diff --git a/ruby/shell/word_expander.rb b/ruby/shell/word_expander.rb
index f1bbf7c..ebf5870 100644
--- a/ruby/shell/word_expander.rb
+++ b/ruby/shell/word_expander.rb
@@ -25,7 +25,7 @@ module Shell
           expand_braces(expanded)
         end
         .flat_map do |word|
-          if word =~ /[*?\[]/
+          if /[*?\[]/.match?(word)
             glob_words = expand_globs(word)
             glob_words.empty? ? [word] : glob_words
           else
@@ -142,8 +142,13 @@ module Shell
             output << run_command_substitution(cmd)
           when "$"
             if line[i + 1] == "("
-              cmd, i = read_dollar_paren(line, i + 2)
-              output << run_command_substitution(cmd)
+              if line[i + 2] == "("
+                expr, i = read_arithmetic(line, i + 3)
+                output << expand_arithmetic(expr)
+              else
+                cmd, i = read_dollar_paren(line, i + 2)
+                output << run_command_substitution(cmd)
+              end
             else
               output << c
               i += 1
@@ -200,8 +205,13 @@ module Shell
             output << run_command_substitution(cmd)
           when "$"
             if line[i + 1] == "("
-              cmd, i = read_dollar_paren(line, i + 2)
-              output << run_command_substitution(cmd)
+              if line[i + 2] == "("
+                expr, i = read_arithmetic(line, i + 3)
+                output << expand_arithmetic(expr)
+              else
+                cmd, i = read_dollar_paren(line, i + 2)
+                output << run_command_substitution(cmd)
+              end
             else
               output << c
               i += 1
@@ -269,6 +279,35 @@ module Shell
       raise ArgumentError, "Unmatched $(...)"
     end
 
+    def read_arithmetic(line, start_index)
+      output = +""
+      i = start_index
+      depth = 1
+      while i < line.length
+        c = line[i]
+        if c == "("
+          depth += 1
+          output << c
+        elsif c == ")"
+          depth -= 1
+          if depth.zero?
+            if line[i + 1] == ")"
+              return [output, i + 2]
+            else
+              depth += 1
+              output << c
+            end
+          else
+            output << c
+          end
+        else
+          output << c
+        end
+        i += 1
+      end
+      raise ArgumentError, "Unmatched $((...))"
+    end
+
     def run_command_substitution(command)
       stdout, status = Open3.capture2("/bin/sh", "-c", command)
       raise Errno::ENOENT, command unless status.success?
@@ -276,6 +315,151 @@ module Shell
       stdout.tr("\n", " ")
     end
 
+    def expand_arithmetic(expr)
+      tokens = tokenize_arithmetic(expr)
+      rpn = arithmetic_to_rpn(tokens)
+      evaluate_rpn(rpn).to_s
+    end
+
+    def tokenize_arithmetic(expr)
+      tokens = []
+      i = 0
+      while i < expr.length
+        c = expr[i]
+        if c.match?(/\s/)
+          i += 1
+          next
+        end
+        if c.match?(/\d/)
+          j = i + 1
+          j += 1 while j < expr.length && expr[j].match?(/\d/)
+          tokens << [:number, expr[i...j].to_i]
+          i = j
+          next
+        end
+        if c.match?(/[A-Za-z_]/)
+          j = i + 1
+          j += 1 while j < expr.length && expr[j].match?(/[A-Za-z0-9_]/)
+          name = expr[i...j]
+          value = ENV[name]
+          value = (value.nil? || value.empty?) ? 0 : value.to_i
+          tokens << [:number, value]
+          i = j
+          next
+        end
+        if c.match?(%r{[+\-*/()%]})
+          tokens << [:op, c]
+          i += 1
+          next
+        end
+        raise ArgumentError, "Invalid arithmetic expression: #{expr}"
+      end
+      tokens
+    end
+
+    def arithmetic_to_rpn(tokens)
+      output = []
+      ops = []
+      prev_type = nil
+      tokens.each do |type, value|
+        if type == :number
+          output << [:number, value]
+          prev_type = :number
+          next
+        end
+
+        op = value
+        if op == "("
+          ops << op
+          prev_type = :lparen
+          next
+        end
+        if op == ")"
+          while (top = ops.pop)
+            break if top == "("
+            output << [:op, top]
+          end
+          raise ArgumentError, "Unmatched ) in arithmetic expression" if top != "("
+          prev_type = :rparen
+          next
+        end
+
+        if op == "-" && (prev_type.nil? || prev_type == :op || prev_type == :lparen)
+          op = "u-"
+        elsif op == "+" && (prev_type.nil? || prev_type == :op || prev_type == :lparen)
+          op = "u+"
+        end
+
+        while !ops.empty? && precedence(ops.last) >= precedence(op)
+          output << [:op, ops.pop]
+        end
+        ops << op
+        prev_type = :op
+      end
+
+      while (top = ops.pop)
+        raise ArgumentError, "Unmatched ( in arithmetic expression" if top == "("
+        output << [:op, top]
+      end
+      output
+    end
+
+    def precedence(op)
+      case op
+      when "u+", "u-"
+        3
+      when "*", "/", "%"
+        2
+      when "+", "-"
+        1
+      else
+        0
+      end
+    end
+
+    def evaluate_rpn(rpn)
+      stack = []
+      rpn.each do |type, value|
+        if type == :number
+          stack << value
+          next
+        end
+
+        case value
+        when "u+"
+          raise ArgumentError, "Invalid arithmetic expression" if stack.empty?
+          stack << stack.pop
+        when "u-"
+          raise ArgumentError, "Invalid arithmetic expression" if stack.empty?
+          stack << -stack.pop
+        else
+          b = stack.pop
+          a = stack.pop
+          raise ArgumentError, "Invalid arithmetic expression" if a.nil? || b.nil?
+          stack << apply_operator(a, b, value)
+        end
+      end
+      raise ArgumentError, "Invalid arithmetic expression" unless stack.length == 1
+      stack[0]
+    end
+
+    def apply_operator(a, b, op)
+      case op
+      when "+"
+        a + b
+      when "-"
+        a - b
+      when "*"
+        a * b
+      when "/"
+        (b == 0) ? 0 : a / b
+      when "%"
+        (b == 0) ? 0 : a % b
+      else
+        raise ArgumentError, "Invalid arithmetic expression"
+      end
+    end
+
     def expand_braces(word)
       # Simple, non-nested brace expansion: pre{a,b}post -> preapost, prebpost
       match = word.match(/(.*?)\{([^{}]*)\}(.*)/)
diff --git a/ruby/test/shell_test.rb b/ruby/test/shell_test.rb
index 664c895..bd451ef 100644
--- a/ruby/test/shell_test.rb
+++ b/ruby/test/shell_test.rb
@@ -75,10 +75,13 @@ class ShellTest < Minitest::Test
   end
 
   def test_expands_arithmetic
-    skip "arithmetic expansion not implemented"
     assert_equal "3", `#{A1_PATH} -c 'echo $((1 + 2))'`.chomp
   end
 
+  def test_expands_arithmetic_with_variables
+    assert_equal "3", `A1_NUM=2 #{A1_PATH} -c 'echo $((A1_NUM + 1))'`.chomp
+  end
+
   def test_expands_tilde_user
     user = Etc.getlogin
     skip "no login user" unless user