It seems like the best place to start is the place we lose people the most. This post has more components than most will, but it provides context to the foundation of sets and operations used in algebra. Hang in there, more fun stuff later!

Natural Numbers and Whole Numbers

Most of us were first introduced to numbers through counting. So, the two sets (a set is like a group of numbers, but these are generally defined in a list format) include the counting numbers {1, 2, 3, 4, 5,…} which we use to count objects in our early introduction to mathematics. The “Natural numbers” are the counting numbers. The “whole numbers” include zero as well. In higher level mathematics courses, the Natural numbers are used a lot to define solutions and relationships more generally.

Integers

The integers include the Natural numbers, zero, and all negative equivalents to the Natural numbers. What that means is that we don’t just have 2, we also have -2 in the set of integers. The integers are another important set that is used a lot, starting in trigonometry. This is helpful when things are periodic (have a repeating pattern) and you get the same result whether you go forward or backward by one cycle of the pattern.

Real numbers

The Real numbers include all possible values between those integers, as well as the integers. This concept is difficult for a lot of people because it is now abstract. So, let’s break this down a bit. If all numbers between 1 and 2 exist in the real numbers, then I can pick any decimal value of 1 and it will be between 1 and 2. 1.246897531? Yes. 1.98765432? Yes. 1.00000001? Yes. Each of those is in the set of real numbers and so are all possible ones you can come up with. There are so many possibilities that I cannot write them all out, and that is precisely why the set of real numbers is an abstract concept rather than something concrete. A real number has no imaginary parts – we’ll get there, but every number you know prior to algebra is a real number. Any real number that I can write will be in the set of real numbers. After all, it’s the set of real numbers – if the number is real, it’s in the set.

Negative Numbers

This is a concept that is often overcomplicated. The negative numbers are the same as the positive real numbers, except they are all the negative versions of those numbers. So, all negative numbers are a subset of the real numbers.

Rational numbers

Rational numbers have a restriction that makes their set smaller than the real numbers. These are real numbers that can be written as a ratio of real numbers. To be comfortable with this set, we need to first be comfortable with a fraction (a ratio of two real numbers.) If I eat a slice of cake and the cake was cut into 8 evenly-sized slices, I ate 1/8th of the cake. This is the ratio of 1 over 8, 1 cake divided into 8 slices, so 1/8th of cake per slice. The number given by 1/8 is in the rational numbers because it is written as a ratio of two real numbers (1 and 8).

Irrational numbers

The irrational numbers are all the numbers in the real numbers that cannot be written as a ratio of real numbers. The need for this set stems from the values found through things like a square root. The square root of 9 is 3 because , but there is no simple answer for the square root of 2… it’s a decimal value that cannot be written as a ratio. When the two sets (rational and irrational numbers) are combined, we have the real numbers.

Complex numbers

The set of complex numbers includes all the real numbers and numbers with imaginary parts. Complex numbers have a real component and an imaginary component. So, even when the imaginary component is zero, the number is still in the set of complex numbers. This is more useful once you encounter complex numbers; at this stage all you need to know is that they exist and the set of complex numbers includes the set of real numbers.

Operations

Most students memorize PEMDAS, an initialism designed to help you memorize the order of operations. However, out of context, PEMDAS has no meaning. Perhaps there’s a way to make it make sense why the order is what it is.

Parentheses, brackets, and braces

In mathematics, we use parentheses (), brackets [], and braces {} to group operations. Each of these operate like bookends, barriers to protect and hold a set of mathematical operations inside. This is the reason that parentheses, brackets, and braces are listed first in PEMDAS, through the P. If you see parentheses, brackets, or braces – treat them as their own group, to be evaluated first. There are challenges online randomly “What is 4 + 3*(8-6)” etc. Before anything else, we look inside the parentheses and compute 8-6 (because it will overall be less work than the alternative, which is to distribute the 3 to the 8 and the 6). 4 + 3*(8-6) using the parenthesis first approach reduces to 4 + 3*2. The alternative, which has a consistent answer, is 4 + 3*8 – 3*6, which looks easier to you? Right. Parentheses first.

Exponentiation

In practice, the term “powers” might be used as vernacular, but the actual term is exponentiation. We use exponents to represent repeated multiplication in a more concise notation. (You might notice a theme in mathematics where we define new operations to represent repeated operations in a “cleaner” or more concise way.) If I multiply 2 to itself 4 times, that’s . So, this is one of the important cases for understanding why exponentiation falls second in the list. If our problem was , the order becomes required. Parentheses first is the only option (would you really prefer multiplying or $\latex 2^2$?). After the parentheses, we have . Exponentiation is second because changing any of these orders will make a mathematically inconsistent result. . One alternative here would be to expand the exponentiation, , which is mathematically consistent. Either way, we do parentheses first and exponentiation second.

Multiplication and division

These operations are grouped because they are the inverses of each other. If I multiply by 2, then divide by 2, the division “undoes” the multiplication. The operations behave the same and are thus at the same level in terms of order of operations. The reason for the order has more to do with intuition and practice (it also makes the initialism easier to say). This is why the multiplication was done third in our example. After exponentiation, which is repeated multiplication, we next complete the multiplications (and divisions) in the problem. Note: multiplication is repeated addition. If I add 3 to itself 4 times, that’s . You may see there is a bit of a hierarchy defined in mathematical operations which leads to the order of operations here.

Addition and subtraction

The most “basic” or foundational of our operations is last. The first thing we learn after counting is addition. If I have 3 toy cars and you have 4, how many toy cars do we have when we meet up to race them? . I taught my 3 year old to add; once they can count, addition is simply counting multiple groups in a continuum. Subtraction is the same level of addition because it’s (effectively) the same operation with negative numbers. Instead of adding 3 and 4, perhaps you loan me 3 cars of your 4. If you loan me 3 cars, , you’ll only have 1 car left. Addition and subtraction are last because any other operations have to be completed before addition or subtraction can take place. In the end, we are not adding 4 to 3 and then multiplying, we are adding 4 to the result of . If you’re able to start thinking of things in mathematics as subgroups instead of looking at the whole, it helps.

Order of operations

In summary, I look at a computation like a puzzle. Each part of the expression is a puzzle piece, and some of them need to be simplified before being combined. Parentheses define pieces explicitly; always look inside the parentheses first. Determine the value inside the parentheses before looking at the rest of the expression. If there are multiple parentheses, work from the inside -> out. (I’ll add an example below.) Once all parentheses are dealt with, then you can check for any exponentiation (where you have a choice: determine the value of the exponentiated term, or rewrite it as repeated multiplication, your choice to your comfort.) Then, check for multiplication or division (where you do have an option to rewrite multiplication as repeated addition.) Finally, add and subtract as defined by the remaining terms.

Some questions you might have along the way: What if addition is inside the parentheses? We still do the addition inside the parentheses first? Yes. Whatever is inside the parentheses, treat it like its own expression. Determine its value before you complete any other processes. The entire purpose of parentheses, brackets, and braces are to separate operations into groups.

You mean I have choices in arithmetic? You always have choices, but you want to understand the operations well enough to make informed choices. We need to ensure mathematical consistency across all our operations, so informed choices are required. However, one of my greatest frustrations with the way mathematics is taught is the loss of autonomy felt by students when they think there is only “one way” and they have to memorize it. The rigidity causes students to focus on repetition and memorization rather than reasoning, and I believe the reasoning is far more important when you leave the classroom.

If there’s anything I hope for you, dear reader, it’s that this blog helps you take ownership of the mathematics you use and opens the world of mathematics to you in a new light.

All the Best,

-Dr. S.