Classification of Real Numbers: Problems with Solutions

Integers(whole numbers): $Z=\left\{......-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7.......\right\}$

Rational numbers: $Q=\left\{ ......-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7......\frac{3}{2},-\frac{1}{5},\frac{4}{3},....\right\}$

Irrational numbers: $I=\left\{ \sqrt{2},-\sqrt{5},e,\pi ....\right\} $. In the image R\Q represents the set of irrational numbers.

Real numbers: $R=Q\cup I$ - the real numbers are the union of rational and irrational.

Problem 1

Classify the numbers [tex]\frac{7}{5}[/tex], [tex]0[/tex], [tex]-2.4[/tex], [tex]e+1[/tex] and [tex]3\times 10^{6}[/tex].
To which of the sets [tex]\mathbb{N}[/tex], [tex]\mathbb{Z}[/tex], [tex]\mathbb{Q}[/tex], [tex]\mathbb{I}[/tex], [tex]\mathbb{R}[/tex] does each of them belong?
Open the solution to check your answer.

Number	[tex]\mathbb{N}[/tex]	[tex]\mathbb{Z}[/tex]	[tex]\mathbb{Q}[/tex]	[tex]\mathbb{I}[/tex]	[tex]\mathbb{R}[/tex]
[tex]\frac{7}{5}[/tex]	No	No	Yes	No	Yes
[tex]0[/tex]	No	Yes	Yes	No	Yes
[tex]-2.4[/tex]	No	No	Yes	No	Yes
[tex]e+1[/tex]	No	No	No	Yes	Yes
[tex]3\times 10^{6}[/tex]	Yes	Yes	Yes	No	Yes

Problem 2

Which of the following statements is true?

A) [tex]\mathbb{Q}\subseteq \mathbb{I}[/tex]

B) [tex]\mathbb{N}\subseteq \mathbb{Q}[/tex]

C) [tex]\mathbb{Z}\nsubseteq \mathbb{R}[/tex]

D) A, B and C are all true

Problem 3

Is it true that [tex]3\in\mathbb{Z}[/tex]?

Yes

Problem 4

Is it true that [tex]-3\in\mathbb{N}[/tex]?

Yes

Problem 5

Is it true that [tex]\sqrt{2}\in\mathbb{Q}[/tex]?

Yes

Problem 6

[tex]\frac{3}{4}[/tex] is a rational number and [tex]2[/tex] is a natural number.
To which of these sets does [tex]\frac{3}{4}+2[/tex] belong?

[tex]\mathbb{I}[/tex]

[tex]\mathbb{Q}[/tex]

[tex]\mathbb{N}[/tex]

[tex]\mathbb{Z}[/tex]

Problem 7

Is the following statement true?
Every natural number is also an integer.

Yes

Problem 8

Is the following statement true?
Every natural number is also a rational number.

Yes

Problem 9

Is the following statement true?
Every irrational number can be written as a fraction.

Yes

Problem 10

Is it true that [tex]-2\in\mathbb{Q}[/tex]?

Yes

Problem 11

Is it true that [tex]\pi\in\mathbb{I}[/tex]?

Yes

Problem 12

How many of the following numbers are integers?
[tex]0,\ \frac{-4}{2},\ 2^{3},\ \frac{5}{2},\ e,\ \sqrt{2},\ -\sqrt{9}[/tex]

Problem 13

Consider the number [tex]n=\frac{3}{5}-2[/tex].
Which of the following statements is false?

[tex]n[/tex] is an irrational number

[tex]n=-\frac{7}{5}[/tex]

[tex]n[/tex] is a real number

[tex]n[/tex] is a rational number

Problem 14

If [tex]m[/tex] and [tex]n[/tex] are rational numbers, which of the following statements is always true?

[tex]m-n[/tex] is an integer

[tex]m-n=0[/tex]

[tex]m-n[/tex] is a rational number

None of these.

Problem 15

Is the number [tex]3.25[/tex] rational or irrational?

Rational

Irrational

Problem 16

The number [tex]n=2.151515\ldots[/tex] has infinitely many decimal places, and the block [tex]15[/tex] repeats forever.
Is [tex]n[/tex] rational or irrational?

Rational

Irrational

Problem 17

Is the number [tex]\frac{\sqrt{5}}{2}[/tex] rational or irrational?

Rational

Irrational

Problem 18

Is the product of two rational numbers always a rational number?

Yes

Problem 19

Is the product of two irrational numbers always an irrational number?

Yes

It depends.

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Classification of Real Numbers: Problems with Solutions

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