diff --git a/library/core/src/num/complex.rs b/library/core/src/num/complex.rs
index 73b904d81fc53..66126c52fadad 100644
--- a/library/core/src/num/complex.rs
+++ b/library/core/src/num/complex.rs
@@ -1,4 +1,4 @@
-use crate::ops::{Add, Sub};
+use crate::ops::{Add, Neg, Sub};
 
 /// A complex number.
 #[derive(Clone, Copy, Debug, PartialEq, Eq)]
@@ -16,11 +16,46 @@ pub struct Complex<T> {
 impl<T> Complex<T> {
     /// Create a new complex number from a real and imaginary component.
     #[must_use]
-    pub fn new(re: T, im: T) -> Complex<T> {
+    pub const fn new(re: T, im: T) -> Complex<T> {
         Complex { re, im }
     }
 }
 
+#[unstable(feature = "complex_numbers", issue = "154023")]
+impl<T: Default> Default for Complex<T> {
+    fn default() -> Self {
+        Self { re: Default::default(), im: Default::default() }
+    }
+}
+
+#[unstable(feature = "complex_numbers", issue = "154023")]
+impl<T> Complex<T>
+where
+    T: Neg<Output = T>,
+{
+    /// The complex conjugate of a complex number.
+    ///
+    /// The conjugate of `a + bi` is `a - bi`: the imaginary component is negated.
+    /// Geometrically, this is a reflection across the real axis.
+    #[must_use]
+    pub fn conjugate(self) -> Self {
+        Complex { re: self.re, im: -self.im }
+    }
+}
+
+#[unstable(feature = "complex_numbers", issue = "154023")]
+impl<T: Neg> Neg for Complex<T> {
+    type Output = Complex<T::Output>;
+
+    /// Negate a complex number.
+    ///
+    /// The negation of `a + bi` is `-a - bi`: both components are negated.
+    /// Geometrically this is a rotation of 180°.
+    fn neg(self) -> Self::Output {
+        Complex::new(-self.re, -self.im)
+    }
+}
+
 #[unstable(feature = "complex_numbers", issue = "154023")]
 impl<T: Add> Add<Self> for Complex<T> {
     type Output = Complex<T::Output>;
diff --git a/library/coretests/tests/num/complex.rs b/library/coretests/tests/num/complex.rs
index 4260c0a9a27a9..c22c5b9575b3d 100644
--- a/library/coretests/tests/num/complex.rs
+++ b/library/coretests/tests/num/complex.rs
@@ -1,5 +1,16 @@
 use core::num::{Complex, Wrapping};
 
+#[test]
+fn complex_default() {
+    assert_eq!(Complex::<i32>::default(), Complex::new(0, 0));
+    assert_eq!(Complex::<f32>::default(), Complex::new(0.0, 0.0));
+
+    // The default is the additive unit.
+    let a = Complex::new(1, 2);
+    assert_eq!(a + Complex::<i32>::default(), a);
+    assert_eq!(Complex::<i32>::default() + a, a);
+}
+
 #[test]
 fn complex_addition() {
     let a = Complex::new(1, 2);
@@ -42,3 +53,23 @@ fn complex_subtraction() {
     assert_eq!(a - b, Complex::new(a.re - b.re, a.im - b.im));
     assert_eq!(a - 8.0, Complex::new(a.re - 8.0, a.im));
 }
+
+#[test]
+fn complex_conjugate() {
+    assert_eq!(Complex::new(1, 2).conjugate(), Complex::new(1, -2));
+    assert_eq!(Complex::new(1, -2).conjugate(), Complex::new(1, 2));
+
+    assert_eq!(Complex::new(1.0, 2.0).conjugate(), Complex::new(1.0, -2.0));
+    assert_eq!(Complex::new(1.0, -2.0).conjugate(), Complex::new(1.0, 2.0));
+    assert_eq!(Complex::new(1.0, f32::INFINITY).conjugate(), Complex::new(1.0, f32::NEG_INFINITY));
+}
+
+#[test]
+fn complex_negation() {
+    assert_eq!(-Complex::new(1, 2), Complex::new(-1, -2));
+    assert_eq!(-Complex::new(1, -2), Complex::new(-1, 2));
+
+    assert_eq!(-Complex::new(1.0, 2.0), Complex::new(-1.0, -2.0));
+    assert_eq!(-Complex::new(1.0, -2.0), Complex::new(-1.0, 2.0));
+    assert_eq!(-Complex::new(1.0, f32::INFINITY), Complex::new(-1.0, f32::NEG_INFINITY),);
+}