Suppose you have 2 apples, someone gave you 1 apple, how many apples do you have? Probably, you already know, that the answer is 3. This means that 2+1=3.

So, you can think about addition as a process during which you gain something.

Another example: suppose you have $5. Your parents gave you$4. How much money do you have? Answer is \$9, so 5+4=9.

Each number being added is called addend and the result is called sum.

So, in 5+4=9 both 5 and 4 are addends and 9 is sum.

It is easy to add 1-digit numbers, but what to do with numbers like 25, 657, 1984?

There is a good technique.

Let's start from simple example.

Example 1. Calculate 27+31.

Let's write numbers one under another:

$\begin{array}{l@{\,}l@{\,}l} \ & \color{blue}{2}&\color{green}{7} \\ + & \color{blue}{3}&\color{green}{1} \\ \hline & \color{blue}{}&\color{green}{} \\ \end{array}$

Start from right, let's add 7 and 1. Result is 8. Write it under 7 and 1.

Now add 2 and 3. Result is 5. Write 5 under 2 and 3.

$\begin{array}{l@{\,}l@{\,}l} \ & \color{blue}{2}&\color{green}{7} \\ + & \color{blue}{3}&\color{green}{1} \\ \hline & \color{blue}{5}&\color{green}{8} \\ \end{array}$

So, 27+31=58.

Actually same applies for 3-digit numbers and in general for any numbers.

Example 2. Calculate 345+101.

Let's write numbers one under another:

$\begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \color{red}{3}&\color{blue}{4}&\color{green}{5} \\ + & \color{red}{1}& \color{blue}{0}&\color{green}{1} \\ \hline & \color{red}{}&\color{blue}{}&\color{green}{} \\ \end{array}$

Start from right, let's add 5 and 1. Result is 6. Write it under 5 and 1.

Now add 4 and 0. Result is 4. Write 4 under 4 and 0.

Finally, write 3+1=4 under 3 and 1.

$\begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \color{red}{3}&\color{blue}{4}& \color{green}{5} \\ + & \color{red}{1}& \color{blue}{0}&\color{green}{1} \\ \hline & \color{red}{4}&\color{blue}{4}&\color{green}{6} \\ \end{array}$

So, 345+101=446.

Let's do a harder example now.

Example 3. Calculate 948+197.

Let's write numbers one under another:

$\begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \color{red}{9}&\color{blue}{4}& \color{green}{8} \\ + & \color{red}{1}& \color{blue}{9}& \color{green}{7} \\ \hline & & &\color{green}{} \\ \end{array}$

Let's add 8 and 7. Result is 15. Oops! We need 1-digit number here. So, we write 5 and remember 1:

$\begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \color{red}{9}& \overset{\color{green}{+1}}{\color{blue}{4}}& \color{green}{8} \\ + & \color{red}{1}& \color{blue}{9}& \color{green}{7} \\ \hline & & & \color{green}{5} \\ \end{array}$

Now, add 4 and 9. Result is 13. Don't forget about green 1. So, result is 14. We again need 1-digit number, so take 4 and remember 1:

$\begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \overset{\color{blue}{+1}}{\color{red}{9}}& \color{blue}{4}& \color{green}{8} \\ + & \color{red}{1}& \color{blue}{9}& \color{green}{7} \\ \hline & & \color{blue}{4} & \color{green}{5} \\ \end{array}$

Finally add 9 and 1 and don't forget about 1 that we remembered. Result is 11. But this is OK. We are on final step, so just write 11.

$\begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \color{red}{9}&\color{blue}{4}& \color{green}{8} \\ + & \color{red}{1}& \color{blue}{9}& \color{green}{7} \\ \hline & \color{red}{11}& \color{blue}{4}& \color{green}{5} \\ \end{array}$

So, 948+197=1145.

Next example shows how to add numbers that have different number of digits.

If two numbers have different number of digits then take number that has smaller number of digits and add zeros in front of it until number of digits will be same.

For example, suppose you need to add 23 and 5537. We take 2-digit number 23 and place zeros in front of it: 23 becomes 0023. Now we can add 0023 and 5537.

Example 4. Calculate 56+345.

Here we add one zero in front of 56: 056.

Now, we can add 056 and 345 using standard technique.

We will do this example a bit faster.

$\begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \overset{\color{blue}{+1}}{\color{red}{0}}& \overset{\color{green}{+1}}{\color{blue}{5}}& \color{green}{6} \\ + & \color{red}{3}& \color{blue}{4}& \color{green}{5} \\ \hline & \color{red}{4}& \color{blue}{0}& \color{green}{1} \\ \end{array}$

So, 56+345=401.

Example 5. Find 349+23+568.

There is no really difference between adding just two numbers.

First we write 0 in front of 23: 023.

$\begin{array}[l@{\,}l@{\,}l@{\,}l] \ & \color{red}{3}& \color{blue}{4}& \color{green}{9} \\ + & \color{red}{0}& \color{blue}{2}& \color{green}{3} \\ + & \color{red}{5}&\color{blue}{6}& \color{green}{8} \\\hline & & & \\ \end{array}$

Add 9 and 3. Result is 12. Take 2 and remember 1.

$\begin{array}{l@{\,}l@{\,}l@{\,}l} & \color{red}{3} & \overset{\color{green}{+1}}{\color{blue}{4}} & \color{green}{9} \\ + & \color{red}{0} & \color{blue}{2} & \color{green}{3\color{purple}{(2)}} \\ + & \color{red}{5}&\color{blue}{6}&\color{green}{8} \\\hline & & &\color{green}{} \\ \end{array}$

Now, continue adding: 2+8 is 10, so we write and remember 1. But since we already have remembered 1 then now we have 1+1=2 reserved 1s.

$\begin{array}{l@{\,}l@{\,}l@{\,}l} & \color{red}{3} & \overset{\color{green}{+2}}{\color{blue}{4}} & \color{green}{9} \\ + & \color{red}{0} & \color{blue}{2} & \color{green}{3} \\ + & \color{red}{5}&\color{blue}{6}&\color{green}{8} \\\hline & & &\color{green}{0} \\ \end{array}$

Next, work with second column. Take borrowed 2, add 4. Result is 6.

Next, add 2. Result is 8. And add 6. Result is 14. Write 4 and remember 1.

$\begin{array}{l@{\,}l@{\,}l@{\,}l} & \overset{\color{blue}{+1}}{\color{red}{3}} & \color{blue}{4} & \color{green}{9} \\ + & \color{red}{0} & \color{blue}{2} & \color{green}{3} \\ + & \color{red}{5}&\color{blue}{6}&\color{green}{8} \\\hline & &\color{blue}{4} &\color{green}{0} \\ \end{array}$

Finally, work with left-most column. Take remembered 1, add 3. Result is 4.

Next, add 0. Result is 4. And add 5. Result is 9.

$\begin{array}{l@{\,}l@{\,}l@{\,}l} & \color{red}{3} & \color{blue}{4} & \color{green}{9} \\ + & \color{red}{0} & \color{blue}{2} & \color{green}{3} \\ + & \color{red}{5}&\color{blue}{6}&\color{green}{8} \\\hline &\color{red}{9} &\color{blue}{4} &\color{green}{0} \\ \end{array}$

So, 349+23+568=940.

Now, your turn. Take pen and paper and solve following problems.

Exercise 1. Find 23+36.

Exercise 2. Find 58+57.