Algebra, as Praxis

I view mathematical literacy as a form of self defense.

[leibnizstan]

Prof today draws a diagram on the board, stares at it for several minutes, erases it, re-draws it with the arrows reversed… then ends class 8 minutes early as he continues to stare at the (still incorrect) diagram. There’s hope for me in academia after all.

[0re0bun]

Mathematics I haven’t learned yet is mysterious, almost religious in how it tantalizes me.

Mathematics I am learning is frustrating and taxing.

Mathematics I have already learned is trivial, and left as an exercise to the reader.

[algebra-as-praxis]

Exponents, Negatives, and Division

In the previous post I set up the rules of arithmetic for addition, subtraction, multiplication, and addition. If you paid really close attention you would have seen one thing pop out that wasn't ever described, which was an exponent. It just shows up all of a sudden in my explanation of the multiplicative inverse. In the post before that, I gave a very basic explanation of what an exponent is, but then said I'd be covering it in more depth. This is that, at long last.

I: Couple Bits of Terminology

We label mathematical statements with some terminology in order to more accurately communicate what we're talking about.

In an addition problem, like 3 + 5 = 8, 3 and 5 are the addends or the summends (some people differentiate the numbers as: 3 is the addend and 5 is the summend, because 5 is "being added to" 3 like it was an action). Generally people will just refer to both numbers as the addends. The result, 8, is the sum.

In a subtraction problem, like 8 - 5 = 3, 8 is the minuend, 5 is the subtrahend, and the result, 3, is the difference.

In a multiplication problem, like 8 ⋅ 5 = 40, 8 and 5 are the multiplicands or the factors (again, most people are gonna use just one of those words, and the word most commonly chosen is factor because, let's face it, multiplicand is a mouthful; so, from now on IMPORTANTLY!!! we will only use the word factor to mean "a number we're multiplying with another number"), and the result, 40, is the product.

In a division problem, like 40 ÷ 5 = 8, 40 is the dividend, 5 is the divisor, and 8 is the quotient.

And finally, in an exponential problem, like 2³ = 8, 2 is the base, 3 is the exponent, and 8 is the product -- just as it is with multiplication, because exponentiation is multiplication:

BASEᵉˣᵖᵒⁿᵉⁿᵗ = PRODUCT

II: Exponents/Powers

I'm aware that this is going to be repeating myself a bit, but I think it'll be good to read through anyway!

In the same way that multiplication is repeated addition, exponentiation (raising a number to an exponent or power) is, at first glance, repeated multiplication. We'll say it that way because there's some complication later and even then it's basically true, but it's absolutely good enough for your first exposure.

We write exponents as a superscript number, meaning we are writing it smaller and above where we normally would write, kind of at the top-right corner of the thing before it: in 2³, the 3 is the exponent. The number it follows, 2, is the base that the exponent is attached to.

The exponent tells the base that there are that many factors of itself in this term; in this case, 2³, there are three factors of 2, like so: 2 ⋅ 2 ⋅ 2

There can be more than one base in a term, and not every base will have a visible exponent. If there is a base with no exponent attached, assume its exponent is 1, meaning there is just that one factor of that base in the term. Some examples:

5⋅2³ has one factor of 5 and three factors of 2, making it 5⋅2⋅2⋅2

7³⋅5 has three factors of 7, and one factor of 5, making it 7⋅5⋅5⋅5

2⋅3⁴⋅5² has one factor of 2, four factors of 3, and two factors of 5, making it 2⋅3⋅3⋅3⋅3⋅5⋅5

IIa: How to Read Exponents

So how do you say 7³ -- you know, like, out loud, so other people can understand what you're telling them?

First, we can say "7 raised to the exponent of 3." But that's a bit stuffy. We can replace that word "exponent" with "power," as we saw briefly above.

So now we can say "7 raised to the power of 3." Still stuffy. Change the ordering a little and we can then say that it is "7 raised to the 3rd power."

But if we're only ever "raising" a number to a "power," do we have to say that it's being "raised" every time? No. So, we can then say it is "seven to the third power."

But when else do we ever say something is a power? Never! So, finally, we can reduce this to "seven to the third."

7³ - Seven to the third

2⁵ - Two to the fifth

etc.

Also, we have a couple of shorthands for the specific 2nd and 3rd powers, which is useful since these are the exponents you will end up using the most often!

When something is raised to the second power, like 9², we can just say that 9 is being squared -- "nine squared." This is because if you find the area of a square whose sides are a length of 9, you would do 9⋅9, which is 9².

Likewise, if something is raised to the third power, like 5³, we can just say that 5 is being cubed -- "five cubed." This is because if you find the volume of a cube whose sides are all a length of 5, you would do 5⋅5⋅5, which is 5³.

IIb: Raising to the Power of Zero

This is actually one of my favorite lessons to teach!

So, when we have 5³, that means we have 5⋅5⋅5, which is 125. In the following, think of the left side as the exponential form of the term, the middle side as a demonstration of the factors in the term, and the right side as the value of the term.

5³ = 5⋅5⋅5 = 125

If we take one factor of 5 away, we now have two factors of 5, which makes it 5².

5² = 5⋅5 = 25

Now we'll take another factor of 5 away, so we now have one factor of five, which makes it 5¹.

5¹ = 5 = 5 --- WAIT!!!!

Something is funny here. If you didn't notice, our middle section, our demonstration of the factors in the term, has a subtle issue. And that's the word "factors." What are factors? Remember from earlier in I: Couple Bits of Terminology, I said, "IMPORTANTLY!!, that factors are numbers that multiply with other numbers." So, uhhhhhhh.... what number is 5 being multiplied with?

Well, remember way way back in the previous post when I was talking about the Properties of Arithmetic? There was one there that was kind of important: the Identity Property. This states that any number multiplied by the multiplicative identity remains unchanged -- anything times 1 is itself. And any equality in mathematics that works in one direction, must work in the other. This means that if you have a number, it is secretly being multiplied by one.

Every number. All the time. Even the ones you're multiplying by secretly, are secretly multiplied by one.

We just don't write them because, wow, what a hassle it would be to write literally infinite multiplications of 1 every time we want to do math. We'd (literally) never get anything done.

So that 5? Is actually 1⋅5. And, to be honest, the 5¹ is also 1⋅5¹

1⋅5¹ = 1⋅5 = 5.

And this holds true for every row above as well!

1⋅5³ = 1⋅5⋅5⋅5 = 125

1⋅5² = 1⋅5⋅5 = 25

1⋅5¹ = 1⋅5 = 5

This important bit of setup will help us understand what's coming.

Because what if we take another factor of 5 away? Well, we now have zero factors of five: 5⁰

And if we take that factor of 5 away from our demonstration of the factors in the term, we just lose the five from our 1⋅5 leaving us with... 1.

1⋅5⁰ = 1 = 1

Every number (except for zero -- math nerds fight me! we'll see more about this later) raised to the power of zero is 1. This is the zero property of exponents:

x⁰ = 1 while x≠0

IIc: Raising to a Negative Exponent (Pt. 1)

Continuing our trend:

1⋅5³ = 1⋅5⋅5⋅5 = 125

1⋅5² = 1⋅5⋅5 = 25

1⋅5¹ = 1⋅5 = 5

1⋅5⁰ = 1 = 1

What if we take another factor of 5 out? Well, there aren't any other factors of 5 in the term, so how would we do that?

If we look at the results of our experiment so far, from one row to the next (top to bottom), we might see that removing a factor of 5 is effectively the same as dividing by 5:

125 ÷ 5 = 25, 25 ÷ 5, = 5, and 5 ÷ 5 = 1.

So the process of removing a factor is the same as dividing by that factor. What's more, we know that every time we remove a factor, the exponent reduces by 1. So, from 5⁰ to the next row where we remove a factor, we'll get 5⁻¹, and we'll have to divide by 5 to get what that is!

1⋅5⁻¹ = 1 ÷ 5 = 1/5

So we get the fraction one-fifth as the result. We'll talk more about why this becomes a fraction in a moment, but first! Notice that this means that 5⁻¹ = 1/5, whereas 5¹ = 5. 5 is actually a fraction whose denominator is 1: 5¹ = 5/1. Comparing the results from 5¹ and 5⁻¹, which are 5/1 and 1/5, we see that these are fractions that have the same numbers but are sort of upside down.

When you take a fraction and flip it over like that, you have found its reciprocal. So:

A negative exponent is the reciprocal of the result you get from a positive exponent!

Okay, so, why would any of that be? Well...

III. Some Facts About Division

Fractions are division problems. Whenever you see a fraction, you are actually looking at a division problem. Reading 6/2 as "six halves" or "six divided by two" makes no mathematical difference; these two things have the same value, which is 3.

IIIa. Division Sign

You ever wonder why the division sign looks like that? I mean. Probably not, right? Why does the addition sign look like that? We made it up! But it turns out, there's a specific reason for the division sign's appearance!

It turns out the division sign, ÷, is just a way of formatting a fraction on a single line! We can do that too with the slash, like in 1/5, but 1÷5 is in fact the same thing! It turns out the dot on the top of the horizontal line in the division sign is the numerator of a fraction, and the dot on bottom is the denominator! So it's telling you to take the number to the left and put it on top, and the number on right and put it on the bottom!

So, yeah, you can think of the fraction line (the "vinculum") as the actual division sign, and the commonly-used division sign as a formatting shorthand to make it easier to write on, for example, a printing press or typewriter.

IIIb. The Word "Negative"

What does the word "negative" mean?

The first instinct of most people is, "less than zero." This is not correct. That is the definition of a "negative number." But that's not the definition of "negative."

The root of "negative" is "negate." To negate something is to reverse the effect of that thing; to cancel, undo, reverse, or revert. A negative stands in opposition to a positive. Simply put:

"Negative" means "opposite"

A negative number is the opposite of that number. What do we mean by that? We mean that it stands on the opposite side of the number line, with zero being at the center. 6 and -6 are both 6 away from zero, but in opposite directions from zero.

Subtraction is negative addition -- the opposite of addition. To subtract is to add a negative number. 9 - 5 is the same as 9 + -5. Here we can see the concept of "negative" can affect operations, and the negative operation is always the inverse operation. The inverse of addition is subtraction.

IIIc. Negative Exponents

Likewise, the inverse of multiplication is... division! But in what way is it negative? It's not that the numbers we're dividing become negative. Remember we have described negative exponents as "removing factors." This is the thing that is becoming negative: the exponent on the divisor!

6 ÷ 2 = 6 ⋅ 2⁻¹ = 6 ⋅ (1/2)

So this is where we get to finish our "table" from earlier:

1⋅5³ = 1⋅5⋅5⋅5 = 125

1⋅5² = 1⋅5⋅5 = 25

1⋅5¹ = 1⋅5 = 5

1⋅5⁰ = 1 = 1

1⋅5⁻¹ = 1÷5 = 1/5

1⋅5⁻² = 1÷(5⋅5) = 1/25

1⋅5⁻³ = 1÷(5⋅5⋅5) = 1/125

[algebra-as-praxis]

Terms, Operations, and the Properties of Arithmetic

As a math tutor, I find the mathematical concept of "terms" to be one of the earliest stumbling blocks in algebra. I talked about terms a bit in my previous post, Factors, Terms, and Exponents, but there's a lot more to talk about and other good ways to view them.

I. Operations

The typical and original way we're taught to think about mathematical statements is as operations -- that is, pairs of numbers with an operator between them, something like 2 + 3.

In this framework, we would call 2 and 3 operands (the numbers to be acted upon), and the plus sign between them the operator (the way we're going to act upon those numbers).

Ia. Operators and Operands

We learn the four basic operators, + - × ÷, in grade school. This continues to be most people's framework for math throughout the rest of their lives.

While this framework is fine for basic arithmetic, it is pretty limited and runs into problems. It even comes into conflict with the way we talk about other operations we learn about later.

For example, exponents. Is an exponent an operand, or an operator?

If we call an exponent an operator, then we have to acknowledge that, because of exponents, there are in fact infinitely many operators because an exponent can be any number.

If we call an exponent an operand, then what is the operator? Can we simply consider the fact that it's written in superscript enough of a sign to call that the operator? None of the other operators act this way, so why the exception?

Ib. Order of Operations

We learn this thing in grade school called the Order of Operations. You can shorten it to OoO, like an angry ghost. Or maybe OoOps, like a math mistake -- which, let's be real, OoOps is kind of wrong and thus kind of a mistake.

There are a couple of these that get taught and the one you learn is basically up to your teacher, school, or district. The order goes:

Parentheses Exponents Multiplication Division Addition Subtraction

or PEMDAS for short. You probably remember the mnemonic, "Please Excuse My Dear Aunt Sally," to keep the order in mind.

Another popular version is BODMAS (Brackets Order Division Multiplication Addition Subtraction), which acknowledges that there are other kinds of brackets than just parentheses available in math, but then also throws the word "order" at you, which is another word used to talk about exponents. I don't know the mnemonic for this one, so how about.. uh.. "Boy, ostriches do make annoying sounds."

Both of these are often taught without including a couple of important caveats, though plenty of teachers do include this I'm not trying to throw them under the bus. Those caveats are:

addition and subtraction happen simultaneously;

multiplication and division also happen simultaneously; and

you always work left-to-right.

So, if you were to write it out again, it would look like:

Parentheses Exponents Multiplication/Division Addition/Subtraction (L->R)

Okay. But why? Why are we required to do it this way?

We actually don't cover why until we enter algebra! This is a thing that irks me quite a bit, because in many ways it sets kids up for failure to teach them order of operations not as an additional tool alongside the reasons, but just instead of those reasons.

Ic. Ambiguity

Despite all the caveats we still run into some major ambiguity when we try to do multi-step problems, particularly when division gets involved.

We all know the engagement bait that goes around the internet, where a problem that looks like 4 - 9 × 3 + 16 ÷ 2 × 4 comes up and people argue for weeks or months about what the answer actually is!

If you follow PEMDAS with the caveats, P E M/D A/S (L->R), you will get 4 - 27 + 32 = 9. But then someone else will pop up and say the answer is actually 4 - 27 + 2 = -21. What happened? Are they wrong?

It turns out, the caveat of L->R (working left to right) is not actually a rule of mathematics. It is a convenient model to get students to work through problems that have all been carefully crafted to never have this kind of ambiguity in it. But you can multiply and divide in any order, and you can also add and subtract in any order! ... Sort of. We're getting there.

Because of this, people who have learned algebra realize they can do 2 × 4 before they do the division, thus they get 16 ÷ 8. But they've remembered incorrectly! It's not that they are wrong that they can multiply and divide in any order; it's that they forgot how to apply that rule. The first answer of 9 is actually more correct than the second answer of -21, despite the pedantic half-rememberings of the former algebra student.

Not correct. Just more correct.

See the thing is... Neither of these answers are correct. The correct answer is:

SYNTAX ERROR

This math problem is not written correctly! It does not convey clearly what operations in what order it wants done! The person who wrote it is the wrong one -- deliberately so! People who post questions like these only want one thing and it's disgusting.

And that thing is engagement. They want you to click, to comment, and to react, all in order to boost their post, which boosts their visibility in the social media algorithm, which earns them clout and followers and, as they hope for in the end, money.

So, you may have noticed that I've gone this entire time so far without using the word "term." Let's break this apparent taboo and finally get to the reason why Order of Operations really is a big OoOps.

II. Properties of Arithmetic

Also called the Properties of Real Numbers and a couple of other things whatever get off my back, this is the list of the basic interactions of numbers in arithmetic when dealing with real numbers (don't ask what "real numbers" means right now, I have a headache already lol, go read about Dedekind cuts on your own free time sickos).

The properties almost all have a version for addition/subtraction and another for multiplication/division, but we will only see addition and multiplication examples in the properties for reasons we'll discuss afterward. We'll start to see a certain word pop up immediately in these definitions.

Associative: We put two things in association when we are about to perform an operation on them. This is about the order in which you are allowed to perform the operations in a problem.

Addition: (a + b) + c = a + (b + c) You may perform the addition of terms in any order.

Multiplication: (ab)c = a(bc) You may perform the multiplication of factors within a term in any order.

Commutative: To "commute" means to move back and forth. This property is about reordering things in a problem.

Addition: a + b = b + a You may rearrange terms into any order.

Multiplication: ab = ba You may rearrange factors within a term in any order.

Identity: There must exist some number that, when used for an operation, effectively does nothing.

Addition: a + 0 = a Adding zero does not change the value of a number. We call zero the additive identity. (This property is sometimes called the "zero property of addition.")

Multiplication: 1a = a Multiplying by one does not change the value of a number. We call one the multiplicative identity. (This property is sometimes called the "unit property of multiplication," where "unit" just means 1.)

Inverse: There must exist some number that, when used for an operation, negates another number, which brings us back to that operational identity.

Addition: a + (-a) = a - a = 0 The negation of a number in addition is the negative number. Thus, adding a number to its own negative will negate the two values, leaving you with zero (the additive identity). This also identifies what subtraction actually is; we'll get to that later!

Multiplication: a ⋅ a⁻¹ = a ÷ a = 1 (while a ≠ 0) The negation of a number in multiplication is actually the negative exponent of that number, also known as its reciprocal. Thus, multiplying a number by its own reciprocal will negate the two values, leaving you with one (the multiplicative identity). This also identifies what division actually is; we'll get to that later!

Distributive: This is the only property of arithmetic that does not have two versions, as it requires both the addition and multiplication operations to work.

When multiplying a value with more than one term, by another value with more than one term, you will multiply all the terms in the first value by all the terms in the second value and add all of their products together.

In this example, we will see how a single term multiplies into a multi-term value: a(b + c) = ab + ac

In this example, which is where FOIL (First Outer Inner Last) comes from, we will have a first value with two terms, a+b, and a second value with two terms, c+d, and we will multiply those two values together: (a + b)(c + d) = ac + ad + bc + bd

In this last example, we'll see how you can continue to expand the number of terms in each of these values, by highlighting all the resulting terms that have a given factor from the first value: (a + b + c)(d + e + f + g + h) = ad + ae + af + ag + ah + bd + be + bf + bg + bh + cd + ce + cf + cg + ch

In the next post, we'll finally explore all the hanging threads.

[algebra-as-praxis]

Factors, Terms, and Exponents

When you learn exponents in school, the language we tend to get taught about them is actually very unclear and can lead you to make mistakes and misunderstand the concept entirely. I intend to clean this up from the bottom floor.

I: Types of Math Problems

(we're gonna gloss over things like proofs and logic, this is just about algebra and arithmetic)

A math problem is more accurately called a statement. Statements can be expressions, equations, or inequalities. Both equations and inequalities are comparisons.

Any statement can either be true or false. Unless we are otherwise asked to figure out the factual nature of the statement, typically we just assume any statement given to us in math class is true.

An expression is any statement about how you'd like numbers to interact, but excluding the actual result of that interaction. For example, if I said I was going to buy two sandwiches at $4 each, we would understand the cost of that as being found with the expression 2 × 4.

Algebraically, if we have some unknown value but we know things we want to do with it, we can use an expression to show that interaction without bothering to find the unknown value yet. If I have x dollars in my pocket and I know every piece of candy I want to buy is 50 cents, I could write the expression x/0.5 to show the amount of candy I could buy; then, when I finally check my pocket to see how much money I have, I could solve it. And I could use that same problem any day, since I have the setup for the problem already.

An equation is a statement that sets two expressions equal in value to each other. An equation is most easily identified by the inclusion of an equal sign: =

Equations are split between its sides -- the expression on the left side of the equal sign, and the expression on the right side of the equal sign. So, in the equation 3x + 4 = 17 + 5x we can see the left side is 3x + 4 and the right side is 17 + 5x, and the equal sign is simply telling us that these two expressions have the same value.

An inequality is a statement that tells you that two expressions are not equal, and in which way they are not equal. Inequalities can tell you that one side is greater than the other, or simply that they aren't equal. They can also tell you that they may be equal but also one side might be greater.

You can use > to mean the left side is greater than the right side, or < to mean the left side is less than the right side. You can also use either of these with a line underneath to show that they are "greater than or equal to" or "less than or equal to," like so: ≤ ≥

And if you use any inequality sign and put a slash through it, or even the equal sign with a slash through it, you are negating that sign, meaning it will be the exact opposite of what it normally means: an equal sign with a slash through it (≠) means "not equal to," whereas a less than sign with a slash through it (≮) means "not less than" (which is equivalent to "greater than or equal to").

Interrupting for a moment: There's this thing called the Law of Trichotomy ("trichotomy" means "three distinct possibilities"), which states that in any comparison of two values, exactly one of three things is true: the first value may be equal to the second value (x = y), or it may be greater than that value (x > y), or it may be less than that value (x < y). This also accurately implies that these are the only three comparative states of values.

This can be confusing the first time you hear it since you know about "greater than or equal to," which sounds like it implies two at once. But remember, the "or" does not mean "both at once." It in fact means "only one of these is true, but they are both possible."

II: Values, Factors, Terms

Every expression has a value. All values are made of one or more terms. And all terms are made of two or more factors.

Once you've performed all of the operations in an expression, you are going to be left with a single number. That number is the expression's value. But notice to find the value we had to actually do all of those operations. The value, therefore, is the result of all of those operations on those numbers.

The expression is split up into smaller values, called terms, which are groups of multiplication and division. Terms add/subtract with other terms.

So, for example, if we have the expression 3 × 12 + 7, there are two terms: 3 × 12 is the first term, and 7 is the second term. Notice that the two terms are separated by a plus sign: +. This is because all terms add/subtract, so you can think of the plus sign as being the thing that separates a problem into its individual terms.

Terms can have subterms. For example, in the term 3(17 - 2), there is a parenthetical statement, 17 - 2, which contains two terms, 17 and -2. Because those two terms exist inside of a term, they are this term's subterms. Likewise, we can call the term that contains these two terms their superterm.

Because terms are groupings of multiplication and division, the numbers you are multiplying and dividing by are that term's factors. Factors multiply in the same way that terms add. So, in the expression 3 × 12 + 7, there are two terms, 3 × 12 and 7, and in the first term you have two factors, 3 and 12; likewise in the second term you have "two" factors, 1 and 7.

IIa: Terms of 0, and Factors of 1

There are what I call "null operations," that is, an operation using a number that has no effect on a problem's value. For addition and subtraction, it's 0. You can add or subtract 0, and it doesn't do anything. For multiplication and division, it's 1; if you multiply or divide by 1, nothing changes.

Because of this, every problem you've ever done technically has an infinite number of these "plus zero" and "times one" operations hiding everywhere (zero property of addition and one property of multiplication). We just don't usually write it cuz... they don't tend to do anything.

But in the example above, we have the term 7. But to describe that number as factors, we said 1 × 7. This is because factors always multiply. But if there's no number next to 7, then what would it be multiplying? It must be 1 × 7, then.

This will come up soon, and is actually very important.

III: Exponents

We use multiplication as a stand-in for repeated addition. 3 + 3 + 3 + 3 + 3 is annoying to write, so instead we notice that we wrote 3 five times, and thus we can say 5 × 3. ----- "Wait why'd you write it in this order?", you might ask of me. "Five times three" has a missing word. It is actually "Five times of three." It's telling me what I have five of. Yes, it is totally acceptable to write 3 × 5, you can multiply numbers within a term in any order (associative/commutative properties of multiplication) but the language itself would imply 5 × 3.

So, like we use multiplication to simplify repeated addition, we need a way to talk about repeated multiplication. That's where exponents come in. Instead of 7 × 7 × 7 × 7, we can write 7⁴, read "seven to the fourth power," which gets shortened to "seven to the fourth." The four, written in superscript (super- as in above, -script as in writing, so "writing above"), is the exponent; the number it is next to is called the base of that exponent. Exponents indicate how many times a particular factor (the base of the exponent) exists in a term.

Notice this definition is different from what you're taught in school. In school, the definition you're given is "multiplied by itself this many times." But that's not a good description, and in fact the language there doesn't even accurately describe what's going on (if you multiply something by itself x times, you have x+1 of that factor, which is not what is going on).

IIIa: Prime Factorization

What we might consider the best way to represent the factors that make up a term is called prime factorization, where we list the prime numbers we'd have to multiply together to get the value of a term. Obviously this doesn't work for every number, but any integer (whole number) or rational number (fraction, terminating decimal, or repeating decimal) can be expressed this way.

Let's try this with the number 144:

144 = (4 × 36) = (2 × 2) × (6 × 6) = 2 × 2 × (2 × 3) × (2 × 3)

Rearranging the factors so all the common factors are next to each other: 2 × 2 × 2 × 2 × 3 × 3. There are four 2's, and two 3's. Therefore, the prime factorization is 2⁴ × 3².

We're gonna continue this next time, going deeper into exponents.

[algebra-as-praxis]

First post!! I wanted to show you how deep a rabbit hole you can dig with a very simple concept like, "What does 'four' mean?"

I have a number line.

So, I'd like you to imagine that you've sat down across from me on your first day in class, and I put some kind of math worksheet with this number line on it that gives you a very simple-sounding task: "Circle four."

If you are like nearly every one of the older students I've had, you'll circle the number 4 on the line. And I'd say something like, "That is a perfectly reasonable response to this request. There is nothing wrong with what you've done. But aren't you suspicious about how simple this question sounds? Did you see that there was a twist coming?"

On a number line, the place where we've written 4 is not actually the value 4. It is more like the position 4; we would call this the ordinal number (a number used to track the positional order of things; another example of an ordinal number is the word "fourth," which tells you that there are three other things before that one).

So, if you just circled the number 4, you definitely circled the position, but the thing you circled isn't worth 4. If you were going to circle the thing that has the value (or cardinality, since we're counting individual objects) of 4, it would look something more like this:

This is because in order to have the value of 4, you need everything that came before it! Imagine counting out candy while your kid sister steals and eats the candies you counted already. When you get to your final number, that number is not how many candies you have, but rather how many you counted cuz your sister's got 'em all! For the value of your candies to stay the same, you need all of them to be there!

"Four" could also look, less straightforwardly, like this:

The distance from the start of the circle to the end is four; and since we're circling the line, the length of the line that is circled is four. If we imagine the circle starts (from the left) at the position of 2.3, it would have to end at the position (from the right) 6.3. If we find the difference of the numbers at these positions, we would be doing the equation 6.3 - 2.3 = 4.

A quick summary:

The position of a number is shown with an ordinal number.

The actual physical quantity of the individual items is shown with a cardinal number, which we more typically call the value.

We can find a value inside the ordinals by finding the distance between any two ordinals, via subtraction.

Okay... But notice we've used two words here that seem to imply the same thing: difference and distance. Let's clear this up real quick:

Difference is the result of subtraction. Generally, if we want to find the difference between two things, we subtract the first thing from the second thing:

second - first = difference

But that's not a rule, so much as a norm. In reality, if you wanted to find the difference of two numbers, the person asking had best be very clear which way they want you to do it, because if you reverse this you will get the negative of the result above!

For example, if you were asked to find the difference between 6 and 15, you would do 15 - 6 = 9. But if it's not clear and you instead did 6 - 15, you get -9.

If, however, someone wanted to know how far apart these two numbers are, they would not be asking the difference but rather the distance between these numbers. The result would therefore be the absolute value of either one of these subtractions:

|15 - 6| = |9| = 9, |6 - 15| = |-9| = 9 Same result!

This is in actuality the real use of absolute values. Absolute value is typically shown as "the distance from zero," which is kind of true? But it's really for finding distances of any kind -- in fact, it's later used to mean exactly that when you get into complex algebra (that is, algebra with imaginary numbers) and linear algebra (matrices and vectors).

But that's for later.