Arithmetic Part 1 : What is the Order of Operations?

A greyscale photo of a pyramid.

Photo by Simon Matzinger on Unsplash

Arithmetic is the study of numbers and operations, especially addition, subtraction, multiplication, and division. It is a very old and fundamental field — some of the first known records of arithmetic are from the ancient Egyptians and Babylonians, although there are a few artifacts that suggest arithmetic is much older (the Ishango bone, a tool found in Africa that is likely over 20,000 years old, has marks on its handle that some think could be mathematical tally marks, although we aren’t sure). We practically all use arithmetic in our daily lives for basic tasks like buying/selling items and telling time.

Arithmetic is all about manipulating numbers: combining them, splitting them, and rearranging them. The key to doing this is in following the order of operations, which is very much like grammar, but for math. It tells you in what order to perform operations to properly solve an expression, equation, or inequality. For a bit of history around the order of operations, check out The History of Arithmetic.

Let’s walk through some of the rules.

In the United States, the order of operations is expressed as PEMDAS:

(P)arentheses

(E)xponents

(M)ultiplication

(D)ivision

(A)ddition

(S)ubtraction

In other places around the world, you might learn BODMAS or BIDMAS:

(B)rackets

(O)rders / (I)ndeces

(D)ivision

(M)ultiplication

(A)ddition

(S)ubtraction

Following any of these acronyms will ultimately get you the same result. Now let’s look at each component, starting with addition and subtraction. Addition combines the values of the numbers you are adding, where subtraction allows you to find the difference between them. Addition is commutative, meaning that \( 3 + 7 = 7 + 3\). Subtraction isn’t because \(3 – 7\) does not equal \(7\ – 3\). If you keep the negative sign together with the number it is in front of, however, you can switch the order of the numbers: \( 3\ – 7 = -7 + 3\).

Multiplication is like addition, only working with groups (factors). For example, \(3*2\) is adding \(2\) groups of \(3\) together to get \(6\). \(2*3\) is adding \(3\) groups of \(2\) together to also get \(6\). Multiplication is commutative \((2*3 = 3*2)\), but division is not (\(\frac{2}{3}\) does not equal \(\frac{3}{2}\)). Division is the opposite, or inverse, of multiplication: \(\frac{6}{2} \)divides \(6\) into \(2\) groups of \(3\), and \(\frac{6}{3}\) divides \(6\) into \(3\) groups of \(2\). Remember, multiplication and division are factor-based.

Raising a number to an exponent is simply multiplying a number by itself as many times as the exponent indicates. For example, \(2^4 = 2*2*2*2\). Remember that \(2*x^2 = 2x^2\), while \((2x)^2 = 4x^2\). If an exponent is applied to parentheses that includes addition or subtraction, remember to distribute fully. For example, \((x + 2)^2 = (x+2)(x+2) = x^2 + 4x + 4\). When you’re undoing an even exponent, remember to add \( \pm \) in front of your solution. If you’re undoing an odd exponent, keep the existing sign. For example, the \(\sqrt{9} = \pm 3\) and \( \sqrt[3]{-27}= -3\). This is because raising a number to an even power always produces a positive result and raising a number to an odd power keeps its sign. This means that the square root of a negative number will be a complex number (part real, part imaginary).

The logarithm is the inverse (opposite) of the exponent operation. A logarithm of a number tells you the exponent to which a chosen number (the base) must be raised in order to equal a given number. For example, log base \(3\) of \(9\) asks what exponent you would need to raise \(3\) to in order to get \(9\). \(3^2 = 9\), so \(log_{3} \ 9 = 2\). If you see a logarithm with no base written, the base is assumed to be \(10\).

Parentheses aren’t technically a numerical operation — they enclose any operations that you should perform before operations outside the parentheses. For example, \((2+5)*2 = 7*2 = 14\). This is different than \(2 + 5 * 2 = 2 + 10 = 12\). If you have parentheses inside parentheses, start with the innermost set. For example, \(5(3+2(2)) = 5(3 + 4) = 5(7) = 35\).

The absolute value operation gives the magnitude of a number or expression — essentially, it tells you how far a number is away from \(\, 0\) on the number line. This means it doesn’t matter if the number itself is positive or negative. For example, \(3\) is \(3\) units away from \(0\) in the positive direction, and \(-3 \,\)is \(3\) units away from\(\, 0\) in the negative direction. That’s why \( |3|\) and\( |-3|\) both\(\, = 3\).