One of the most common errors made by the unwary when attempting to evaluate a mathematical problem involving two or more different types of arithmetic operations (e.g. addition, subtraction, multiplication etc.) is to evaluate the operations in the wrong order and arrive at completely the wrong answer. In mathematics, the rules governing which operations should be carried out first are well established.
Confusion can arise for a number of reasons, including the proliferation of different mnemonics (of which BODMAS is only one) that all refer to the same set of rules, and a degree of ambiguity as to the interpretation of the rules themselves. An over reliance on learning a particular mnemonic without fully understanding the rules can often lead to problems. The situation is not helped by the fact that electronic calculators, programming languages and various software packages such as spreadsheet programs often have a different ordering scheme from that used in mathematics (and of course from each other). Here is a simple example of an expression that includes operators of two different types:
2 × 3 + 4 = 6 + 4 = 10
Here we have a multiplication operation followed by an addition operation. Since multiplication is always carried out before addition (unless brackets are used to alter the order of operations - more about that shortly) the correct answer is derived as shown above. The rules are fairly easy to apply here, but before we go any further it might be useful to state exactly what the rules are, i.e. what BODMAS stands for:
B = Brackets
O = Order
D = Division
M = Multiplication
A = Addition
S = Subtraction
Before we even start to look at the meaning of each of these terms, it should be pointed out that perhaps the most important rule is not even covered by the mnemonic. That rule simply states that, in the absence of any other rule being applicable, operations are evaluated from left to right. Of course if the expression we are looking at consists entirely of addition operations, even breaking this simple rule would not really matter, since addition is associative. This means that the order in which the individual addition operations are carried out does not matter. The same applies to an expression involving only multiplication operations. As soon as we start to look at expressions involving subtraction or division however, using the left-to-right rule becomes critical to achieving a correct answer. When several different types of operation are involved in the same expression, we must apply the BODMAS rules and ensure that the left-to-right rule is adhered to. We can now look at each of the BODMAS terms in more detail.
The "B" in BODMAS stands for brackets. Note that brackets are often called parentheses, especially in North America, and some variations on the BODMAS mnemonic use "P" (for parentheses ) instead of "B". They may use alternative letters for other rules as well, so expect to see variations like PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction ). The BBC's Bitesize educational pages seem to favour BIDMAS (Brackets, Indices, Division and Multiplication, Addition and Subtraction ). In this page we will stick to BODMAS, but as long as you understand the principles you can use whatever mnemonic you find easiest to remember (or just whatever you think sounds best!). This rule states that operations enclosed by brackets are evaluated first. Consider the following expression:
2 × (3 + 4) = 2 × 7 = 14
This expression should look somewhat familiar since you saw almost the same expression earlier on this page. The numbers have not changed, but the brackets around the "3 + 4" means that the addition operation is carried out before the multiplication operation. This means we end up with "2 × 7" rather than "6 + 4", and the answer is fourteen and not ten. Brackets are often used to force evaluation of an operation that would not otherwise take precedence. Because the use of brackets to elevate the priority level of an operation is almost universally understood, brackets are sometimes used (somewhat controversially) to reinforce the existing order of operations. Consider the following expression:
4 + (2 × 3) = 4 + 6 = 10
The use of the brackets here is actually unnecessary, since the multiplication operation has a higher priority than, and should therefore automatically be evaluated before, the addition operation. Brackets are sometimes used (or perhaps misused) in order to "remove ambiguity" from the evaluation of an expression. While this may be a legitimate way of drawing attention to the correct order of operations when first teaching this topic, it should generally be discouraged in the long term and should certainly not be used as an alternative to a thorough understanding of the rules governing the order of operations. There may well be a need to re-evaluate the use of brackets when working with mathematical expressions in computer programming languages and spreadsheet programs, since such applications have their own rules governing operator precedence.
After brackets, the operations with the next highest precedence fall under the heading of "Order". The term "Order" refers to exponents (or indices ), which means that a term is expressed as something to the power of something else ("52" is the same as saying "five to the power of two") or as the root of something ("√5" is the same as saying "the square root of five"). Incidentally, there is a clear relationship between roots and powers, since roots can be expressed as reciprocal powers. For example, the square root of sixteen is four. Four is the number which, if we raised it to the power of two, would give us sixteen. Four can also be expressed as sixteen to the power of one divided by two, i.e. 4 = 16 1/2. To look at another example, the cubed root of twenty-seven is three. Three is the number which, if we raised it to the power of three, would give us twenty-seven. Three can also be expressed as twenty-seven to the power of one divided by three, i.e. 3 = 27 1/3. Consider the following expression:
4 + 52 = 4 + 25 = 29
In this example the exponentiation 52 is evaluated first to give twenty-five, which is then added to four to give the answer twenty-nine. Contrast this with the following:
(4 + 5)2 = 92 = 81
Here, the brackets force the evaluation of the addition operation first, giving us four plus five equals nine. The result is then raised to the power of two to give the answer eighty-nine. So far the BODMAS mnemonic has been relatively easy to apply, but a degree of confusion can arise from this point on. The next two letters in the mnemonic (D and M) stand for Division and Multiplication. One possible (and as it happens incorrect) interpretation is that division takes priority over multiplication. This is in fact not so. Division and multiplication take equal precedence, and are evaluated in strict left-to-right order. The following expression serves to illustrate the point:
12 × 6 ÷ 2 = 72 ÷ 2 = 36
In the above example, the multiplication operation is evaluated first simply because it appears first in the left-to-right order of operations. Just to prove a point, if we evaluate the division operation first, we will get the following:
12 × 6 ÷ 2 = 12 × 3 = 36
Obviously, the order of evaluation here has made no difference to our answer, even though it breaks the rule of strict left-to-right evaluation when the operators have equal precedence. Don't think for one moment, however, that you can ignore the left-to-right rule just because operators have equal precedence. Lets look at another example, this time involving two division operations:
64 ÷ 4 ÷ 2 = 16 ÷ 2 = 8
If we evaluate the second division operation first, this happens:
64 ÷ 4 ÷ 2 = 64 ÷ 2 = 32
Obviously, using the wrong order of evaluation here has given us a completely different (and incorrect) result.
Division and multiplication take priority over addition and subtraction, but are equal in status to each other. Likewise, addition and subtraction have equal status, regardless of the order in which their representative letters appear in the BODMAS mnemonic. However, in expressions involving both addition and subtraction operations, the left-to-right ordering of the operations is important. Consider the following expression:
11 - 5 + 8 = 6 + 8 = 14
Here, the subtraction operation is performed first because it appears before the addition operation. If we were to misinterpret the BODMAS mnemonic by assuming that addition took priority over subtraction, the outcome would be different (and incorrect):
11 - 5 + 8 = 11 - 13 = -2
The left-to-right ordering is also important for expressions involving two or more subtraction operations. Consider the following expression:
31 - 16 - 5 = 15 - 5 = 10
Here, the first subtraction operation (31 - 16) is performed first because it appears before the second subtraction operation. If we were to carry out the second subtraction operation (16 - 5) first, the outcome would be very different (and incorrect):
31 - 16 - 5 = 31 - 11 = 20
The above examples are not exhaustive, but they illustrate the importance of correctly interpreting and following the BODMAS rules. The left-to-right rule is especially important when it comes to expressions involving division or subtraction, because both of these operations are non-commutative (i.e. they must be evaluated in strict left-to-right order).
The evaluation of complex mathematical expressions, especially those that involve different operators, requires an understanding of the rules of operator precedence. Whether you use BODMAS, BIDMAS, PEMDAS, or some other mnemonic to remember those rules, it is important to understand them thoroughly. When tasked with evaluating a complex expression, we can re-write the expression as many times as is necessary to eliminate the high order operations, leaving an expression that consists purely of addition and / or subtraction operations. From there, it is simply a case of evaluating the expression by working from left to right. Once you have learned the rules and how to apply them, they should become second nature and you will find the use of mnemonics is no longer necessary.