Before starting, we need to deal with any and all negatives. When subtracting two terms (minuend-subtrahend=difference), negatives can show up in one of the following three ways:
First, we must draw a clear distinction between subtraction and differencing. Subtraction is the operation of loss in mathematics, and represents the action of subtracting a specified amount (the subtrahend) from a known initial amount (the minuend). We start the subtraction by determining how to deal with negatives if they exist.
Long subtraction is the technique that we use to find the difference in between two whole numbers (ignoring their sign, magnitude, and order). Subtraction determines whether we should sum (as done in long addition) or difference the terms (long subtraction), and whether we should make the final answer positive or negative.
When subtracting with a negative minuend, sum the two values and make the final answer negative.
Subtracting from a negative minuend (or from an initial deficit) makes it more negative (or pushes it into a further deficit). That is why we simply sum the two values together and make the answer negative.
A double negative is a positive, so subtracting by a negative is the same as adding by the positive value of the same number.
Subtracting with a negative subtrahend (or subtracting a loss) from a value is a gain by that same amount. This fundamental principle of double negatives boils down to the saying: 'the loss of a loss is a gain.'
When both the minuend and subtrahend are negative, both of the effects from A and B are applied.
Note that the final answer is not made negative by this flip like the flip done in the next step.
Only the whole value (positive value without the negative) of each term is needed to find the difference between them with long subtraction. So once handled, negatives are ignored for the entirety of the long subtraction process (steps 2-9).
The first step in long subtraction is to arrange the two terms with the larger in top.
Similar to long addition: whole numbers are right aligned, decimal places are aligned vertically, and a decimal point is placed just to the right of whole numbers as needed.
If the subtrahend (or the value that we are subtracting by) is larger than the minuend (the first value), then the final answer will be negative. In the last step we drew the distinction between performing the operation of subtraction and taking a difference. This distinction becomes clear when we set up the two examples 49-9 and 9-49 for long subtraction. The two are identical, as is difference yielded. This is because the relative difference between 49 and 9 is 40 regardless of sign or order.
However, once we apply the principals of subtraction and take into account the fact that the second term (the subtrahend) is larger than the first (the minuend), we can deduce that the final answer will be negative. We can reduce the chance of forgetting to make the final difference negative by placing a negative in front of where the final answer will go.
Remember that we need to deal with the negatives before starting the process of long subtraction. We use the rules from step 1 to determine that the 2 becomes positive and the 1.235 remains negative.
A minus sign (-) is placed to the left of the subtrahend to indicate that the operation being performed is subtraction. An equals bar is placed below the subtrahend to separate it from the final answer.
As with the long addition process, zeros can be placed to the left and right of any real number without affecting its value. We need to make sure that when we place a zero to the right of a number, that it is also to the right of the decimal place (or the factor of the number will be changed).
The example 9-49 will be omitted until step 10; because, as mentioned in the last step, it is the exact same process and answer as the example 49-9 above.
This distinction can also be seen in the following example in which neither negatives nor order is considered in the ordering of terms, only that 2 is larger than 1.235.
If either the minuend or subtrahend contain a decimal point, then place one in the answer row below the equals bar so it vertically aligned with the others.
Because subtraction is the operation of loss, it becomes just as important to find out if the final answer is negative as it is to find its numerical value. The time we spend performing step 1 is critical, for improper handling of negatives will result in either the wrong sign in front of the final answer or the wrong operation being applied entirely.
In step 2 we compare the order and magnitude of the two terms and find the sign of the final answer. Note that if you apply a negative from step 2 to an answer that is already negative from step 1, the two negatives cancel, and the final answer is positive.
The 7 types of errors commonly introduced while performing subtraction are:
The most common errors introduced in this process have to do with negatives (error types 1, 2, and 7). Performing steps 1 an 2 twice over (in double-checking) and placing a negative in front of where the final answer will go before starting step 5 provides us with the best chance of eliminating these errors.
In long subtraction, we move right-to-left, column by column, differencing the top and bottom digits. We start by choosing the column and comparing the two values.
We can only perform the subtraction if the top number is larger than the bottom. If the bottom number is larger, we must borrow 10 from the digit to left.
The base ten number system means that a digit has a value that is ten times the value of the digit directly to the right of it. A quick example of this is: 200 is ten times the value of 20, 20 is ten times the value of 2, 2 is ten times the value of 0.2, and so on. This is important because when we borrow from the digit to the left, it is worth ten times the digit to its right. This is why we add 10 to the current digit while only subtracting 1 from the digit we are borrowing from.
If a digit does not contain a number, then it assumes the value zero.
If the value of the digit to be borrowed from is zero, then 10 must be borrowed from the next digit to the left (and so on). The final value of this digit after being 'passed through' is 9.
Focusing on the same column, we subtract the two values and place the difference in the answer row below.
The difference yielded from this step will always be a positive number between 0 and 9.
Once the difference is placed, repeat steps 5 and 6 using the next column to the left.
That completes the process of long subtraction. Continue to apply steps 5-7 for every column in the problem, checking the difference found.
Once finished, what you are left with is the difference between the two values. We will use this difference in the next step to find the final answer.
In this last step, we finish the problem by writing the final answer. The primary concern in this step is whether the final answer is negative or not. Now we determine whether the final answer is negative in steps 1 and 2, but this is the time were we make sure everything fits together logically. A good double check to perform at this stage is to add the final answer with the subtrahend (including the negatives) and see if it equals the minuend.
Applying this check with a few of our examples we see that:
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