diff --git a/base/floatfuncs.jl b/base/floatfuncs.jl
index 7a15934ed58a3..85db2a2ae92e3 100644
--- a/base/floatfuncs.jl
+++ b/base/floatfuncs.jl
@@ -170,17 +170,17 @@ function _round(x, r::RoundingMode, digits::Integer, sigdigits::Nothing, base)
 end
 
 hidigit(x::Integer, base) = ndigits0z(x, base)
-function hidigit(x::Real, base)
+function hidigit(x::AbstractFloat, base)
     iszero(x) && return 0
-    fx = float(x)
     if base == 10
-        return 1 + floor(Int, log10(abs(fx)))
+        return 1 + floor(Int, log10(abs(x)))
     elseif base == 2
         return 1 + exponent(x)
     else
-        return 1 + floor(Int, log(base, abs(fx)))
+        return 1 + floor(Int, log(base, abs(x)))
     end
 end
+hidigit(x::Real, base) = hidigit(float(x), base)
 
 function _round(x, r::RoundingMode, digits::Nothing, sigdigits::Integer, base)
     h = hidigit(x, base)
diff --git a/base/irrationals.jl b/base/irrationals.jl
index 5f23988f5195b..5b4ab8351996c 100644
--- a/base/irrationals.jl
+++ b/base/irrationals.jl
@@ -125,6 +125,8 @@ for op in Symbol[:+, :-, :*, :/, :^]
 end
 *(x::Bool, y::AbstractIrrational) = ifelse(x, Float64(y), 0.0)
 
+_round(x::Irrational, r::RoundingMode) = _round(float(x), r)
+
 macro irrational(sym, val, def)
     esym = esc(sym)
     qsym = esc(Expr(:quote, sym))
diff --git a/test/rounding.jl b/test/rounding.jl
index c7b0c9dde2864..a9c183228b5b8 100644
--- a/test/rounding.jl
+++ b/test/rounding.jl
@@ -265,8 +265,13 @@ end
 
 # custom rounding and significant-digit ops
 @testset "rounding to digits relative to the decimal point" begin
+    @test round(pi) ≈ 3.
     @test round(pi, digits=0) ≈ 3.
     @test round(pi, digits=1) ≈ 3.1
+    @test round(pi, digits=3, base=2) ≈ 3.125
+    @test round(pi, sigdigits=1) ≈ 3.
+    @test round(pi, sigdigits=3) ≈ 3.14
+    @test round(pi, sigdigits=4, base=2) ≈ 3.25
     @test round(10*pi, digits=-1) ≈ 30.
     @test round(.1, digits=0) == 0.
     @test round(-.1, digits=0) == -0.