Arithmetic Operators

Part I · Module 4 · Chapter 4.1 · Arithmetic Operators

1. The Six

puts "7 + 3   = [expr {7 + 3}]"
puts "7 - 3   = [expr {7 - 3}]"
puts "7 * 3   = [expr {7 * 3}]"
puts "7 / 3   = [expr {7 / 3}]"
puts "7 % 3   = [expr {7 % 3}]"
puts "2 ** 10 = [expr {2 ** 10}]"

7 + 3   = 10
7 - 3   = 4
7 * 3   = 21
7 / 3   = 2
7 % 3   = 1
2 ** 10 = 1024

Add, subtract, multiply, divide, remainder, power. Every expression is braced, for the reasons in Chapter 3.8.

2. Unary Minus and Plus

set a 5

puts "negate:        [expr {-$a}]"
puts "negate twice:  [expr {- -$a}]"
puts "unary plus:    [expr {+$a}]"

negate:        -5
negate twice:  5
unary plus:    5

Note the space in - -$a. Without it, --$a is not a decrement — Tcl has no ++ or -- operators at all. Counting up is incr, which is Chapter 3.9.

3. Division Is Covered Elsewhere

7 / 3 is 2 because both operands are whole numbers. That is important enough to have its own chapter — Chapter 3.10 — and this chapter does not repeat it.

The one-line version: if you want a fraction, make one operand a real number before dividing.

4. The Remainder, Including Negatives

% gives what is left after whole-number division:

puts "7 % 3 = [expr {7 % 3}]"

7 % 3 = 1

With negatives it is less obvious, so here is the complete table rather than one example:

foreach pair {{7 3} {-7 3} {7 -3} {-7 -3}} {
    lassign $pair a b
    puts [format "%3s %% %3s = %3s      %3s / %3s = %3s" \
        $a $b [expr {$a % $b}] $a $b [expr {$a / $b}]]
}

  7 %   3 =   1        7 /   3 =   2
 -7 %   3 =   2       -7 /   3 =  -3
  7 %  -3 =  -2        7 /  -3 =  -3
 -7 %  -3 =  -1       -7 /  -3 =   2

The remainder takes the sign of the divisor. -7 % 3 is 2, not -1.

This follows from Tcl rounding division toward negative infinity, which Chapter 3.10 §6 covers. C truncates toward zero instead and gives -1 here, so code translated from C needs checking.

% works on whole numbers only:

puts [catch {expr {7.5 % 2}} m]
puts $m

1
can't use floating-point value as operand of "%"

If you need the remainder of a real number, fmod() is in Chapter 4.8.

5. The Power Operator Has Two Surprises

It associates to the right. Every other arithmetic operator groups left to right. ** does not:

puts "2 ** 3 ** 2  = [expr {2 ** 3 ** 2}]"
puts "(2 ** 3) ** 2 = [expr {(2 ** 3) ** 2}]"
puts "2 ** (3 ** 2) = [expr {2 ** (3 ** 2)}]"

2 ** 3 ** 2  = 512
(2 ** 3) ** 2 = 64
2 ** (3 ** 2) = 512

2 ** 3 ** 2 is 2 to the power 9, not 8 squared. This matches ordinary mathematical notation, and it is the opposite of what you get by reading left to right.

Unary minus binds tighter than it does in most languages:

puts "-2 ** 2 = [expr {-2 ** 2}]"

-2 ** 2 = 4

Tcl computed (-2) ** 2, which is 4. Python and several other languages compute -(2 ** 2), which is -4.

6. Powers Follow the Integer Rule Too

puts "2 ** -1    = [expr {2 ** -1}]"
puts "2.0 ** -1  = [expr {2.0 ** -1}]"
puts "4 ** 0.5   = [expr {4 ** 0.5}]"

2 ** -1    = 0
2.0 ** -1  = 0.5
4 ** 0.5   = 2.0

2 ** -1 is one half, which as a whole number is 0. Two whole-number operands give a whole-number answer, exactly as with division.

The fix is the same: make one operand real, or use sqrt() and the other functions in Chapter 4.8.

One more, because the error is clear and worth recognising:

puts [catch {expr {(-8) ** (1.0/3)}} m]
puts $m

1
domain error: argument not in valid range

Raising a negative number to a fractional power has no real answer, and Tcl says so rather than returning something meaningless.

7. Big Numbers Still Work

From Chapter 3.5 §8:

puts [expr {2 ** 64}]
puts [expr {3 ** 40}]

18446744073709551616
12157665459056928801

Whole-number arithmetic has arbitrary precision, so powers do not overflow. If you want the real-number version, write a real operand.

8. Common Mistakes

Expecting / to give a fraction. Chapter 3.10.

Expecting -7 % 3 to be -1. It is 2. §4.

Reading 2 ** 3 ** 2 left to right. It is right-associative. §5.

Expecting -2 ** 2 to be -4. Tcl negates first. §5.

Using % on a real number. Use fmod(). §4.

Looking for ++ or --. They do not exist. Use incr.

9. Exercises

Exercise 1 — the six. Compute, predict first: 9 + 4, 9 - 4, 9 * 4, 9 / 4, 9 % 4, 9 ** 2.

Exercise 2 — the sign table. Without running, fill in -9 % 4, 9 % -4 and -9 % -4. Then check with the §4 loop. Which rule did you have to apply?

Exercise 3 — wrap an index. Write one expression that turns any integer i into a value between 0 and 6, and test it with i equal to 3, 10, -1 and -8. You should not need an if.

Exercise 4 — two power surprises. Predict 3 ** 2 ** 2 and -3 ** 2, then check. Rewrite both with parentheses so that a reader does not need to know the rules.

Exercise 5 — fix a fraction. expr {5 ** -2} gives 0. Produce 0.04 in two different ways.

Exercise 6 — the remainder of a real number. expr {7.5 % 2} fails. Find the answer another way, then say why Tcl refuses the operator form.

10. Remember

• Six operators: + - * / % **, all inside expr.
• There is no ++ or --. Use incr.
• Whole-number division discards the fraction — Chapter 3.10.
• % takes the sign of the divisor. -7 % 3 is 2, where C gives -1.
• That makes $i % $n safe for wrapping an index, including negative i.
• % rejects real numbers. Use fmod().
• ** associates to the right: 2 ** 3 ** 2 is 512.
• Unary minus binds tighter than **: -2 ** 2 is 4, not -4.
• Parenthesise every power expression. Two surprises in one operator is enough.
• 2 ** -1 is 0, by the same whole-number rule as division.

Next: the order Tcl applies all of these in, and the habit that makes the order stop mattering.