental Theorem of Algebra
Euclid’s Lemma
Galois Correspondence
The Lindemann–Weierstrass theorem generalizes Lindemann’s 1882 proof of the transcendence of e by showing that exponentials of algebraic numbers are linearly independent. It directly implies that e, as e^1, is transcendental. This theorem is a cornerstone of transcendence theory. Read the theorem.
Which proof established that ? is transcendental?
Lindemann’s proof using euler’s identity and exponential functions
Cantor’s diagonal argument
Galois theory on field extensions
Fermat’s Last Theorem methods
Ferdinand von Lindemann proved ? is transcendental by showing that e^{?} is transcendental for any non-zero algebraic ?, and then applying it to ? = i? to show ? itself cannot satisfy any algebraic equation. This approach relies on exponential function properties. Details here.
Which of the following is guaranteed to be transcendental by the Gelfond–Schneider theorem?
2^?2
?2^?2
?^2
e + ?
The Gelfond–Schneider theorem states that if a is algebraic (?0,1) and b is algebraic irrational, then any value of a^b is transcendental. Here a=2 (algebraic) and b=?2 (algebraic irrational), so 2^?2 is transcendental. Learn more.
Which of the following describes the number e?
Transcendental
Algebraic irrational
Rational
Algebraic integer
It was proved in 1873 by Charles Hermite that e cannot satisfy any non-zero polynomial equation with rational coefficients, establishing its transcendence. This was the first important transcendence proof. e is not algebraic or rational. Source.
Consider the Liouville constant L = ?_{n=1}^? 10^{-n!}. What is its classification?
Transcendental
Algebraic irrational
Rational
Algebraic integer
Liouville constructed this constant as a specific example of a transcendental number, by demonstrating it can be approximated extremely well by rationals. This shows it cannot satisfy any algebraic equation. It was the first explicit transcendental number. Read about Liouville’s constant.
What is the Lebesgue measure of the set of algebraic numbers on the real line?
Zero
One
Infinity
Undefined
Algebraic numbers form a countable set, and any countable subset of ? has Lebesgue measure zero. This implies almost all real numbers in the measure sense are transcendental. It’s a fundamental result in real analysis. Details.
What does the Gelfond–Schneider theorem assert about a^b when a is algebraic ?0,1 and b is algebraic irrational?
a^b is transcendental
a^b is algebraic
a^b is rational
a^b is an integer
The Gelfond–Schneider theorem provides a broad class of transcendental numbers by showing that exponential expressions a^b are transcendental whenever a is an algebraic number other than 0 or 1 and b is an algebraic irrational. This result generalizes many known examples. More about it.
Which constant’s transcendence remains an open problem in mathematics?
Euler–Mascheroni constant ?
?
e
Liouville’s constant
The Euler–Mascheroni constant ? is defined by a limit involving the harmonic series minus logarithm, but whether it is algebraic or transcendental is still unknown. In contrast, ?, e, and Liouville’s constant have known transcendence status. This remains a famous open problem. Read about ?.
Which of the following numbers has an unknown status regarding being algebraic or transcendental?
e + ?
2^?2
?^e
e^?
It is a major open problem whether e + ? is algebraic or transcendental. In contrast, 2^?2 is known transcendental by Gelfond–Schneider, and ?^e or e^? have status settled via other results. e^? = (?1)^?i also has known properties. Discussion.
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Study Outcomes
Understand Algebraic vs Transcendental Definitions -
Understand the formal criteria that distinguish transcendental numbers from algebraic numbers.
Identify Transcendental Number Examples -
Identify well-known transcendental numbers such as π and e and distinguish them from algebraic numbers through defining properties.
Analyze Polynomial Root Criteria -
Analyze the polynomial-root criteria in number theory quiz questions to determine if a given number is algebraic or transcendental.
Apply Classification Skills -
Apply your knowledge to accurately classify numbers as algebraic or transcendental in challenging quiz scenarios.
Recall Historical Proofs and Context -
Recall key historical milestones and proofs that established the transcendence of numbers like π and e.
Evaluate Math Trivia Statements -
Evaluate fun math trivia statements and number theory facts to test your comprehension of transcendental concepts.
Cheat Sheet
Algebraic vs. Transcendental Definitions -
Algebraic numbers are roots of nonzero polynomials with integer coefficients, while transcendental numbers are not solutions to any such polynomial. Mnemonic: think "trans" as "beyond algebraic," a must-know for any number theory quiz.
Liouville's Theorem & Constant -
Liouville's Theorem (1844, Journal of Number Theory) shows that certain numbers can't be approximated "too well" by rationals, leading to the first explicit transcendental number. Liouville's constant L = Σn=1∞10−n! has 1s at factorial positions, guaranteeing transcendence and providing a classic example for any transcendental number quiz. Use the memory hook "factorial floors" to recall why L escapes all integer-coefficient polynomials.
Lindemann - Weierstrass Theorem -
Proved in 1882 (Cambridge University Press), this theorem establishes that eα is transcendental for any nonzero algebraic α, which directly implies π is transcendental since eiπ + 1 = 0. This landmark result closes the famous problem of squaring the circle and underpins many examples in math trivia and number theory quizzes. Recall "EW" (Euler to Weierstrass) for the flow from ex to transcendence.
Gelfond - Schneider Theorem -
Gelfond and Schneider (1934, Journal für die reine und angewandte Mathematik) show that if a≠0,1 is algebraic and b is an irrational algebraic number, then any value of ab is transcendental. For example, 2√2 (the Gelfond - Schneider constant) is one of the most famous transcendental numbers examples, solving Hilbert's seventh problem. Remember the phrase "a-to-b, rational set you free" to link algebraic bases, irrational exponents, and transcendence.
Almost All Reals Are Transcendental -
Measure theory proofs (see standard real analysis texts, Cambridge University Press) show that the set of algebraic numbers is countable, so almost all real numbers are indeed transcendental. This striking fact provides a fun math trivia tidbit for your number theory quiz: pick a random real, and you almost surely get a transcendental. To cement this, recall "countable algebraic vs. uncountable real" as a quick proof sketch.
