Number Systems: From Naturals to Complex Numbers

Assumption is the mother of all screw-ups, Anonymous.

Numbers, Numbers, and More Numbers

Numbers form the foundation of mathematics, serving as building blocks for counting, measuring, and describing the world around us. They can be categorized into various sets, each extending the previous one to address limitations like negativity, fractions, or incompleteness.

Natural Numbers

Natural numbers are the most basic numbers, used primarily for counting (“there are six books on the table”) and ordering (“this is the second largest city in the world”). Natural numbers date back to ancient civilizations like the Babylonians and Egyptians, who used them for trade and astronomy. Natural numbers are countable and infinite, as they can be listed in a sequence without end.

A natural number is a mathematical object. It is a number that occurs commonly in nature. As such, it is a whole, non-negative number,that is, a member of the sequence starting from 0 and increasing by 1 each time: 0, 1, 2, 3, and so on. They are obtained by starting at 0 and repeatedly adding 1. The set of natural numbers is denoted by ℕ. In set‐builder notation: ℕ = {0, 1, 2, 3, …}.

Note that the inclusion of 0 is a matter of convention; some texts include it, while others start at 1.

Natural numbers are generated recursively: start with 0, then repeatedly add 1 (the successor function). They are discrete, meaning there are gaps between them (e.g., you can’t find a natural number between 1 and 2).

Properties and operations

• Closure. Natural numbers are closed under addition and multiplication, $\forall a, b \in \mathbb{N}, a + b \in \mathbb{N}$ and $a \times b \in \mathbb{N}$, e.g., $2 + 3 = 5, 2 \times 3 = 6$.
• Not closed under subtraction or division: Subtraction can yield negatives (e.g., $2 - 3 = -1 \notin \mathbb{N}, 4 - 6 \notin \mathbb{N}$), and division can yield fractions (e.g., $1 \div 2 = 0.5 \notin \mathbb{N}, 2 \div 3 \notin \mathbb{N}$).
• Ordering: They form a well-ordered set under the usual less-than relation, meaning every non-empty subset has a least element.

Integers

To handle debts, directions, and other scenarios requiring negatives, we extend natural numbers to include their opposites. An integer number is a whole number that may be positive, negative, or zero, but has not fractional or decimal part. The set of integer numbers is denoted by ℤ. Thus, ℤ = { ···, -3, -2, -1, 0, 1, 2, 3, ···}.

Integers are symmetric around zero, meaning for every integer n, there exists an integer −n.

They are used in contexts like accounting (profits/losses, e.g., -💲15 loss is written as -15, +💲20 profit is written as 20), direction (positive/negative movement, e.g., 3 + 2: start at 3, move 2 units to the right, landing on 5; 3 +(-4): start at 3, move 4 units to the left, landing on -1), and many areas of mathematics (e.g., solving equations like $x + 3 = 0$). This models temperature (e.g., -5°C below freezing), elevations (e.g., -100m below sea level), or timelines (e.g., the Theravada tradition claims that the death of the Buddha occurred in -544 or -543 BCE).

Properties and Operations

• Closure: Closed under addition, subtraction, and multiplication $\forall a, b \in \mathbb{N}, a + b \in \mathbb{Z}, a - b \in \mathbb{Z}$, and $a \times b \in \mathbb{Z}$, e.g., $2 - 3 = -1 \in \mathbb{Z}, -2\times 3 = -6 \in \mathbb{Z}$.
• Not closed under division: Division may yield non-integers, e.g., $1 \div 2 = 0.5 \notin \mathbb{Z}, 2 \div 3 \notin \mathbb{Z}$.
• Discrete: Like naturals, integers are spaced evenly (unit steps), with no fractions between them.
• The integers form an integral domain: a commutative ring with unity and no zero divisors (if a b = 0, then a or b is 0).

Rational Numbers

Rational numbers address the need for parts of wholes, such as fractions in measurements or ratios. A rational number is any number that is of the form p/q where p and q are integers and q is not equal to 0. The set or rational numbers is denoted by the doublestruck capital letter ℚ. In set-builder notation: ℚ = {$\frac{p}{q}|~ p, q \in ℤ, q \ne 0$}.

Rational numbers can have either a terminating (e.g., $\frac{1}{2} = 0.5$) or repeating decimal (e.g., $\frac{1}{3} = 0.333...$) representation. Every rational has a unique reduced form, e.g., $ \frac{2}{4} = \frac{1}{2}$.

Rational numbers are used for measurements, ratios, and finance, e.g., a carpenter cuts a plank that is $2\frac{3}{4}$ meters long; a recipe might require $\frac{1}{2}$ cup of oil and $\frac{3}{4}$ teaspoon of salt; a classroom has 12 boys and 18 girls, so the ratio of boys to girls is $\frac{12}{18} =\frac{2}{3} = \frac{2}{3}$; a map uses a scale of 1:100,000, so the ratio “map distance : real distance” is $\frac{1}{100000}$, interest rates (e.g., 4.5% = $\frac{9}{20}$) etc.

Properties and Operations

• Closure: Closed under addition, subtraction, multiplication, and division (except by zero).
• Density: $\mathbb{Q}$ is dense in itself —between any two rationals, there’s another rational (e.g., between $\frac{1}{2}$ and $\frac{3}{4}$, $\frac{5}{8}$).
• Countable: Despite filling “gaps” between integers, rationals are countable (they can be listed via a diagonal argument).
• Field under the usual addition and multiplication: full inverses (every $q \in \mathbb{Q}$ has an additive inverse -q, and every non-zero $q \in \mathbb{Q}$ has a multiplicative inverse 1/q) and (no zero divisors, if ab = 0 with $a, b \in \mathbb{Q}$, then a = 0 or b = 0).

Rational numbers fill in “gaps” between integers but still form a countable set. In calculus, ℚ is dense in ℝ: between any two real numbers (even ones very close together, e.g., 1.414 and 1.415 are approximations of $\sqrt{2}$), you can always find a rational number (e.g., 14145).

Irrational Numbers

Not all numbers fit neatly as fractions —some defy exact ratio representation. An irrational number is any number that is not a rational number, i.e., it cannot be expressed as a fraction ratio of two integers. The decimal representation of an irrational number goes on forever without repeating -non-terminating and non-repeating. The following are examples of irrational numbers: $\sqrt{2} = 1.41421356...$ (the first irrational discovered by the Pythagoreans, who were quite horrified by it), $\pi = 3.14159265...$ (the ratio of a circle’s circumference to its diameter), $\sqrt{3}$, e (Euler’s number, the base of natural logarithms), and φ (the golden ration, a number usually found in art, architecture, and nature).

Properties

• Uncountable: There are uncountably many irrationals, far more than rationals.
• Density: Like rationals, irrationals are dense in $\mathbb{R}$ —between any two reals, there’s an irrational.
• Transcendental vs. Algebraic: Some irrationals are algebraic (roots of polynomials, e.g., $\sqrt{2} $), others transcendental (not, e.g., $\pi$, e).
• Not a standalone structure (not closed under addition/multiplication, e.g., √2 + (-√2) = 0 rational).

Real Numbers

Together, rationals and irrationals form the real numbers ℝ, a complete ordered field It is denoted by ℝ. $\mathbb{R}$ includes all points on the number line: positives, negatives, zero, fractions, and irrationals. It is uncountable and continuous —no gaps.

Key Properties

• Ordered Field: The real numbers form a complete ordered field, a field with a total order and the least upper bound property.
• Completeness (Least Upper Bound Property): Every non-empty subset of $\mathbb{R}$ bounded above has a least upper bound (supremum) in $\mathbb{R}$. This fails for $\mathbb{Q}$; e.g., the set $\{ x \in \mathbb{Q} \mid x^2 < 2 \}$ has supremum $\sqrt{2} \notin \mathbb{Q}$.
• Density: Both $\mathbb{Q}$ and irrationals are dense in $\mathbb{R}$.
• Archimedean Property: No “infinitely large” (given any y > 0, repeatedly adding some smaller positive x will eventually exceed y) or “infinitely small” positives; for any positive real $x, y$, some integer n satisfies n x > y.

Complex Numbers

Complex numbers are fundamental in various fields of mathematics, physics, and engineering. They provide a complete number system where every polynomial equation has a solution.

A complex number is specified by an ordered pair of real numbers (a, b) ∈ ℝ² and expressed or written in the form z = a + bi, where a and b are real numbers, and i is the imaginary unit, defined by the property $i^2 = -1 \iff[\text{or equivalently}] i = \sqrt{-1}, \boxed{\mathbb{C}= \{ a + bi ∣a, b ∈ ℝ\} }, e.g., 2 + 5i, 7\pi + i\sqrt{2}.$

Powers of $i$

The powers of $i$ cycle with period 4: $i^0 = 1, i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1$, and so on. Therefore, general formula: For any integer $n$: $\boxed{i^n = i^{n \mod 4}}$, e.g., $i^{23} = i^{23 \mod 4} = i^3 = -i, i^{100} = i^{100 \mod 4} = i^0 = 1, i^{-1} = \frac{1}{i} = \frac{1}{i} \cdot \frac{-i}{-i} = \frac{-i}{-i^2} = \frac{-i}{1} = -i$

Fundamental Theorem of Algebra

ℂ is algebraically closed: every non-constant polynomial with complex coefficients has at least one root in ℂ.

Equality, Real & Imaginary Parts

Two complex numbers are equal if and only if their their real and imaginary are equal: a + bi = c + di  ⟺  a = c and b = d.

Real and Imaginary Parts

In the complex number z = a + bi: a is called the real part of the complex number, denoted as ℜ(z) = a. b is called the imaginary part, denoted as ℑ(z) = b. Example: For z = 3 +4i, the real part is ℜ(z) = 3 and the imaginary part is ℑ(z) = 4.

Key observations:

• The imaginary part is a real number (it’s $b$, not $bi$).
• Real numbers: $a + 0i = a, \Im(z) = 0$, e.g., $5, -3, \pi$. The real numbers $\mathbb{R}$ are contained within the complex numbers $\mathbb{C}$: $\mathbb{R} \subset \mathbb{C}$
• Pure imaginary numbers: $0 + bi = bi, \Re(z) = 0$, e.g., $2i, -7i, i\sqrt{3}$
• Zero: $0 + 0i = 0, \Re(z) = \Im(z) = 0$.

Geometric Interpretation

Complex numbers can be visualized as points (or vectors) in a two-dimensional plane called the complex plane (or Argand plane).

We often use z, w, ζ to denote complex numbers. Each complex number a + bi corresponds to a point (a, b) in the plane, where the real part “a” represents the coordinate for the horizontal axis (real axis) and the imaginary part “b” represents the coordinate for the vertical axis (imaginary axis).

Therefore, we can identify $\mathbb{C}$ with $\mathbb{R}$ as sets, $z = a + bi \in \mathbb{C} \longleftrightarrow (a, b) \in \mathbb{R}^2$. In this sense: $\mathbb{C} \cong \mathbb{R}^2$, {(a, b): a, b ∈ ℝ} $\longleftrightarrow$ {(a, b) or a +bi: a, b ∈ ℝ}

a + i·0, 0 + i·b, 0 + i·1, 0 + i·(-1), 0 + i·(-b) are often abbreviated (or written) as a, ib or bi, i, -i, and -ib respectively. We embed $\mathbb{R}$ into $\mathbb{C}$ via a ↦ a + i0. In this sense, the real numbers are contained within the complex numbers. In other words, we identify a real number a with the complex number a + i·0. Besides, numbers of the form 0 + ib = ib are called purely imaginary numbers.

Arithmetic of Complex Numbers

• Addition. Add corresponding parts: (a +bi) + (c +di) = (a +c) + (b +d)i, e.g., (1 + 3i) + (3 + 4i) = 4 + 7i. Geometrically: Vector addition (parallelogram rule).
• Subtraction. Subtract corresponding parts: (a +bi) − (c +di) = (a −c) + (b −d)i, e.g., (1 + 3i) - (3 + 4i) = -2 -i.
• Multiplication. We can multiply complex numbers by applying the distributive property and using i² = −1: $(a + bi)(c + di) = ac + adi + bci + bdi^2 = ac + adi + bci - bd = (ac - bd) + (ad + bc)i$. Formula: $\boxed{(a + bi)(c + di) = (ac - bd) + (ad + bc)i}$, e.g., (1 + 3i)·(4 + 2i) = 4 + 6i² + 12i + 2i =[i²=-1] 4 -6 + 14i = -2 + 14i.

Complex conjugates & Modulus

The complex conjugate of a complex number z = a + bi is denoted by $\bar z$ and defined by $\bar z = a -bi$, e.g, $\overline{3 + i} = 3 -i$, $\overline{\sqrt{2}-\frac{\pi}{3}i } = \sqrt{2} +\frac{\pi}{3}i, \overline{3} = -3, \overline{2i} = -2i$.

If you plot the complex number on the complex plane, the conjugate is a reflection across the real axis (x-axis) changing (or flipping) the sign of the imaginary component while the real part remains unchanged.

The modulus (or absolute value) of $z = a + bi$ is: $\boxed{|z| = \sqrt{a^2 + b^2}}$, e.g., $|3 + 4i| = \sqrt{9 + 16} = 5, |1 - i| = \sqrt{1 + 1} = \sqrt{2}, |5| = 5, |-3i| = 3$. Geometrically, this is the distance from the origin to the point z = (a, b) in the complex plane (by the Pythagorean theorem).

The modulus of a complex number is the square root of the product of the number itself and its conjugate: $\boxed{|z| = \sqrt{z·\bar z}}$.

$z·\bar z = (a + bi)(a -bi) = a^2 + b^2, ∣z∣ = \sqrt{a^2 + b^2} = \sqrt{z·\bar z}$, e.g., z = 3 + 4i, $z·\bar z = (3 + 4i)(3 - 4i) = 3^2 + 4^2 = 9+16 = 25.$

Properties

• Involution: $\overline{\overline{z}} = z$.
• The conjugate of the sum of two complex numbers is the sum of their conjugates: $\overline {z_1 + z_2} = \overline {(a + bi) + (c+di)} = \overline{(a + c) + (b+d)i} = (a + c) - (b+d)i = (a -bi) + (c-di) = \overline {z_1}+\overline {z_2}$
• Conjugate of difference: $\overline{z_1 - z_2} = \bar{z_1} - \bar{z_2}$.
• Conjugate of products: $\overline {z_1 · z_2} = \overline {z_1}· \overline {z_2}$.
• Conjugate of quotient, $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}$.
• Conjugate of power, $\overline{z^n} = (\bar{z})^n$.
• Real part formula, $\Re(z) = \frac{z + \bar{z}}{2}$.
• Imaginary part formula, $\Im(z) = \frac{z - \bar{z}}{2i}$.
• $z$ is real, $z = \bar{z} \iff \Im(z) = 0$. A complex number is real if and only if its imaginary part is zero, making it equal to its conjugate.
• $z$ is pure imaginary, $z = -\bar{z} \iff \Re(z) = 0$. If a complex number equals its negative conjugate, then it is purely imaginary.
• Non-negativity: |z| ≥ 0
• Zero only at zero: ∣z∣ = 0   ⟺   z = 0
• The modulus of the product of two complex numbers is equal to the product of their moduli: $|z \cdot w| = \sqrt{(zw) \cdot \overline{(zw)}} = \sqrt{zw \cdot \bar{z}\bar{w}} = \sqrt{(z\bar{z})(w\bar{w})} = \sqrt{z\bar{z}} \cdot \sqrt{w\bar{w}} = |z| \cdot |w| \quad \blacksquare$
• Triangle inequality: |z + w| ≼ |z| + |w|

$|z + w|^2 = (z + w)\overline{(z + w)} = (z + w)(\bar{z} + \bar{w}) = z\bar{z} + z\bar{w} + w\bar{z} + w\bar{w} = |z|^2 + |w|^2 + z\bar{w} + \overline{z\bar{w}} = |z|^2 + |w|^2 + 2\Re(z\bar{w})$

Since $\Re(z\bar{w}) \leq |z\bar{w}| = |z||w|$: $|z + w|^2 \leq |z|^2 + 2|z||w| + |w|^2 = (|z| + |w|)^2$

Taking square roots: $|z + w| \leq |z| + |w| \quad \blacksquare$

• By induction, it can be demonstrated that $|z_1 + z_2 + \cdots z_n| \le |z_1| + |z_2| \cdots |z_n|$
• Reverse triangle inequality for complex numbers. Start with the triangle inequality applied to $z_1 - z_1$ and $z_2: |z_1| = |z_1 - z_2 + z_2| \le |z_1 - z_2| + |z_2| \leadsto |z_1| - |z_2| \le |z_1 - z_2| \leadsto |z_1 - z_2| \ge |z_1| - |z_2|$.

By symmetry (swap $z_1$ and $z_2$, and note $|z_1 - z_2| = |z_2 -z_1|$), we also have: $|z_1 - z_2| \ge |z_2| - |z_1|$. Combining these two inequalities gives: $\boxed{|z_1 - z_2| \ge \big||z_2| - |z_1|\big|}$

Division of Complex Numbers

To divide complex numbers, z₂≠0 (x₂ and y₂ are not simultaneously zero), we multiply both the numerator and the denominator by the complex conjugate of the denominator. $\frac{z_1}{z_2} = \frac{x_1+iy_1}{x_2+iy_2} = \frac{x_1+iy_1}{x_2+iy_2}\frac{x_2-iy_2}{x_2-iy_2}=\frac{x_1x_2 + y_1y_2 + i(x_2y_1-x_1y_2)}{x_2²-y_2²} = \frac{x_1x_2 + y_1y_2}{x_2²-y_2²} + i(\frac{x_2y_1-x_1y_2}{x_2²-y_2²})$

Example: Calculate $\frac{2+i}{1-3i}$

$\frac{2+i}{1-3i}$ =[To simplify the expression, multiply both numerator and denominator by the conjugate of the denominator:] $\frac{2+i}{1-3i}·\frac{1+3i}{1+3i} = \frac{-1+7i}{10} =[\text{Thus, the simplified form is:}] \frac{-1}{10}+\frac{7}{10}i$.

$(2 + i)(1 + 3i) = 2 + 6i + i + 3i^2 = 2 + 7i - 3 = -1 + 7i, (1 - 3i)(1 + 3i) = 1 - 9i^2 = 1 + 9 = 10$

Multiplicative Inverse

For a non-zero complex number $z = a + bi \neq 0$, the multiplicative inverse $z^{-1}$ is defined by $z^{-1} = \frac{1}{z} = \frac{\bar{z}}{|z|^2} = \frac{a - bi}{a^2 + b^2} = \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i$

Verification: $z \cdot z^{-1} = z \cdot \frac{\bar{z}}{|z|^2} = \frac{z\bar{z}}{|z|^2} = \frac{|z|^2}{|z|^2} = 1 \quad \checkmark$

Algebraic Properties of $\mathbb{C}$

Complex numbers form a field under addition and multiplication:

Property	Addition	Multiplication
Closure	$z_1 + z_2 \in \mathbb{C}$	$z_1 \cdot z_2 \in \mathbb{C}$
Commutativity	$z_1 + z_2 = z_2 + z_1$	$z_1 \cdot z_2 = z_2 \cdot z_1$
Associativity	$(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$	$(z_1 \cdot z_2) \cdot z_3 = z_1 \cdot (z_2 \cdot z_3)$
Identity	$z + 0 = z$	$z \cdot 1 = z$
Distributivity	$z_1·(z_2 + z_3) = z_1·z_2 + z_1·z_3$
Inverse	$z + (-z) = 0$ where −z = −a −ib	$z \cdot z^{-1} = 1$ (for $z \neq 0$)

z = a + ib $\ne 0$ and $z^{-1}=\frac{\bar{z}}{|z|^2}$ (for $z \neq 0$), $z \cdot z^{-1} = 1$.