A Common Math Trap that Even Catches Experts
2020-10-24 | 6 min read
A recent puzzle going around social media asks what 8/2(2+2) is. Some people reason that 8/2 is 4, and (2+2) is 4, so the product is 16. Other people interpret the problem as 8 divided by the quantity 2(2+2). This reduces to 8/8, so these people argue the answer is 1. According to the current order of operations "PEMDAS" convention (Parentheses, Exponents, Multiplication & Division, Addition & Subtraction), the expression in parentheses is evaluated first, and then multiplication and division proceed left to right:
This seems straightforward enough, but suppose the expression was written like this:
8 / 2(2+2)
Here the writer uses whitespace to emphasize the close binding of 2(2+2). Surely a reader could see this as 8/8, or 1. But no rule specifies how much space around a division operator indicates how values are to be grouped.
What about letters?
What if the operands to the right of the division sign are all letters? For example, consider 7/xyz. Using PEMDAS, this should be 7 divided by x and then multiplied by y and z, which is 7yz/x. However, the convention of treating the expression xyz as a complete term is understandably strong.
But what about 7/x•y•z? Is the x•y•z part not equal to xyz? We usually tell students they are the same, do we not? And if we insist that concatenated letters (for example, xyz) are somehow glued into a single term, then what about xyz²? Does the addition of an exponent dissolve the glue?
That symbol binding glue is even stronger if multiple consecutive symbols form a well-known expression, such as πr². According to PEMDAS, 8πr²/2πr² should be 4π2r4, but the usual convention would be to reduce this to 4.
Even simple things can be tricky
Everybody knows that any real number (other than zero) divided by itself is one. For example, 6 ÷ 6 = 1, 2.7 ÷ 2.7 = 1, and 3/5 ÷ 3/5 = 1.
Actually, no. This is the trap mentioned in the title. If we use PEMDAS, multiplication and division proceed left to right and 3/5 ÷ 3/5 should be interpreted as 3/5 divided by 3 and then divided by 5, which is 1/25. Of course, most people would see the 3/5 as the fraction three-fifths on both sides of the division operator. Indeed, many videos and worksheets found online treat fraction division this way. It is the common convention, but it is wrong according to PEMDAS.
Another variation to consider is what happens when the ÷ operator is replaced by the /. Of course, both are supposed to mean “divided by,” and the common interpretation of 1/2/3/4 is 1 divided by 2 then 3 then 4, or 1/24. Again, spacing seems to matter: 1 / 2/3 / 4 is more likely to be interpreted as “one divided by two-thirds then divided by 4,” which is 3/8. On the other hand, 1/2 / 3/4 would probably be seen as “one-half divided by three-fourths,” which is 2/3. It is obviously problematic to say that the value of 1/2/3/4 is either 1/24 or 3/8 or 2/3, depending on the spacing.
What about scientific notation? Everybody agrees that 6×108 = 600,000,000 and 3×107 = 30,000,000. Therefore, 6×108 ÷ 3×107 should be 2×101 or 20, right? Not according to PEMDAS, which says:
6×108 ÷ 3×107
= 6×100,000,000 ÷ 3×107
= 6×100,000,000 ÷ 3×10,000,000
= 600,000,000 ÷ 3×10,000,000
Obviously, there is a huge difference between 20 and 2×1015!
It is easy to avoid the ambiguity by making the center division line horizontal. Nobody would disagree with:
----------- = 2/3
You can also use parentheses to avoid ambiguity: (2/3) ÷ (2/3) = 1 and (6×108) ÷ (3×107) = 20.
Context matters, and students may lack the firm foothold that experienced engineers and teachers take for granted. Proceed carefully to ensure that students are not tripped up by ambiguous or confusing notation.