diff --git a/src/boolean/BooleanT.js b/src/boolean/BooleanT.js
index f2792ed..6f4ede6 100644
--- a/src/boolean/BooleanT.js
+++ b/src/boolean/BooleanT.js
@@ -1,6 +1,7 @@
 import {Type} from '#core/Type.js'
 
 export const BooleanT = new Type(n => typeof n === 'boolean', {
+   typeName: 'BooleanT',
    zero: false,
    one: true
 })
diff --git a/src/boolean/type.js b/src/boolean/type.js
index 989bd20..63b2ae7 100644
--- a/src/boolean/type.js
+++ b/src/boolean/type.js
@@ -1,5 +1,5 @@
 import {BooleanT} from './BooleanT.js'
-import {match} from '#core/helpers.js'
+import {match} from '#core/TypePatterns.js'
 import {Returns, Type, TypeOfTypes, Undefined} from '#core/Type.js'
 import {NumberT} from '#number/NumberT.js'
 
diff --git a/src/complex/Complex.js b/src/complex/Complex.js
index d934fd1..9dc0630 100644
--- a/src/complex/Complex.js
+++ b/src/complex/Complex.js
@@ -1,5 +1,5 @@
-import {Type} from '#core/Type.js'
-import {match} from '#core/helpers.js'
+import {Returns, Type} from '#core/Type.js'
+import {match} from '#core/TypePatterns.js'
 
 const isComplex = z => z && typeof z === 'object' && 're' in z && 'im' in z
 
@@ -66,6 +66,12 @@ function complexSpecialize(ComponentType) {
 export const Complex = new Type(isComplex, {
    specialize: complexSpecialize,
    specializesTo,
+   from: [match( // can promote any non-complex type T to Complex(T) as needed
+      // but watch out, this should be tried late, because it can preclude
+      // other more reasonable conversions like bool => number.
+      T => !T.complex,
+      (math, T) => Returns(Complex(T), r => math.complex(r, math.zero(T)))
+   )],
    refine: function(z, typer) {
       const reType = typer(z.re)
       const imType = typer(z.im)
diff --git a/src/complex/__test__/arithmetic.spec.js b/src/complex/__test__/arithmetic.spec.js
new file mode 100644
index 0000000..966843b
--- /dev/null
+++ b/src/complex/__test__/arithmetic.spec.js
@@ -0,0 +1,78 @@
+import assert from 'assert'
+import math from '#nanomath'
+import {Complex} from '../Complex.js'
+import {NumberT} from '#number/NumberT.js'
+
+const cplx = math.complex
+
+describe('complex arithmetic operations', () => {
+   it('computes absquare of complex numbers', () => {
+      assert.strictEqual(math.absquare(cplx(3, 4)), 25)
+      assert.strictEqual(math.absquare(cplx(cplx(2, 3), cplx(4,5))), 54)
+      assert.strictEqual(math.absquare(cplx(true, true)), 2)
+   })
+   it('adds complex numbers', () => {
+      const z = cplx(3, 4)
+      const add = math.add
+      assert.deepStrictEqual(add(z, cplx(-1, 1)), cplx(2, 5))
+      assert.deepStrictEqual(add(z, cplx(true, false)), cplx(4, 4))
+      assert.deepStrictEqual(add(cplx(false, true), z), cplx(3, 5))
+      assert.deepStrictEqual(
+         add(cplx(z, z), cplx(cplx(0.5, 0.5), z)),
+         cplx(cplx(3.5, 4.5), cplx(6, 8)))
+      assert.deepStrictEqual(add(z, 5), cplx(8, 4))
+      assert.deepStrictEqual(add(true, z), cplx(4, 4))
+      assert.deepStrictEqual(add(cplx(z,z), 10), cplx(cplx(13, 4), z))
+      assert.deepStrictEqual(
+         add(cplx(z,z), cplx(10,20)), cplx(cplx(13, 4), cplx(23, 4)))
+   })
+   it('conjugates complex numbers', () => {
+      const conj = math.conj
+      const z = cplx(3, 4)
+      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)
+      assert.deepStrictEqual(mult(z, z), cplx(-7, 24))
+      assert(math.equal(mult(z, math.conj(z)), 25))
+      const q0 = cplx(cplx(1, 1), math.zero(Complex(NumberT)))
+      const q1 = cplx(cplx(1, 0.5), cplx(0.5, 0.75))
+      assert.deepStrictEqual(
+         mult(q0, q1), cplx(cplx(0.5, 1.5), cplx(1.25, 0.25)))
+      assert.deepStrictEqual(
+         mult(q0, cplx(cplx(2, 0.1), cplx(1, 0.1))),
+         cplx(cplx(1.9, 2.1), cplx(1.1, -0.9)))
+   })
+   it('subtracts complex numbers', () => {
+      const z = cplx(3, 4)
+      const sub = math.subtract
+      assert.deepStrictEqual(sub(z, cplx(-1, 1)), cplx(4, 3))
+      assert.deepStrictEqual(sub(z, cplx(true, false)), cplx(2, 4))
+      assert.deepStrictEqual(sub(cplx(false, true), z), cplx(-3, -3))
+      assert.deepStrictEqual(
+         sub(cplx(z, z), cplx(cplx(0.5, 0.5), z)),
+         cplx(cplx(2.5, 3.5), cplx(0, 0)))
+      assert.deepStrictEqual(sub(z, 5), cplx(-2, 4))
+      assert.deepStrictEqual(sub(true, z), cplx(-2, -4))
+      assert.deepStrictEqual(sub(cplx(z,z), 10), cplx(cplx(-7, 4), z))
+      assert.deepStrictEqual(
+         sub(cplx(z,z), cplx(10,20)), cplx(cplx(-7, 4), cplx(-17, 4)))
+   })
+})
diff --git a/src/complex/__test__/complex.spec.js b/src/complex/__test__/complex.spec.js
deleted file mode 100644
index a9d4725..0000000
--- a/src/complex/__test__/complex.spec.js
+++ /dev/null
@@ -1,12 +0,0 @@
-import assert from 'assert'
-import math from '#nanomath'
-
-describe('complex type operations', () => {
-   it('converts to number', () => {
-      assert.deepStrictEqual(math.complex(3), {re: 3, im: 0})
-      assert.deepStrictEqual(math.complex(NaN), {re: NaN, im: NaN})
-      assert.deepStrictEqual(math.complex(3, -1), {re: 3, im: -1})
-      assert.deepStrictEqual(math.complex(false, true), {re: false, im: true})
-      assert.throws(() => math.complex(3, false), RangeError)
-   })
-})
diff --git a/src/complex/__test__/type.spec.js b/src/complex/__test__/type.spec.js
new file mode 100644
index 0000000..6ea0875
--- /dev/null
+++ b/src/complex/__test__/type.spec.js
@@ -0,0 +1,39 @@
+import assert from 'assert'
+import math from '#nanomath'
+
+const cplx = math.complex
+
+describe('complex type operations', () => {
+   it('converts to number', () => {
+      assert.deepStrictEqual(cplx(3), {re: 3, im: 0})
+      assert.deepStrictEqual(cplx(NaN), {re: NaN, im: NaN})
+      assert.deepStrictEqual(cplx(3, -1), {re: 3, im: -1})
+      assert.deepStrictEqual(cplx(false, true), {re: false, im: true})
+      assert.throws(() => cplx(3, false), RangeError)
+   })
+   it('calculates the argument of a complex number', () => {
+      assert.strictEqual(math.arg(cplx(1, Math.sqrt(3))), Math.PI/3)
+      assert.strictEqual(math.arg(cplx(true, true)), Math.PI/4)
+   })
+   it('detects associates of a complex number', () => {
+      const z = cplx(3, 4)
+      const assoc = math.associate
+      assert(assoc(z, z))
+      assert(assoc(z, cplx(-3, -4)))
+      assert(assoc(z, cplx(-4, 3)))
+      assert(assoc(z, cplx(4, -3)))
+      assert(!assoc(z, math.conj(z)))
+      const b = cplx(true, true)
+      assert(assoc(b, cplx(-1, 1)))
+      assert(assoc(cplx(1, 1), b))
+      assert(assoc(cplx(-1, -1), b))
+      assert(assoc(cplx(1, -1), b))
+      assert(assoc(cplx(-1, 1), b))
+      assert(!assoc(b, cplx(false, true)))
+      assert(!assoc(cplx(0, 1), b))
+   })
+   it('computes cis of an angle', () => {
+      assert(math.equal(math.cis(0), 1))
+      assert(math.equal(math.cis(Math.PI/3), cplx(0.5, Math.sqrt(3)/2)))
+   })
+})
diff --git a/src/complex/all.js b/src/complex/all.js
index 9795516..023c144 100644
--- a/src/complex/all.js
+++ b/src/complex/all.js
@@ -1,2 +1,5 @@
 export * as typeDefinition from './Complex.js'
+export * as arithmetic from './arithmetic.js'
+export * as relational from './relational.js'
 export * as type from './type.js'
+export * as utilities from './utils.js'
diff --git a/src/complex/arithmetic.js b/src/complex/arithmetic.js
new file mode 100644
index 0000000..d50b993
--- /dev/null
+++ b/src/complex/arithmetic.js
@@ -0,0 +1,69 @@
+import {Complex} from './Complex.js'
+import {promoteBinary, promoteUnary} from './helpers.js'
+import {ResolutionError} from '#core/helpers.js'
+import {match} from '#core/TypePatterns.js'
+import {ReturnsAs} from '#generic/helpers.js'
+
+export const absquare = match(Complex, (math, C) => {
+   const compAbsq = math.absquare.resolve([C.Component])
+   const R = compAbsq.returns
+   const add = math.add.resolve([R,R])
+   return ReturnsAs(add, z => add(compAbsq(z.re), compAbsq(z.im)))
+})
+
+export const add = promoteBinary('add')
+
+export const conj = match(Complex, (math, C) => {
+   const neg = math.negate.resolve(C.Component)
+   const compConj = math.conj.resolve(C.Component)
+   const cplx = math.complex.resolve([compConj.returns, neg.returns])
+   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]) => {
+   const conj = math.conj.resolve(W.Component)
+   if (conj.returns !== W.Component) {
+      throw new ResolutionError(
+         `conjugation on ${W.Component} returns other type (${conj.returns})`)
+   }
+   const mWZ = math.multiply.resolve([W.Component, Z.Component])
+   const mZW = math.multiply.resolve([Z.Component, W.Component])
+   const sub = math.subtract.resolve([mWZ.returns, mZW.returns])
+   const add = math.add.resolve([mWZ.returns, mZW.returns])
+   const cplx = math.complex.resolve([sub.returns, add.returns])
+   return ReturnsAs(cplx, (w, z) => {
+      const real = sub(mWZ(     w.re,  z.re), mZW(z.im, conj(w.im)))
+      const imag = add(mWZ(conj(w.re), z.im), mZW(z.re,      w.im))
+      return cplx(real, imag)
+   })
+})
+
+export const negate = promoteUnary('negate')
+export const subtract = promoteBinary('subtract')
diff --git a/src/complex/helpers.js b/src/complex/helpers.js
new file mode 100644
index 0000000..84e828f
--- /dev/null
+++ b/src/complex/helpers.js
@@ -0,0 +1,20 @@
+import {Complex} from './Complex.js'
+import {match} from '#core/TypePatterns.js'
+import {ReturnsAs} from '#generic/helpers.js'
+
+export const promoteUnary = name => match(
+   Complex,
+   (math, C) => {
+      const compOp = math.resolve(name, C.Component)
+      const cplx = math.complex.resolve([compOp.returns, compOp.returns])
+      return ReturnsAs(cplx, z => cplx(compOp(z.re), compOp(z.im)))
+   })
+
+export const promoteBinary = name => match(
+   [Complex, Complex],
+   (math, [W, Z]) => {
+      const compOp = math.resolve(name, [W.Component, Z.Component])
+      const cplx = math.complex.resolve([compOp.returns, compOp.returns])
+      return ReturnsAs(
+         cplx, (w, z) => cplx(compOp(w.re, z.re), compOp(w.im, z.im)))
+   })
diff --git a/src/complex/relational.js b/src/complex/relational.js
new file mode 100644
index 0000000..a8c58f3
--- /dev/null
+++ b/src/complex/relational.js
@@ -0,0 +1,24 @@
+import {Complex} from './Complex.js'
+import {BooleanT} from '#boolean/BooleanT.js'
+import {Returns} from '#core/Type.js'
+import {match, Any, Optional} from '#core/TypePatterns.js'
+
+export const indistinguishable = match(
+   [Complex, Complex, Optional([Any, Any])],
+   (math, [W, Z, T]) => {
+      let WComp = W.Component
+      let ZComp = Z.Component
+      if (T.length === 0) { // no tolerances
+         const same = math.indistinguishable.resolve([WComp, ZComp])
+         return Returns(
+            BooleanT, (w, z) => same(w.re, z.re) && same(w.im, z.im))
+      }
+      const [[RT, AT]] = T
+      const same = math.indistinguishable.resolve([WComp, ZComp, RT, AT])
+      return Returns(
+         BooleanT,
+         (w, z, [[rT, aT]]) => {
+            return same(w.re, z.re, rT, aT) && same(w.im, z.im, rT, aT)
+         })
+   })
+
diff --git a/src/complex/type.js b/src/complex/type.js
index 9881bc5..e83aabb 100644
--- a/src/complex/type.js
+++ b/src/complex/type.js
@@ -1,7 +1,8 @@
 import {Complex} from './Complex.js'
-import {match} from "#core/helpers.js"
 import {Returns} from "#core/Type.js"
-import {Any} from "#core/TypePatterns.js"
+import {Any, match} from "#core/TypePatterns.js"
+import {BooleanT} from '#boolean/BooleanT.js'
+import {NumberT} from '#number/NumberT.js'
 
 export const complex = [
    match(Any, (math, T) => {
@@ -25,3 +26,30 @@ export const complex = [
       return Returns(Complex(T), (r, m) => ({re: r, im: m}))
    })
 ]
+
+export const arg = match(
+    Complex(NumberT), Returns(NumberT, z => Math.atan2(z.im, z.re)))
+
+/* Returns true if w is z multiplied by a complex unit */
+export const associate = match([Complex, Complex], (math, [W, Z]) => {
+    if (Z.Component.complex) {
+        throw new Error(
+            `The group of units of type ${Z} is not yet implemented`)
+    }
+    const eq = math.equal.resolve([W, Z])
+    const neg = math.negate.resolve(Z)
+    const eqN = math.equal.resolve([W, neg.returns])
+    const mult = math.multiply.resolve([Z, Z])
+    const eqM = math.equal.resolve([W, mult.returns])
+    const negM = math.negate.resolve(mult.returns)
+    const eqNM = math.equal.resolve([W, negM.returns])
+    const iZ = math.complex(math.zero(Z.Component), math.one(Z.Component))
+    return Returns(BooleanT, (w, z) => {
+        if (eq(w, z) || eqN(w, neg(z))) return true
+        const iz = mult(iZ, z)
+        return eqM(w, iz) || eqNM(w, negM(iz))
+    })
+})
+
+export const cis = match(NumberT, Returns(Complex(NumberT), t => ({
+    re: Math.cos(t), im: Math.sin(t)})))
diff --git a/src/complex/utils.js b/src/complex/utils.js
new file mode 100644
index 0000000..2fef03a
--- /dev/null
+++ b/src/complex/utils.js
@@ -0,0 +1,9 @@
+import {Complex} from './Complex.js'
+import {match} from '#core/TypePatterns.js'
+import {ReturnsAs} from '#generic/helpers.js'
+
+export const isReal = match(Complex, (math, C) => {
+    const eq = math.equal.resolve([C.Component, C.Component])
+    const add = math.add.resolve([C.Component, C.Component])
+    return ReturnsAs(eq, z => eq(z.re, add(z.re, z.im)))
+})
diff --git a/src/core/Implementations.js b/src/core/Implementations.js
new file mode 100644
index 0000000..c53ea3e
--- /dev/null
+++ b/src/core/Implementations.js
@@ -0,0 +1,13 @@
+export class Implementations {
+   constructor(impOrImps) {
+      if (Array.isArray(impOrImps)) {
+         this.matchers = impOrImps
+      } else this.matchers = [impOrImps]
+   }
+}
+
+export class ImplementationsGenerator {
+   constructor(f) {
+      this.generate = f
+   }
+}
diff --git a/src/core/Type.js b/src/core/Type.js
index 088b9e7..555b547 100644
--- a/src/core/Type.js
+++ b/src/core/Type.js
@@ -1,6 +1,4 @@
 import ArrayKeyedMap from 'array-keyed-map'
-import {match} from './helpers.js'
-import {Passthru} from './TypePatterns.js'
 
 // Generic types are callable, so we have no choice but to extend Function
 export class Type extends Function {
@@ -12,9 +10,8 @@ export class Type extends Function {
       // let the proxy out of this function, never `this`.
       const rewired = new Proxy(this, {
          apply: (target, thisForCall, args) => {
-            const callThrough = thisForCall ?? target
-            if (callThrough.specialize) return callThrough.specialize(...args)
-            throw new TypeError(`Type ${callThrough} is not generic`)
+            if (target.specialize) return target.specialize(...args)
+            throw new TypeError(`Type ${target} is not generic`)
          },
          get: (target, prop, receiver) => {
             if (prop === 'isAproxy') return true
@@ -87,13 +84,3 @@ export const whichType = typs => Returns(TypeOfTypes, item => {
    }
    throw new TypeError(errorMsg)
 })
-
-export const typeOf = match(Passthru, math => whichType(math.types))
-
-// bootstrapping order matters, but order of exports in a module isn't
-// simply the order that the items are listed in the module. So we make
-// an explicitly ordered export of implementations for this sake:
-
-export const bootstrapTypes = {
-   Type, Undefined, TypeOfTypes, typeOf
-}
diff --git a/src/core/TypeDispatcher.js b/src/core/TypeDispatcher.js
index 202afe1..2b997e6 100644
--- a/src/core/TypeDispatcher.js
+++ b/src/core/TypeDispatcher.js
@@ -1,11 +1,14 @@
 import ArrayKeyedMap from 'array-keyed-map'
 
+import {ResolutionError, isPlainFunction, isPlainObject} from './helpers.js'
+import {Implementations, ImplementationsGenerator} from './Implementations.js'
+import {bootstrapTypes} from './type.js'
+import {Returns, whichType, Type} from './Type.js'
 import {
-   Implementations, ImplementationsGenerator, Matcher, ResolutionError,
-   isPlainFunction, isPlainObject, match, types
-} from './helpers.js'
-import {bootstrapTypes, Returns, whichType, Type} from './Type.js'
-import {matched, needsCollection, Passthru} from './TypePatterns.js'
+   matched, needsCollection, Passthru, Matcher, match
+} from './TypePatterns.js'
+
+const underResolution = Symbol('underResolution')
 
 export class TypeDispatcher {
    constructor(...specs) {
@@ -14,7 +17,6 @@ export class TypeDispatcher {
       this._behaviors = {} // maps key to a map from type vectors to results
       this._fallbacks = {} // maps key to a catchall result
       // bootstrap the instance
-      this.merge({types})
       this.merge(bootstrapTypes)
       for (const spec of specs) this.merge(spec)
    }
@@ -261,6 +263,7 @@ export class TypeDispatcher {
       if (!(key in this)) {
          throw new ReferenceError(`no method or value for key '${key}'`)
       }
+      if (!Array.isArray(types)) types = [types]
 
       const generatingDeps = this.resolve._genDepsOf?.length
       if (generatingDeps) this._addToDeps(key, types)
@@ -269,6 +272,10 @@ export class TypeDispatcher {
       // Return the cached resolution if it's there
       if (behave.has(types)) {
          const result = behave.get(types)
+         if (result === underResolution) {
+            throw new ResolutionError(
+               `recursive resolution of ${key} on ${types}`)
+         }
          if (generatingDeps
              && typeof result === 'object'
              && !(result instanceof Type)
@@ -328,13 +335,14 @@ export class TypeDispatcher {
       let theBehavior = () => undefined
       let finalBehavior
       this.resolve._genDepsOf.push([key, types])
+      behave.set(types, underResolution)
       try { // Used to make sure not to return without popping _genDepsOf
          if (!('returns' in item)) {
             // looks like a factory
             try {
                theBehavior = item(
                   DependencyRecorder(this, '', this, []),
-                  matched(template))
+                  matched(template, this))
             } catch (e) {
                e.message = `Error in factory for ${key} on ${types} `
                   + `(match data ${template}): ${e.message}`
diff --git a/src/core/TypePatterns.js b/src/core/TypePatterns.js
index eb73afd..14f41c0 100644
--- a/src/core/TypePatterns.js
+++ b/src/core/TypePatterns.js
@@ -1,4 +1,5 @@
 import {Type, Undefined} from './Type.js'
+import {isPlainFunction} from './helpers.js'
 
 export class TypePattern {
    match(typeSequence, options={}) {
@@ -64,12 +65,35 @@ class SequencePattern extends TypePattern {
    }
 }
 
+class PredicatePattern extends TypePattern {
+   constructor(predicate) {
+      super()
+      this.predicate = predicate
+   }
+   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]
+   }
+   sampleTypes() {
+      throw new Error('sampleTypes() not yet implemented for PredicatePattern')
+   }
+   equal(other) {
+      return super.equal(other) && this.predicate === other.predicate
+   }
+}
+
 export const pattern = patternOrSpec => {
    if (patternOrSpec instanceof TypePattern) return patternOrSpec
    if (patternOrSpec instanceof Type) {
       return new MatchTypePattern(patternOrSpec)
    }
    if (Array.isArray(patternOrSpec)) return new SequencePattern(patternOrSpec)
+   if (isPlainFunction(patternOrSpec)) {
+      return new PredicatePattern(patternOrSpec)
+   }
    throw new TypeError(`Can't interpret '${patternOrSpec}' as a type pattern`)
 }
 
@@ -150,9 +174,16 @@ class PassthruPattern extends TypePattern {
 export const Passthru = new PassthruPattern()
 
 // returns the template just of matched types, dropping any actual types
-export const matched = (template) => {
-   if (Array.isArray(template)) return template.map(matched)
-   return template.matched ?? template
+export const matched = (template, math) => {
+   if (Array.isArray(template)) {
+      return template.map(pattern => matched(pattern, math))
+   }
+   if (template.matched) {
+      let convert = template.convertor
+      if (!convert.returns) convert = convert(math, template.actual)
+      return convert.returns || template.matched
+   }
+   return template
 }
 
 // checks if the template is just pass-through or needs collection
@@ -163,3 +194,12 @@ export const needsCollection = (template) => {
    }
    return 'actual' in template
 }
+
+export class Matcher {
+   constructor(spec, facOrBehave) {
+      this.pattern = pattern(spec)
+      this.does = facOrBehave
+   }
+}
+
+export const match = (spec, facOrBehave) => new Matcher(spec, facOrBehave)
diff --git a/src/core/__test__/TypeDispatcher.spec.js b/src/core/__test__/TypeDispatcher.spec.js
index 37e5618..c843e59 100644
--- a/src/core/__test__/TypeDispatcher.spec.js
+++ b/src/core/__test__/TypeDispatcher.spec.js
@@ -4,8 +4,8 @@ import * as booleans from '#boolean/all.js'
 import * as generics from '#generic/all.js'
 import * as numbers from '#number/all.js'
 import {NumberT} from '#number/NumberT.js'
-import {match, ResolutionError} from "#core/helpers.js"
-import {Any} from "#core/TypePatterns.js"
+import {ResolutionError} from "#core/helpers.js"
+import {match, Any} from "#core/TypePatterns.js"
 import {Returns, NotAType} from "#core/Type.js"
 import {plain} from "#number/helpers.js"
 
diff --git a/src/core/__test__/helpers.spec.js b/src/core/__test__/helpers.spec.js
index 10849d6..46576eb 100644
--- a/src/core/__test__/helpers.spec.js
+++ b/src/core/__test__/helpers.spec.js
@@ -1,9 +1,8 @@
 import assert from 'assert'
-import {
-   Matcher, match, isPlainObject, isPlainFunction, Implementations
-} from '../helpers.js'
+import {isPlainObject, isPlainFunction} from '../helpers.js'
+import {Implementations} from '../Implementations.js'
 import {Type, Undefined, TypeOfTypes} from '../Type.js'
-import {TypePattern} from '../TypePatterns.js'
+import {match, Matcher, TypePattern} from '../TypePatterns.js'
 
 describe('Core helpers', () => {
    it('defines what Matchers are', () => {
diff --git a/src/core/helpers.js b/src/core/helpers.js
index dc05182..8d7b2b2 100644
--- a/src/core/helpers.js
+++ b/src/core/helpers.js
@@ -1,41 +1,3 @@
-import {pattern, Passthru} from './TypePatterns.js'
-
-export class Matcher {
-   constructor(spec, facOrBehave) {
-      this.pattern = pattern(spec)
-      this.does = facOrBehave
-   }
-}
-
-export class Implementations {
-   constructor(impOrImps) {
-      if (Array.isArray(impOrImps)) {
-         this.matchers = impOrImps
-      } else this.matchers = [impOrImps]
-   }
-}
-
-export const match = (spec, facOrBehave) => new Matcher(spec, facOrBehave)
-
-export class ImplementationsGenerator {
-   constructor(f) {
-      this.generate = f
-   }
-}
-
-// the archetypal example of needing an ImplementationsGenerator:
-// each TypeDispatcher must have a types property, which will be a
-// plain object of types. This must be a different object for each
-// TypeDispatcher, but the same object regardless of the types vector
-// passed to resolve. So an ordinary factory won't work, because it
-// would make a new plain object for each different types vector that
-// the property `types` was resolved with. And just a plain object
-// wouldn't work, because then every TypeDispatcher would have the same
-// collection of types (and modifying the types in one would affect them
-// all). Hence we do:
-
-export const types = new ImplementationsGenerator(() => match(Passthru, {}))
-
 export class ResolutionError extends TypeError {
    constructor(...args) {
       super(...args)
diff --git a/src/core/type.js b/src/core/type.js
new file mode 100644
index 0000000..7452dd4
--- /dev/null
+++ b/src/core/type.js
@@ -0,0 +1,26 @@
+import {ImplementationsGenerator} from './Implementations.js'
+import {Type, TypeOfTypes, Undefined, whichType} from './Type.js'
+import {match, Passthru} from './TypePatterns.js'
+
+export const typeOf = match(Passthru, math => whichType(math.types))
+
+// And now the types object itself: This property provides the archetypal
+// example of needing an ImplementationsGenerator. Each TypeDispatcher must
+// have a types property, which will be a plain object of types. This object
+// must be different for each TypeDispatcher, but within a TypeDispatcher,
+// it must be the same object regardless of the types vector passed to resolve.
+// So an ordinary factory won't work, because it would make a new plain object
+// for each different types vector that the property `types` was resolved with.
+// And just a plain object wouldn't work, because then every TypeDispatcher
+// would have the same collection of types (and modifying the types in one
+// would affect them all). Hence we do:
+
+export const types = new ImplementationsGenerator(() => match(Passthru, {}))
+
+// bootstrapping order matters, but order of exports in a module isn't
+// simply the order that the items are listed in the module. So we make
+// an explicitly ordered export of implementations for this sake:
+
+export const bootstrapTypes = {
+   types, Type, Undefined, TypeOfTypes, typeOf
+}
diff --git a/src/coretypes/relational.js b/src/coretypes/relational.js
index 4192dc1..a45cadf 100644
--- a/src/coretypes/relational.js
+++ b/src/coretypes/relational.js
@@ -1,5 +1,5 @@
-import {match} from '#core/helpers.js'
 import {TypeOfTypes, Undefined} from '#core/Type.js'
+import {match} from '#core/TypePatterns.js'
 import {boolnum} from '#number/helpers.js'
 
 export const indistinguishable = [
diff --git a/src/coretypes/utils.js b/src/coretypes/utils.js
index 4386ac1..e89b74c 100644
--- a/src/coretypes/utils.js
+++ b/src/coretypes/utils.js
@@ -1,6 +1,5 @@
-import {match} from '#core/helpers.js'
 import {NotAType, Returns, TypeOfTypes} from '#core/Type.js'
-import {Any} from "#core/TypePatterns.js"
+import {match,Any} from "#core/TypePatterns.js"
 import {boolnum} from "#number/helpers.js"
 
 export const zero = [
diff --git a/src/generic/__test__/utils.spec.js b/src/generic/__test__/utils.spec.js
index e178a2d..f8b6d53 100644
--- a/src/generic/__test__/utils.spec.js
+++ b/src/generic/__test__/utils.spec.js
@@ -1,5 +1,6 @@
 import assert from 'assert'
 import math from '#nanomath'
+import {NumberT} from '#number/NumberT.js'
 
 describe('generic utility functions', () => {
    it('tests whether an element is zero', () => {
@@ -9,5 +10,16 @@ describe('generic utility functions', () => {
       assert(isZero(false))
       assert(!isZero(true))
       assert(isZero(undefined))
+      assert(isZero(math.types.Complex(NumberT).zero))
+      assert(isZero(math.complex(-2e-16, 4e-17)))
+      assert(!isZero(math.complex(true)))
+   })
+   it('tests whether an element is real', () => {
+      const {isReal} = math
+      assert(isReal(Infinity))
+      assert(isReal(false))
+      assert(isReal(math.types.Complex(NumberT).one))
+      assert(isReal(math.complex(-3.25, 4e-16)))
+      assert(!isReal(math.complex(3, 4)))
    })
 })
diff --git a/src/generic/arithmetic.js b/src/generic/arithmetic.js
index fe1556f..cf67fdc 100644
--- a/src/generic/arithmetic.js
+++ b/src/generic/arithmetic.js
@@ -1,7 +1,7 @@
-import {match} from '#core/helpers.js'
 import {Returns} from '#core/Type.js'
-import {Any} from '#core/TypePatterns.js'
+import {match, Any} from '#core/TypePatterns.js'
 
+export const conj = match(Any, (_math, T) => Returns(T, a => a))
 export const square = match(Any, (math, T) => {
    const mult = math.multiply.resolve([T, T])
    return Returns(mult.returns, a => mult(a, a))
diff --git a/src/generic/config.js b/src/generic/config.js
index 9f421ec..72f8ced 100644
--- a/src/generic/config.js
+++ b/src/generic/config.js
@@ -1,5 +1,5 @@
-import {ImplementationsGenerator, match} from '#core/helpers.js'
-import {Passthru} from '#core/TypePatterns.js'
+import {ImplementationsGenerator} from '#core/Implementations.js'
+import {match, Passthru} from '#core/TypePatterns.js'
 
 export const config = new ImplementationsGenerator(
     () => match(Passthru, {relTol: 1e-12, absTol: 1e-15}))
diff --git a/src/generic/relational.js b/src/generic/relational.js
index 810c06a..bd23f92 100644
--- a/src/generic/relational.js
+++ b/src/generic/relational.js
@@ -1,7 +1,6 @@
 import {ReturnsAs} from './helpers.js'
-import {match} from '#core/helpers.js'
 import {Returns} from '#core/Type.js'
-import {Any, Passthru, matched} from '#core/TypePatterns.js'
+import {Any, Passthru, match, matched} from '#core/TypePatterns.js'
 import {boolnum} from '#number/helpers.js'
 
 export const equal = match([Any, Any], (math, [T, U]) => {
@@ -18,7 +17,7 @@ export const equal = match([Any, Any], (math, [T, U]) => {
       return boolnum(() => false)(math)
    }
    // Get the type of the first argument to the matching checker:
-   const ByType = matched(exactChecker.template).flat()[0]
+   const ByType = matched(exactChecker.template, math).flat()[0]
    // Now see if there are tolerances for that type:
    const typeConfig = math.resolve('config', [ByType])
    if ('relTol' in typeConfig) {
diff --git a/src/generic/utils.js b/src/generic/utils.js
index d4c8f87..d696d6c 100644
--- a/src/generic/utils.js
+++ b/src/generic/utils.js
@@ -1,8 +1,10 @@
 import {ReturnsAs} from './helpers.js'
-import {ResolutionError, match} from '#core/helpers.js'
-import {Returns} from '#core/Type.js'
-import {Passthru} from "#core/TypePatterns.js"
+import {ResolutionError} from '#core/helpers.js'
+import {Passthru, match} from '#core/TypePatterns.js'
+import {boolnum} from '#number/helpers.js'
 
+// Most types are real. Have to make sure to redefine on all non-real types
+export const isReal = match(Passthru, boolnum(() => true))
 export const isZero = match(Passthru, (math, [T]) => {
    if (!T) { // called with no arguments
       throw new ResolutionError('isZero() requires one argument')
diff --git a/src/nanomath.js b/src/nanomath.js
index c133cbf..ddeeeba 100644
--- a/src/nanomath.js
+++ b/src/nanomath.js
@@ -5,6 +5,16 @@ import * as numbers from './number/all.js'
 import * as complex from './complex/all.js'
 import {TypeDispatcher} from '#core/TypeDispatcher.js'
 
-const math = new TypeDispatcher(booleans, coretypes, generics, numbers, complex)
+// At the moment, since we are not sorting patterns in any way,
+// order matters in the construction. Patterns that come later in
+// the following list will be tried earlier. (The rationale for that
+// ordering is that any time you merge something, it should supersede
+// whatever has been merged before.)
+// Hence, in building the math instance, we put generics first because
+// they should only kick in when there are not specific implementations,
+// and complex next becausewe want its conversion (which converts _any_
+// non-complex type to complex, potentially making a poor overload choice)
+// to be tried last.
+const math = new TypeDispatcher(generics, complex, booleans, coretypes, numbers)
 
 export default math
diff --git a/src/number/NumberT.js b/src/number/NumberT.js
index c3816a0..415a37d 100644
--- a/src/number/NumberT.js
+++ b/src/number/NumberT.js
@@ -1,9 +1,10 @@
 import {Type} from '#core/Type.js'
-import {match} from '#core/helpers.js'
+import {match} from '#core/TypePatterns.js'
 import {BooleanT} from '#boolean/BooleanT.js'
 
 export const NumberT = new Type(n => typeof n === 'number', {
     from: match(BooleanT, math => math.number.resolve([BooleanT])),
+    typeName: 'NumberT', // since used before ever put in a TypeDispatcher
     one: 1,
     zero: 0,
     nan: NaN
diff --git a/src/number/helpers.js b/src/number/helpers.js
index 4ab75af..1d6bf9b 100644
--- a/src/number/helpers.js
+++ b/src/number/helpers.js
@@ -1,7 +1,7 @@
 import {NumberT} from './NumberT.js'
 
-import {match} from '#core/helpers.js'
 import {Returns} from '#core/Type.js'
+import {match} from '#core/TypePatterns.js'
 
 export const plain = f => match(
    Array(f.length).fill(NumberT), Returns(NumberT, f))
diff --git a/src/number/relational.js b/src/number/relational.js
index 8839c37..a4820a6 100644
--- a/src/number/relational.js
+++ b/src/number/relational.js
@@ -1,6 +1,5 @@
-import {match} from '#core/helpers.js'
 import {Returns} from '#core/Type.js'
-import {Optional} from '#core/TypePatterns.js'
+import {match, Optional} from '#core/TypePatterns.js'
 import {boolnum} from './helpers.js'
 import {NumberT} from './NumberT.js'
 
diff --git a/src/number/type.js b/src/number/type.js
index ddb47cc..d8e6806 100644
--- a/src/number/type.js
+++ b/src/number/type.js
@@ -1,7 +1,7 @@
 import {plain} from './helpers.js'
 import {BooleanT} from '#boolean/BooleanT.js'
-import {match} from '#core/helpers.js'
 import {Returns} from '#core/Type.js'
+import {match} from '#core/TypePatterns.js'
 import {NumberT} from '#number/NumberT.js'
 
 const num = f => Returns(NumberT, f)
diff --git a/src/number/utils.js b/src/number/utils.js
index 08372c4..01fdbe4 100644
--- a/src/number/utils.js
+++ b/src/number/utils.js
@@ -2,7 +2,7 @@ import {plain, boolnum} from './helpers.js'
 import {NumberT} from './NumberT.js'
 
 import {Returns} from '#core/Type.js'
-import {match} from '#core/helpers.js'
+import {match} from '#core/TypePatterns.js'
 
 export const clone = plain(a => a)
 export const isnan = match(NumberT, boolnum(isNaN))