Adding and Subtracting Multi-Digit Numbers

When working with large numbers up to six digits, the best way to add or subtract is by using the standard vertical algorithm. The most important rule is to line up the numbers by their place values: ones under ones, tens under tens, and so on.

Multi-Digit Addition (Regrouping)

To add large numbers, start from the right (the ones column) and work your way to the left. If a column adds up to 10 or more, you must regroup (or carry) the extra value to the next column.

Example: $345{,}678 + 87{,}543$

Write the numbers vertically, aligning the place values:

\begin{array}{r@{\quad}l} 345{,}678 \\ + \phantom{0}87{,}543 \\ \hline \end{array}

1. Ones: $8 + 3 = 11$. Write $1$, carry $1$.
2. Tens: $7 + 4 + 1 \text{ (carried)} = 12$. Write $2$, carry $1$.
3. Hundreds: $6 + 5 + 1 = 12$. Write $2$, carry $1$.
4. Thousands: $5 + 7 + 1 = 13$. Write $3$, carry $1$.
5. Ten Thousands: $4 + 8 + 1 = 13$. Write $3$, carry $1$.
6. Hundred Thousands: $3 + 1 \text{ (carried)} = 4$.

The final answer is $433{,}221$.

Multi-Digit Subtraction (Borrowing)

For subtraction, you also start from the right. If the top digit is smaller than the bottom digit, you need to regroup (or borrow) $1$ from the column to the left, which adds $10$ to your current column.

Example: $500{,}000 - 234{,}567$

Subtracting across zeros requires borrowing across multiple columns.

\begin{array}{r@{\quad}l} 500{,}000 \\ - 234{,}567 \\ \hline \end{array}

1. We cannot subtract $7$ from $0$. We must borrow from the $5$ in the hundred thousands place, turning it into a $4$.
2. The zeros to the right become $9$s all the way down to the tens place, and the ones place becomes $10$.
3. Now, subtract column by column from right to left:
   • Ones: $10 - 7 = 3$
   • Tens: $9 - 6 = 3$
   • Hundreds: $9 - 5 = 4$
   • Thousands: $9 - 4 = 5$
   • Ten Thousands: $9 - 3 = 6$
   • Hundred Thousands: $4 - 2 = 2$

The final answer is $265{,}433$.

Mixed Operations

When a problem has both addition and subtraction, just solve it step-by-step from left to right.

Example: $23{,}456 + 78{,}901 - 45{,}123$

Step 1: Add the first two numbers. $23{,}456 + 78{,}901 = 102{,}357$

Step 2: Subtract the third number from that result. $102{,}357 - 45{,}123 = 57{,}234$

The final answer is $57{,}234$.