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Complex numbers and the Argand plane

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The nested number sets

Read from the smallest set to the largest. Every number in one row also belongs to every row below it. Each next set includes additional numbers.

  1. Natural numbers

    Counting numbers 1,2,3,…1,2,3,\ldots. This source starts at 1, so zero is not included in N\mathbb N here.

  2. Integers

    The counting numbers, their negatives and zero: …,−1,0,1,…\ldots,-1,0,1,\ldots.

  3. Rational numbers

    Numbers p/qp/q, where p and q are integers and q≠0q\ne0. Every integer n is rational because n=n/1n=n/1.

  4. Real numbers

    Rational and irrational numbers together. Irrationals such as 2\sqrt2, π\pi and ee are real but not rational; their decimal representations neither terminate nor recur.

  5. Complex numbers

    Numbers a+iba+ib, with a and b real and i2=−1i^2=-1. Every real a is also complex: a=a+0ia=a+0i. A complex number need not have a nonzero imaginary part.

Why is each inclusion strict?

A proper subset is contained in a larger set without being equal to it. These examples show what each larger set adds.

  • Natural numbers are a proper subset of integers

    N⊊Z\mathbb N\subsetneq \mathbb Z

    00 — Zero is an integer, but it is not natural under this source’s convention.

  • Integers are a proper subset of rational numbers

    Z⊊Q\mathbb Z\subsetneq \mathbb Q

    12\tfrac12 — One half is rational, but it is not an integer.

  • Rational numbers are a proper subset of real numbers

    Q⊊R\mathbb Q\subsetneq \mathbb R

    2\sqrt2 — The positive square root of 2 is real and irrational, so it is not rational.

  • Real numbers are a proper subset of complex numbers

    R⊊C\mathbb R\subsetneq \mathbb C

    ii — The imaginary unit is complex but not real: no real number has square −1.

Compare with the nested regions

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Nested regions show N strictly inside Z, Z inside Q, Q inside R, and R inside C. The real region outside Q represents irrational numbers. This source uses natural numbers starting at 1; every real a is complex as a+0i.
Nested regions show N strictly inside Z, Z inside Q, Q inside R, and R inside C. The real region outside Q represents irrational numbers. This source uses natural numbers starting at 1; every real a is complex as a+0i.
Nested regions show N strictly inside Z, Z inside Q, Q inside R, and R inside C. The real region outside Q represents irrational numbers. This source uses natural numbers starting at 1; every real a is complex as a+0i.
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