Long Addition Calculator

column k: total = digit_k(a₁) + … + digit_k(aₙ) + carry_in written digit = total mod 10 carry out = floor(total ÷ 10)

a₁ … aₙ

The addends: between two and ten whole numbers, each no larger than one trillion. The first column is the ones column of every number at once, which is why the numbers have to be written with their digits lined up before this procedure means anything.

column k

The k-th place from the right, counting the ones column as zero. Every column holds one digit from each addend, and a number too short to reach that far contributes a zero rather than being skipped — the place value of a digit is decided by the column it sits in, not by the number it came from.

carry

The amount carried into the next column left. It is whatever is left over once the digit written down has been taken out — the tens part of the total. A column can carry out at most 9, and the highest column may still carry one place beyond the width of the longest addend, which is what makes 9 + 9 into a two-digit answer.

sum

The total, and also the number the carry chain is measured against: the chain has one entry per column, so its length equals the number of digits in the sum. Leading zeros are never written, so 007 is the answer 7.

Use this when the addition has to be shown rather than just answered: teaching the carrying step, checking a total that was worked by hand, or adding several numbers at once where the intermediate carries are the part that goes wrong. Adding numbers to find their average is the next question after a total, and the average page takes the same list.

1. 1234 + 567 + 89

   1. Ones column: 4 + 7 + 9 = 20, so write 0 and carry 2
   2. Tens column: 3 + 6 + 8 + 2 = 19, so write 9 and carry 1
   3. Hundreds column: 2 + 5 + 0 + 1 = 8, so write 8 and carry 0
   4. Thousands column: 1 + 0 + 0 + 0 = 1, so write 1 and carry 0
   5. The digits written, from the highest column down, are 1, 8, 9, 0
   6. The sum is 1890 and the carry chain is 2, 1, 0, 0

   The three addends have four, three and two digits, so the two shorter ones run out of digits in the higher columns and contribute nothing there. The chain has four entries and the sum has four digits — that match is the assertion the page is built around, and it holds on every input, including the ones where nothing carries.

2. 9 + 9, where a new column appears

   1. Ones column: 9 + 9 = 18, so write 8 and carry 1
   2. There is no column left, so the carry opens one: write 1 and carry 0
   3. The digits are 1 and 8
   4. The sum is 18, so the carry chain is 1, 0

   Two single-digit addends and a two-digit answer. The carry chain has an entry for the column that the addends never had, which is why the chain is one longer than the widest addend. Reading 1, 0 as a subtraction from the answer would be a mistake — the entries run from the ones column upwards, so the first one is the ones column's carry.

3. A total with no carries at all

   1. Ones column: 0 + 0 + 0 = 0, write 0, carry 0
   2. Tens column: 0 + 0 + 0 = 0, write 0, carry 0
   3. Hundreds column: 1 + 2 + 3 = 6, write 6, carry 0
   4. The sum is 600 and the carry chain is 0, 0, 0

   Nothing carries here, and the zeros are still printed. A chain that stopped at the first non-zero entry would be unreadable against the columns, and the point of the reading is the correspondence between chain entries and places. This example also separates its addends with spaces rather than semicolons; both are accepted, and the spaces avoid the comma, which some countries read as a decimal point.

Only whole numbers are accepted. A decimal point would require the addends to be aligned on it first, with zeros filled in to equal lengths, which is a different procedure from adding more columns — so decimals are refused rather than silently treated as whole numbers. Negative addends are refused for the same kind of reason: a minus sign is not a digit, and it needs a column and a rule of its own. Between two and ten addends are accepted; a single number has no column to write, and past ten the arithmetic is better served by a sum over a list. Each addend is capped at one trillion and the sum at the largest exactly representable integer, because the digits are read one place at a time and beyond that the written digits stop matching the answer. This page adds; it does not subtract, multiply or divide, each of which has its own column procedure.

What is long addition?

Adding numbers the way they are written down: line the digits up by place, work from the ones column leftwards, and carry whatever is left over into the next column. It is the same sum a calculator gives, with the intermediate steps kept instead of discarded.

Why start at the ones column?

Because the carry travels leftwards. Each column needs the carry coming into it, and that comes from the column on its right, so the ones column has to be settled before the tens column can be. Starting anywhere else would mean going back once the carry arrives.

What does the carry chain tell me?

It lists what each column carried into the next one, starting from the ones column. Its length is the number of digits in the sum, so it doubles as a check: if a carry went missing, or a column was dropped, the two lengths stop matching.

Why can the answer have more digits than the longest addend?

Because the last column can still carry. Adding 9 and 9 fills the ones column and leaves a carry with nowhere to go, so a new column opens to hold it, and 9 + 9 becomes 18 rather than 8. That extra column is the reason the carry chain can be one entry longer than the widest addend.

Can I add decimals?

Not on this page. Decimals can be added in columns too, but only after being aligned on the decimal point and padded with zeros, and that changes what a column means — from a place value to a position after the point. The page refuses the input rather than treating 1.5 as 15.

Does the order of the addends matter?

The sum is the same in any order. The carry chain is not: a different order changes which digit each column receives, and a column whose digits happen to total 9 will carry where another arrangement does not. The intermediate working is order-dependent even though the answer is not.

• Base — how a digit's worth comes from the column it sits in, which is what makes the ones column the right place to start — Wolfram MathWorld (United States)
• Binary — the identical carry procedure in base two, where a single 1 + 1 already carries and the chain is easiest to watch — Wolfram MathWorld (United States)
• Additive Inverse — the opposite of a number, and the reason a negative addend cannot simply be written into the column — Wolfram MathWorld (United States)