In algebra, we adopt a new way of thinking when it comes to subtracting numbers.  Here is the definition of subtraction we will use from now on:

Definition: \(a - b = a + \left( { - b} \right)\)

That is, subtracting a number is the same as adding its opposite.  There is an easy way to add negative numbers: subtract the smaller number from the bigger number, and keep the sign of the bigger number.  Let’s try it out.  To find \(9 + \left( { - 11} \right)\), we subtract the smaller number \(\left( 9 \right)\) from the bigger number and keep the sign of the bigger number \(\left(  -  \right)\) Then the answer is

\(9 + \left( { - 11} \right) =  - \left( {11 - 9} \right) =  - 2\)

Again, if we want to find \(8 + \left( { - 2} \right)\), we subtract the smaller number \(\left( 2 \right)\) from the bigger number \(\left( 8 \right)\) and keep the sign of the bigger number \(\left(  +  \right)\). Then the answer is

\(8 + \left( { - 2} \right) = 8 - 2 = 6\)

Similarly, to add \( - 2 + 8\), we do the same thing.  Subtract the smaller number \(\left( 2 \right)\) from the bigger number \(\left( 8 \right)\) and keep the sign of the bigger number \(\left(  +  \right)\). Then the answer is

\( - 2 + 8 = 8 - 2 = 6\)

And doesn’t it make sense that \(8 + \left( { - 2} \right) =  - 2 + 8\)?

Let’s try a few more complex examples now.

Example:  Evaluate \(\left( { - 23} \right) + 16 + 42\)

Solution:  We add and subtract each term in order.  So first we evaluate \(\left( { - 23} \right) + 16\). We subtract the smaller number \(\left( 16 \right)\) from the bigger number \(\left( 23 \right)\) and keep the sign of the bigger number \(\left(  -  \right)\). Then we have

\(\left( { - 23} \right) + 16 =  - \left( {23 - 16} \right) =  - 7\)

Now we finish the problem by computing \( - 7 + 42\) By the same procedure as above, we have

\( - 7 + 42 = 42 - 7 = 35\)

Let’s run the whole problem through without interruption now.  We have

\(\begin{array}{l}
\left( { - 23} \right) + 16 + 42 =  - \left( {23 - 16} \right) + 42\\
 =  - 7 + 42\\
 = 42 - 7\\
 = 35
\end{array}\)

With this method, you should never again get confused about whether we are moving left or right on some number line, or whether we’re adding negatives, or subtracting positive, or whatever!  Just follow the same method when you’re adding or subtracting numbers with different signs.

Example:  Evaluate \(\left( { - 26} \right) - \left( { - 27} \right) + \left( { - 41} \right)\)

\begin{array}{l}
\left( { - 26} \right) - \left( { - 27} \right) + \left( { - 41} \right) = \left( { - 26} \right) + \left( {27} \right) + \left( { - 41} \right)\\
 = 27 - 26 + \left( { - 41} \right)\\
 = 1 + \left( { - 41} \right)\\
 =  - \left( {41 - 1} \right)\\
 =  - 40
\end{array}

Bellow you can download some free math worksheets and practice.