The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers

¶ This setting extends The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness: its conventions and notation are in force, as extended by the clause identification below.

¶ $\mathbb{C}$, its operations $+$ and $\cdot$, and $i$ are as in The Complex Numbers: Pairs of Real Numbers with Their Addition and Multiplication, and the Imaginary Unit §complex, The Complex Numbers: Pairs of Real Numbers with Their Addition and Multiplication, and the Imaginary Unit §operations and The Complex Numbers: Pairs of Real Numbers with Their Addition and Multiplication, and the Imaginary Unit §imaginary-unit. $\mathbb{C}$, with $+$, $\cdot$ and the zero and unit given in The Complex Numbers Form a Field Containing the Real Numbers: i Squared Is -1 and Not Real, the Canonical Form a + bi and Its Arithmetic, and Numerals §field, is a field by that clause, so the notation of Commutative Rings, Fields and Ordered Fields: Standard Notation §rings and Commutative Rings, Fields and Ordered Fields: Standard Notation §fields applies to it: negatives, differences, powers, finite sums and products, reciprocals and quotients. Its zero and its unit are written $0$ and $1$.

¶ As for $\mathbb{N}_{0}$ in The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §numbers-only and for $\mathbb{R}$ in The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §numbers-only, a complex number is used as a number only, never as a set or as an ordered pair: its description as a pair of real numbers is used only in The Complex Numbers: Pairs of Real Numbers with Their Addition and Multiplication, and the Imaginary Unit and The Complex Numbers Form a Field Containing the Real Numbers: i Squared Is -1 and Not Real, the Canonical Form a + bi and Its Arithmetic, and Numerals.

¶ Let $\iota:\mathbb{R}\to\mathbb{C}$ be the map of The Complex Numbers Form a Field Containing the Real Numbers: i Squared Is -1 and Not Real, the Canonical Form a + bi and Its Arithmetic, and Numerals §embedding. It is injective and preserves $0$, $1$, sums, products, negatives and reciprocals, by The Complex Numbers Form a Field Containing the Real Numbers: i Squared Is -1 and Not Real, the Canonical Form a + bi and Its Arithmetic, and Numerals §embedding; hence it preserves finite sums and products and powers, by Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals §homomorphism and Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals §reindexing. The element of $\mathbb{C}$ that an $n\in\mathbb{N}_{0}$ denotes by Commutative Rings, Fields and Ordered Fields: Standard Notation §numerals is $\iota$ of the element of $\mathbb{R}$ that $n$ denotes, by The Complex Numbers Form a Field Containing the Real Numbers: i Squared Is -1 and Not Real, the Canonical Form a + bi and Its Arithmetic, and Numerals §numerals; the latter is the real number with which $n$ is identified, by The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §embedding and The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §identification, so the two readings of $n$ in $\mathbb{C}$, by Commutative Rings, Fields and Ordered Fields: Standard Notation §numerals and by the clause identification below, agree.

¶ A real number, and a set introduced as a subset of $\mathbb{R}$, is identified with its image under $\iota$; likewise an element of $\mathbb{N}_{0}$, $\mathbb{Z}$ or $\mathbb{Q}$, and a set introduced as a subset of one of these, is identified with its image under the composite of $\iota$ with the map by which The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §identification identifies it with a real number or a subset of $\mathbb{R}$. This is done wherever a complex number or a subset of $\mathbb{C}$ is required, in the senses listed in The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals §identification with $\mathbb{C}$ as the larger set; conversely, where a real number or a subset of $\mathbb{R}$, or an element or a subset of one of the smaller sets, is required, such as an operand of $\le$ or $<$, a complex number or a subset of $\mathbb{C}$ of this form stands for the element or subset of which it is the image. In this sense $\mathbb{N}\subseteq\mathbb{N}_{0}\subseteq\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}\subseteq\mathbb{C}$.

¶ By the clause embedding, The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §agreement, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §union, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §composition, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §membership, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §intersection, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §difference and Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §inclusion, this is well defined, and equality, $0$, $1$, sums, products, finite sums and products, powers, negatives, differences, reciprocals, quotients, membership in subsets, unions, intersections, set differences and inclusions agree whether they are formed in the smaller set or in $\mathbb{C}$. This setting introduces no order and no absolute value on $\mathbb{C}$: in it $\le$, $<$ and $|x|$ are those of $\mathbb{R}$ and are applied to real numbers only, unless an item adopting it defines otherwise; bounds, suprema and infima of a subset of $\mathbb{R}$ are formed in $\mathbb{R}$.

¶ Every $z\in\mathbb{C}$ is $z=a+bi$ for exactly one choice of real numbers $a$ and $b$, by The Complex Numbers Form a Field Containing the Real Numbers: i Squared Is -1 and Not Real, the Canonical Form a + bi and Its Arithmetic, and Numerals §canonical read through the clause identification. Read through the clause identification in the same way, for all real numbers $a$, $b$, $c$ and $d$, $i\neq a$, by The Complex Numbers Form a Field Containing the Real Numbers: i Squared Is -1 and Not Real, the Canonical Form a + bi and Its Arithmetic, and Numerals §not-real, and

$$(a+bi)+(c+di)=(a+c)+(b+d)i,\qquad (a+bi)(c+di)=(ac-bd)+(ad+bc)i,$$

by The Complex Numbers Form a Field Containing the Real Numbers: i Squared Is -1 and Not Real, the Canonical Form a + bi and Its Arithmetic, and Numerals §arithmetic. Finally $i^{2}=-1$, by The Complex Numbers Form a Field Containing the Real Numbers: i Squared Is -1 and Not Real, the Canonical Form a + bi and Its Arithmetic, and Numerals §imaginary-unit, with $i^{2}=i\cdot i$ by Arithmetic and Order of the Natural Numbers §digits, Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality §exponents and Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality §product.