Hierarchy of the Numbers

Number Theory Level 2

$\mathbb{N}$ is the set of natural numbers,
$\mathbb{Q}$ is the set of rational numbers,
$\mathbb{R}$ is the set of real numbers,
and $\mathbb{Z}$ is the set of integers,

then what is the correct ordering of the containment of these sets in each other?

\mathbb{R} \subset \mathbb{Q} \subset \mathbb{Z} \subset \mathbb{N}

\mathbb{Z} \subset \mathbb{N} \subset \mathbb{Q} \subset \mathbb{R}

\mathbb{Q} \subset \mathbb{N} \subset \mathbb{Z} \subset \mathbb{R}

\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

2 solutions

Ricky Ramcharitar
Jan 18, 2016

$\mathbb{N}$ $\rightarrow$ Natural Numbers: 1,2,3,4,5, ...

$\mathbb{Z}$ $\rightarrow$ Integers: ...-3,-2,-1,0,1,2,3, ...

Contains the Set of Natural Numbers (i.e. $\mathbb{N}$ )

$\mathbb{Q}$ $\rightarrow$ Rational Numbers: Any number that can be written as a fraction --> $\dfrac{m}{n}$

Contains the Sets of Natural Numbers and Integers (i.e. $\mathbb{Z}$ and $\mathbb{N}$ )

$\mathbb{R}$ $\rightarrow$ Real Numbers: Any number that is not complex (i.e. all Rational and Irrational Numbers) --> $\pi, e, -\dfrac{1}{3}, \sqrt{0}, \sqrt{2}, \sqrt{\dfrac{4}{7}}$ , etc...

Contains the Sets of Natural Numbers, Integers, and Rational Numbers (i.e. $\mathbb{N}$ , $\mathbb{Z}$ , and $\mathbb{Q}$ )

Therefore:

$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$

Specifically where m and n are integers. Also, some reals can't be expressed with square roots. Pi, for example, and e, and the Euler-Mascheroni constant. In fact, I'd say "most" reals can't be expressed with square roots. You need to fix your definition.

Kieran Kaempen - 5 years, 4 months ago

I also wrote that the real numbers contain all rational and irrational numbers, not just positive square roots.

I fixed it up a bit to highlight all rational and irrational at the beginning.

Ricky Ramcharitar - 5 years, 4 months ago

Fahim Abid
Jan 16, 2016

As N, Z, Q, R contains natural, integers, rational and real numbers respectively, that is why N ⊂ Z ⊂ Q ⊂ R

Hierarchy of the Numbers

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