dy/dx

FOUNDATIONS 01

Number Systems

Numbers are extended step by step whenever an earlier system cannot represent what mathematics needs.

The main idea

We begin with numbers for counting, then extend them to represent negative values, fractions, irrational quantities, all real values and finally complex numbers.

Each larger number system contains the earlier one:

\[\boxed{\mathbb N\subset\mathbb Z\subset\mathbb Q\subset\mathbb R\subset\mathbb C}\]

For example, every integer is rational, but not every rational number is an integer.

1. Natural numbers \(\mathbb N\)

Natural numbers are used for counting. They start at \(1\), or at \(0\) under another common convention, and continue without end.

\[\mathbb N=\{1,2,3,4,5,\ldots\}.\]
1234567Natural numbers continue toward +∞

2. Integers \(\mathbb Z\)

Integers add zero and negative whole numbers, allowing us to represent values below zero.

\[\mathbb Z=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}.\]
−3−2−10123Extends toward −∞ and +∞

3. Rational numbers \(\mathbb Q\)

A rational number can be written as

\[\frac pq,\qquad p,q\in\mathbb Z,\quad q\ne0.\]

Fractions, integers, terminating decimals and repeating decimals are rational. There are infinitely many rational numbers between any two distinct rational numbers.

01/41/31/22/33/41Zoomed view between 0 and 1Infinitely many more rational numbers lie between these marked values.

4. Irrational numbers

Irrational numbers cannot be written as a fraction of two integers. Examples include \(\sqrt2\), \(\pi\) and \(e\).

They are still ordinary points on the real number line.

01234√2 ≈ 1.414π ≈ 3.142

5. Real numbers \(\mathbb R\)

Rational and irrational numbers together form the real numbers. Every point on the ordinary number line corresponds to exactly one real number.

−2−10123The real number line is continuous and extends toward −∞ and +∞.

6. Complex numbers \(\mathbb C\)

A complex number has the form

\[z=a+bi,\qquad a,b\in\mathbb R,\qquad i^2=-1.\]

Complex numbers require two coordinates, so they are represented on a plane rather than on a one-dimensional number line.

real axisimaginary axisz = 2 + 3i23i0
Key takeaways

Natural numbers count. Integers add negative values and zero. Rational numbers add exact fractions. Rational and irrational numbers together form the real numbers. Complex numbers extend the real numbers with an imaginary direction.

Why this matters for calculus

Most introductory calculus is developed on \(\mathbb R\), because quantities such as time, position and temperature can vary continuously. Later mathematics also studies calculus for complex-valued functions.

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