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Introductory Real Analysis Trivia Quiz

Real analysis studies various fundamental concepts of mathematics such as calculus, geometry, algebra and number theory. Enjoy!

A multiple-choice quiz by Matthew_07. Estimated time: 5 mins.

Author
Matthew_07

Time
5 mins

Type
Multiple Choice

Quiz #
279,852

Updated
Dec 03 21

# Qns
10

Difficulty
Average

Avg Score
7 / 10

Plays
458

One at a Time - Single Page - Timed Game

Question 1 of 10

1. Literally, real analysis means the analysis of real numbers. We often encounter the terms "real numbers" (R), "integers" (Z) and "natural numbers"(N) in real analysis. Here, integers refer to the set of number ..., -2, -1, 0, 1, 2 ... while natural numbers refer to the set of number 1, 2, 3, .... Which of the following statement is true regarding the relationship between these three groups of numbers? Hint

Z is a subset of R, which is a subset of N.

N is a subset of R, which is a subset of Z..

N is a subset of Z, which is a subset of R.

Z is a subset of N, which is a subset of R.

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Question 2 of 10

2. The real number system is characterized by three important axioms. Which of the following is NOT one of them? Hint

Infinite axiom

Algebraic axiom

Completeness axiom

Order axiom

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Question 3 of 10

3. Rational numbers can be expressed as ratio of two integers with repeated or terminating decimals. On the other hand, irrational numbers refer to any real numbers that are not rational. So, is 1 a rational number or an irrational number?

Answer: (Type R for rational or I for irrational)

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Question 4 of 10

4. The square root of 2 is an irrational number. We can prove this statement by first assuming it is a rational number. By doing some analysis, later we find out that surd 2 is indeed an irrational number. Which method are we using here? Hint

Factorization

Induction

Contradiction

Inspection

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Question 5 of 10

5. Given a non-empty subset S of R (real number) on the interval [0, 5]. Then, any numbers greater than 5 is an upper bound of S since it is greater than all of the numbers contained in S. Therefore, we can say that 5.01, 5.1, 6 and 7 are all upper bounds of S. Among all these upper bound, the one with the smallest value is known as the _______ of S. Hint

Infimum

Minimum

Maximum

Supremum

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Question 6 of 10

6. The absolute value of a real non-zero number r, denoted by |r|, is always _______. Hint

Non-negative

Rational

Non-positive

Irrational

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Question 7 of 10

7. A famous theorem states that for any real number r, there exists a unique integer, symbolized by [r], which satisfies the inequality of r - 1 < [r] <= r. If [r]=[1.23], what is the value of r?

Answer: (A number)

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Question 8 of 10

8. The product of any 2 consecutive positive integers is a (an) ____ number. Hint

Odd

Irrational

Prime

Even

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Question 9 of 10

9. The real number system can be extended both ways on the real number line so that it contains negative infinity and positive infinity. Let r be a positive real number. Notice that r + infinity = infinity. Then, r - infinity = negative infinity. Also, r x positive infinity = positive infinity. What is r / positive infinity? Hint

Infinity

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Question 10 of 10

10. The mathematical analysis of real numbers is called real analysis. On the other hand, complex analysis studies the properties of complex numbers.

True

False

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Quiz Answer Key and Fun Facts

1. Literally, real analysis means the analysis of real numbers. We often encounter the terms "real numbers" (R), "integers" (Z) and "natural numbers"(N) in real analysis. Here, integers refer to the set of number {..., -2, -1, 0, 1, 2 ...} while natural numbers refer to the set of number {1, 2, 3, ...}. Which of the following statement is true regarding the relationship between these three groups of numbers?

Answer: N is a subset of Z, which is a subset of R.

Natural numbers are also known as counting numbers. Real numbers are considered as the universal set that includes integers, rational numbers and irrational numbers.

2. The real number system is characterized by three important axioms. Which of the following is NOT one of them?

Answer: Infinite axiom

The name of these three axioms can be memorized by the mnemonics CAO. The algebraic axioms deal with addition, subtraction, multiplication and division of real numbers.

Meanwhile, the completeness axioms refer to maximum and minimum values of a set of numbers. On the other hand, the order axioms lay the rule for the inequalities of two or more integers.

Answer: Rational

Notice that 1 can be written as 1/1, which is a ratio of 2 integers. Therefore, 1 is a rational number. In addition, every integer {..., -2, -1, 0, 1, 2, ...} is a rational number.

Answer: Contradiction

First, we assume that surd 2 is a rational number, m/n, where m and n are positive integers with a greatest common divisor of 1, namely gcd (m, n) = 1.

Squaring both sides, we will get 2 = (m^2)/(n^2). Then, m^2 = 2n^2. Since m^2 is a multiple of 2n^2, then m^2 must be an even number. This implies that m itself is also an even number, since the square of any even number is also even.

So, after finding out m is even, we can express m = 2k. Substituting m = 2k into the equation, we will get (2k)^2 = 2n^2, that is 4k^2 = 2n^2, which can be simplified to 2k^2 = n^2. By using the same argument, we find out that n^2 and n are even also.

We find out that both m and n are even. So, gcd (m, n) is a multiple of 2. This contradicts with our previous assumption gcd (m, n) = 1. We then arrive at our conclusion that surd 2 is indeed an irrational number.

Answer: Supremum

The supremum of a set of numbers defined on an interval is also known as the least upper bound, lub.
On the contrary, the infimum of a set of numbers is also known as the greatest lower bound, glb.

6. The absolute value of a real non-zero number r, denoted by |r|, is always _______.

Answer: Non-negative

A very famous inequality that is derived from the concept of this absolute value is the triangle inequality, which states that |a + b| is less than or equals to |a| + |b|.

7. A famous theorem states that for any real number r, there exists a unique integer, symbolized by [r], which satisfies the inequality of r - 1 < [r] <= r. If [r]=[1.23], what is the value of r?

Answer: 2

Notice that 1 is less than 1.23 which is less than or equals to 2. So, the correct answer is 2.

8. The product of any 2 consecutive positive integers is a (an) ____ number.

Answer: Even

We can prove this statement. Let n be the first integer. So the second integer is (n + 1). Observe that n (n + 1) = n^2 + n.

Now, if n is odd, then n^2 is odd also. Therefore, n^2 + n is an addition of 2 odd numbers which will end up in an even number.

If n is even, n^2 is also even. The addition of n^2 + n, which is the addition of 2 even numbers will also end up in an even number.

Answer: 0

When a number r is divided by positive infinity, you can imagine r is being divided by a very big number, let say 1, 000, 000, 000. Let say r = 1. So, the answer obtained will be very small, and this answer approaches 0.

10. The mathematical analysis of real numbers is called real analysis. On the other hand, complex analysis studies the properties of complex numbers.

Answer: True

A complex number is such as 2+3i, where i is the square root of -1. Here, the imaginary number is 3i, while 2 is the real number. Both of them make up a complex number.
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I hope you enjoyed playing this quiz and learned something as well. Thanks for playing and have a nice day! ;)

Source: Author Matthew_07

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