The imaginary part of a value that does not exist has none

Sources/.editorconfig

@@ -932,6 +932,9 @@ file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed
 [Tests/UnitTests/Convenience/ArcsineOfARealPastOneTest.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
 
+[Tests/UnitTests/Convenience/ImaginaryPartOfNaNTest.cs]
+file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
+
 [AngouriMath/Functions/Compilation/RealProducts.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
 

Sources/AngouriMath/Functions/Evaluation/Evaluation.Continuous/Evaluation.Continuous.Arithmetics.Classes.cs

@@ -826,6 +826,9 @@ public partial record Imf
             // im(prime(x)) has no value at x = -3.
             private static Entity? ImaginaryPart(Entity argument, bool isExact) => argument switch
             {
+                // A number that does not exist has no parts. NaN is a Real, whose imaginary part is 0, so
+                // im(1/0) was 0. https://github.com/asc-community/AngouriMath/issues/2253
+                Complex { IsNaN: true } => Real.NaN,
                 Complex number => number.ImaginaryPart,
                 Conjugatef(var z) => (-new Imf(z)).InnerSimplified(isExact),
                 Sumf(var left, var right) => (new Imf(left) + new Imf(right)).InnerSimplified(isExact),

Sources/Tests/UnitTests/Convenience/ImaginaryPartOfNaNTest.cs

@@ -0,0 +1,44 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using AngouriMath;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Convenience
+{
+    /// <summary>
+    /// The imaginary part of a value that does not exist has none, as its real part has none, where it was 0: <c>NaN</c>
+    /// is a <see cref="Entity.Number.Real"/>, whose imaginary part is 0.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/2253">#2253</a>
+    /// </summary>
+    [Trait("Area", "Convenience")]
+    public sealed class ImaginaryPartOfNaNTest
+    {
+        [Theory]
+        [InlineData("Im(1/0)")]
+        [InlineData("Im(arccos(pi) provided pi < 3)")]
+        [InlineData("Im(1/0) + 1")]
+        [InlineData("Re(1/0)")]
+        public void ItHasNoValue(string expression)
+        {
+            Assert.True(expression.ToEntity().Evaled.IsNaN, $"{expression} evaluates to {expression.ToEntity().Evaled}");
+            Assert.True(expression.ToEntity().Simplify().IsNaN, $"{expression} simplifies to {expression.ToEntity().Simplify()}");
+        }
+
+        [Fact]
+        public void SubstitutedItHasNoValue()
+            => Assert.True("Im(x)".ToEntity().Substitute("x", "1/0".ToEntity()).Evaled.IsNaN);
+
+        [Theory]
+        [InlineData("Im(2 + 3i)", "3")]
+        [InlineData("Im(5)", "0")]
+        [InlineData("Im(-oo)", "0")]
+        public void ANumberHasItsImaginaryPart(string expression, string part)
+            => Assert.Equal(part.ToEntity().Evaled, expression.ToEntity().Evaled);
+    }
+}