diff --git a/src/complex/__test__/arithmetic.spec.js b/src/complex/__test__/arithmetic.spec.js
index d72b335..966843b 100644
--- a/src/complex/__test__/arithmetic.spec.js
+++ b/src/complex/__test__/arithmetic.spec.js
@@ -32,6 +32,21 @@ describe('complex arithmetic operations', () => {
       assert.deepStrictEqual(conj(z), cplx(3, -4))
       assert.deepStrictEqual(conj(cplx(z,z)), cplx(cplx(3, -4), cplx(-3, -4)))
    })
+   it('divides complex numbers', () => {
+      const div = math.divide
+      const z = cplx(3, 4)
+      assert.deepStrictEqual(div(z, cplx(0, 1)), cplx(4, -3))
+      assert(math.equal(div(z, z), 1))
+      // should probably have a quaternion example, but it's intricate
+   })
+   it('inverts complex numbers', () => {
+      const inv = math.invert
+      assert.deepStrictEqual(inv(cplx(0, 1)), cplx(0, -1))
+      assert.deepStrictEqual(inv(cplx(3, 4)), cplx(3/25, -4/25))
+      assert.deepStrictEqual(
+         inv(cplx(cplx(1, 2), cplx(4, 2))),
+         cplx(cplx(1/25, -2/25), cplx(-4/25, -2/25)))
+   })
    it('multiplies complex numbers', () => {
       const mult = math.multiply
       const z = cplx(3, 4)
diff --git a/src/complex/arithmetic.js b/src/complex/arithmetic.js
index 70fafae..d50b993 100644
--- a/src/complex/arithmetic.js
+++ b/src/complex/arithmetic.js
@@ -20,6 +20,31 @@ export const conj = match(Complex, (math, C) => {
    return ReturnsAs(cplx, z => cplx(compConj(z.re), neg(z.im)))
 })
 
+export const divide = [
+   match([Complex, T => !T.complex], (math, [C, R]) => {
+      const div = math.divide.resolve([C.Component, R])
+      const cplx = math.complex.resolve([div.returns, div.returns])
+      return ReturnsAs(cplx, (z, r) => cplx(div(z.re, r), div(z.im, r)))
+   }),
+   match([Complex, Complex], (math, [W, Z]) => {
+      const inv = math.invert.resolve(Z)
+      const mult = math.multiply.resolve([W, inv.returns])
+      return ReturnsAs(mult, (w, z) => mult(w, inv(z)))
+   })
+]
+
+export const invert = match(Complex, (math, C) => {
+   const conj = math.conj.resolve(C)
+   const norm = math.absquare.resolve(C)
+   const div = math.divide.resolve([C.Component, norm.returns])
+   const cplx = math.complex.resolve([div.returns, div.returns])
+   return ReturnsAs(cplx, z => {
+      const c = conj(z)
+      const d = norm(z)
+      return cplx(div(c.re, d), div(c.im, d))
+   })
+})
+
 // We want this to work for complex numbers, quaternions, octonions, etc
 // See https://math.ucr.edu/home/baez/octonions/node5.html
 export const multiply = match([Complex, Complex], (math, [W, Z]) => {
diff --git a/src/core/TypePatterns.js b/src/core/TypePatterns.js
index a19cb2b..14f41c0 100644
--- a/src/core/TypePatterns.js
+++ b/src/core/TypePatterns.js
@@ -72,6 +72,7 @@ class PredicatePattern extends TypePattern {
    }
    match(typeSequence, options={}) {
       const position = options.position ?? 0
+      if (position >= typeSequence.length) return [-1, undefined]
       const actual = typeSequence[position]
       if (this.predicate(actual)) return [position + 1, actual]
       return [-1, Undefined]