A complex number has real coordinates a and b and an imaginary unit satisfying i²=−1.
Key knowledge
A complex number is written z=a+bi with real a and b and i²=−1. Its real part is a and its imaginary part is b. It is represented by (a,b) in the complex plane; addition combines the two components and multiplication uses i²=−1.
The imaginary part is the real coefficient b, not the product bi.
When b=0 the number is real; when a=b=0 it is zero.
The actual selected NCERT chapter inspection is the Math/Information author’s continuation evidence; this editor binds that exact proposal. No new whole-chapter proof or all complex-analysis review is inferred.
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References
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Complex coordinates, equality, addition and multiplication.
The Argand plane is §4.5; §4.4 covers modulus and conjugation. Preserve the valid page-fragment citations. Citation metadata only: complex-number statements, equations, conditions and page-fragment URLs remain unchanged.
2026-10-10 · Compared definitions and operations; retained NCERT's coefficient definition of imaginary part and excluded misleading division restrictions in both textbook discussions.
2026-10-11 · Continuation author/editor source-check scope: The Argand plane is §4.5; §4.4 covers modulus and conjugation. Preserve the valid page-fragment citations. Citation metadata only: complex-number statements, equations, conditions and page-fragment URLs remain unchanged. The actual selected NCERT chapter inspection is the Math/Information author’s continuation evidence; this editor binds that exact proposal. No new whole-chapter proof or all complex-analysis review is inferred. Exact independent adoption review is pending; historical reviews retain their original date and scope.
Foundational review
2026-10-11 · Historical approval retained from its original dated scope: Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication. Continuation 2026-10-11: math-complex-numbers: The Argand plane is §4.5; §4.4 covers modulus and conjugation. Preserve the valid page-fragment citations. Citation metadata only: complex-number statements, equations, conditions and page-fragment URLs remain unchanged. Unchanged claim and inventory decisions are carried forward; this delta is not a new whole-field, whole-document or human certification.
Sources from other places
Agreement within this scope
Complex numbers have real components, imaginary coefficient, plane representation and algebra using i squared equals minus one.
The imaginary part is the coefficient, not its product with i. math-complex-numbers: The Argand plane is §4.5; §4.4 covers modulus and conjugation. Preserve the valid page-fragment citations. Citation metadata only: complex-number statements, equations, conditions and page-fragment URLs remain unchanged. The earlier independent complex-number comparison is retained; this delta corrects the NCERT reference locator/edition, not its algebra.
University of Cape Town, Department of Mathematics and Applied Mathematics
A complex number is written z=a+bi with real a and b and i²=−1. Its real part is a and its imaginary part is b. It is represented by (a,b) in the complex plane; addition combines the two components and multiplication uses i²=−1.
Every nonconstant polynomial with complex coefficients has a complex root. Repeated factorization expresses a degree-n polynomial as a product of n linear factors times its leading coefficient, counting multiplicity.
A function is holomorphic on an open set when it is complex differentiable at every point. For f=u+iv with C¹ real components, holomorphy is equivalent to uₓ=vᵧ and uᵧ=−vₓ.
The one-sided Laplace transform is F(s)=∫₀∞e⁻ˢᵗf(t)dt. For piecewise continuous f of exponential type a, it converges absolutely when Re s>a; under corresponding bounds for f′, L{f′}=sF(s)−f(0).
For a smooth rapidly decreasing f whose derivatives also decrease faster than any power, define f̂(ξ)=∫Rⁿe⁻ⁱˣ·ξf(x)dx. Then f̂ has the same property and f(x)=(2π)⁻ⁿ∫Rⁿeⁱˣ·ξf̂(ξ)dξ.
For aₙ=saₙ₋₁+taₙ₋₂ with t≠0, distinct roots r₁,r₂ of r²−sr−t give aₙ=Ar₁ⁿ+Br₂ⁿ; a repeated root r gives aₙ=(A+Bn)rⁿ. Determine A,B from a₀,a₁ and verify both recurrence and initial values.
For N complex coefficients aⱼ define yₖ=Σ_(j=0)^(N−1)aⱼωᴶᴷ, with ω=e²πⁱ⁄ᴺ. The inverse is aⱼ=N⁻¹Σ_(k=0)^(N−1)yₖω⁻ᴶᴷ. This converts polynomial coefficients to values at the Nth roots of unity.
At a point a where f′(a)≠0, a holomorphic map locally preserves oriented angles: tangent vectors are multiplied by f′(a), rotating by arg f′(a) and scaling by |f′(a)|. A local inverse exists; zero derivatives can destroy conformality.
T(z)=(az+b)/(cz+d), ad−bc≠0, extends to a bijection of C∪{∞}, with T(−d/c)=∞ and T(∞)=a/c when c≠0. It maps generalized circles (lines or circles) to generalized circles. For example (z−i)/(z+i) maps the upper half-plane bijectively to the unit disk.
On y′=λy with fixed h>0, a one-step method y_(k+1)=R(hλ)y_k is absolutely stable for a nonzero start exactly when |R(hλ)|≤1; strict inequality yields decay. Forward Euler has R(z)=1+z and region |1+z|≤1, so real λ<0 requires h≤2/|λ|.
For a complex 2π-periodic Riemann-integrable f, cₙ=(2π)⁻¹∫₋π^πf(x)e⁻ⁱⁿˣdx and s_N=Σ|n|≤N cₙeⁱⁿˣ satisfy ∫|f−s_N|²→0. Parseval gives (2π)⁻¹∫₋π^π|f|²=Σₙ∈Z|cₙ|². This is mean-square convergence, which does not assert pointwise convergence everywhere.