All of us have been taught on how to add, subtract, multiply, và divide in grade school, without using a scientific calculator. These are the chip core of problem solving, which starts from simple arithmetic until it gets complicated with exponentiation & grouping included.
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While lớn perform these mathematical operations seems easy, there has been confusion going around the internet when it comes khổng lồ the order of operations. Và it’s just right to lớn settle this once and for all.
To put simply, the order of operations states which operations take precedence, or are taken care of, before which other operations. In other words, this mathematical rule should tell us how lớn answer an equation with the plus, minus, times, and divide signs put altogether. Because it’s not automatically done from left to right lượt thích in reading.
The most common technique in remembering the order of operations is through the acronym “PEMDAS” which stands for Parentheses, Exponents, Multiplication and Division, and Addition & Subtraction.
What this essentially tells us is the ranks of how the operations are performed: expressions within the parentheses come first; the exponents follow; multiplication và division are next, whichever of the two comes first; and addition và subtraction are last, whichever of the two comes first.
For a much clearer precedence:Parentheses (simplify inside them)ExponentsMultiplication và Division (from left khổng lồ right)Addition và Subtraction (from left khổng lồ right)
Take for example this problem that I found in Quora, which asks for the solution to the equation 6/2(2+1).
Two Casio calculators – the left is an fx-570MS & the right is an fx-570ES – have the same đầu vào equation but different answers. But there is only one real answer. How come?
By following the order of operations or PEMDAS, it is clear that the expression inside the parenthesis, which is the (2+1), is done first. For that, we are left with 6/2*3. The next khổng lồ perform is exponents but since there is none in this case, this should be skipped.
With 6/2*3 left & either of the multiplication or division to vày next, we should get 3*3 và eventually 9. So the answer is 9.
Remember that multiplication and division are in the same rank so whichever operation is in the leftmost, that should be performed first. This is the most common source of confusion for some who answered 1, thinking that the 2*3 part should be done first, hence 6/6 or 1.
Moreover, by simply looking at the original equation, one would answer 1 because he or she could see that the expression 2(2+1) is more tightly bound than 6/2. But that’s not should be the case when the order of operations, a rule of mathematics with origin as far back as the 1500s, is observed.
Another solution is that the division part of the 6/2*3 can be converted into the multiplication of its reciprocal. That gives us 6*(1/2)*3, which is still equal to lớn 9.
So the answer is 9. I am sure about that. Even Microsoft Excel, which follows the order of operations, is sure with that value.
But how come one calculator of the same brand said that the answer is 1?
The difference has something to bởi with the mode or the program of the calculator. For one, not all calculators follow the order-of-operations hierarchy, which leads lớn a different interpretation of the equation.
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In this case, the true size of the equation is 6/2(2+1), which yields 9. However, a calculator could read it as 6/<2(2+1)>, which interprets everything after the division sign as a group. That is how one Casio calculator got the answer khổng lồ be 1.
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