Integers: Intro (Math 0)

The in­te­gers are an ex­ten­sion of the nat­u­ral num­bers into the nega­tives.

Nega­tive numbers

Nega­tive num­bers are num­bers less than zero. They are no­tated as num­bers with a minus sign in front of them, like \(-2\) or \(-15\) or \(-6387\).

When many peo­ple are first in­tro­duced to the con­cept of nega­tive num­bers, one ques­tion they have is, “how do nega­tive num­bers ex­ist? How can we have less than noth­ing of some­thing?”

Usu­ally this ques­tion is an­swered with some­thing about trav­el­ling along the num­ber line and then go­ing in the op­po­site di­rec­tion past zero. But we can also an­swer this ques­tion from the tra­di­tional per­spec­tive of quan­tities, us­ing an­ti­mat­ter.

Arith­metic with anti-cows

Mat­ter and an­ti­mat­ter are sub­stances that an­nihilate each other — they can­cel each other out. In the same way, pos­i­tive and nega­tive quan­tities can­cel each other out to some ex­tent when added to­gether.

cow artwork

Let’s cre­ate a phys­i­cal ex­am­ple of this an­ti­mat­ter phe­nomenon, and con­sider a col­lec­tion of cows. di­a­gram of cows If a nat­u­ral num­ber is how many cows we have, then nega­tive num­bers are the pres­ence of anti-cows. For the sake of easy read­ing, let’s colour a cow red, and an anti-cow blue: di­a­gram of red cow = cow, blue cow = an­ti­cow For ex­am­ple, if you have \(3\) anti-cows, you have \(-3\) cows. If you have \(128\) anti-cows, you have \(-128\) cows, and so on.

When a cow and an anti-cow come into con­tact, they an­nihilate each other and noth­ing is left (ex­cept tech­ni­cally an enor­mous burst of en­ergy that could cause a mag­ni­tude 9 earth­quake, but we won’t get into the nitty-gritty physics of an­ti­mat­ter an­nihila­tion here). di­a­gram of cow/​anti-cow an­nihila­tion More­over, each cow will an­nihilate ex­actly one anti-cow, and vice versa.

This lets us perform sub­trac­tion by adding anti-cows in­stead of tak­ing away cows. If you want to perform \(6 - 4\), you imag­ine you have a col­lec­tion of six cows, and you add four anti-cows to the mix, which an­nihilate four of the cows leav­ing you with two. This is ex­actly the same as if you’d taken away four cows (ex­cept that you’ve now re­leased enough en­ergy to power the en­tire world’s en­ergy de­mands for seven months). di­a­gram of 6 − 4

This form of adding nega­tives is slightly more flex­ible than reg­u­lar sub­trac­tion. In reg­u­lar sub­trac­tion, you couldn’t take away more cows than there already were. But when you add nega­tives, you can keep adding anti-cows af­ter there are no cows left, at which point you just have more and more anti-cows, giv­ing you a more and more nega­tive num­ber. So if you wanted to sub­tract \(4 - 6\), you couldn’t take away six cows from four, but you could add six anti-cows, and four of them would an­nihilate the four cows, leav­ing you with two anti-cows or a fi­nal an­swer of \(-2\). di­a­gram of 4 − 6

Use­ful prop­er­ties of nega­tive numbers

Ex­tend­ing the nat­u­ral num­bers into nega­tives gives us lots of use­ful prop­er­ties when do­ing ar­ith­metic.

Ad­di­tion of nega­tive numbers

Nega­tive num­bers add up just like pos­i­tive ones. If you have \(-3\) and \(-7\), they add up to \(-10\), just like three anti-cows and seven anti-cows would come to­gether to make ten anti-cows. di­a­gram of (-3) + (-7)

Know­ing this, how would you calcu­late \((-6) + (-8) + (-12) + (-20)\)?

Sim­ply add up all the nega­tive quan­tities to­gether.
$$6 + 8 + 12 + 20 = 46 \to (-6) + (-8) + (-12) + (-20) = -46$$

Sub­trac­tion as ad­di­tive inversion

If we rep­re­sent a nat­u­ral num­ber as a num­ber of cows, then the same num­ber of anti-cows is called the ad­di­tive in­verse of that num­ber, be­cause a num­ber and its ad­di­tive in­verse add up to zero (equal num­bers of cows and anti-cows will all an­nihilate each other to give noth­ing), which is the “ad­di­tive iden­tity” (or “the num­ber that changes noth­ing when you add it to some­thing”).

Us­ing ad­di­tive in­verses al­lows us to ex­press sub­trac­tion in the form of ad­di­tion — for ex­am­ple, \(5 - 2\) can be rewrit­ten as \(5 + (-2)\) (which just means that in­stead of tak­ing away two cows, you’re adding two anti-cows). When we do this, sud­denly we can re­ar­range sub­trac­tion op­er­a­tions along with ad­di­tion op­er­a­tions even though sub­trac­tion isn’t com­mu­ta­tive.

Then, if you have a gi­ant ad­di­tion and sub­trac­tion prob­lem you want to solve, you can add up all the num­bers you want to add (into a big col­lec­tion of cows), then add up all the num­bers you want to sub­tract (into a big col­lec­tion of anti-cows), and just do one sub­trac­tion (an­nihilat­ing cow/​anti-cow pairs) to get your an­swer. For ex­am­ple, \(6 - 2 + 7 - 5\) is re­ally just \(6 + (-2) + 7 + (-5)\) which is equal to \(6 + 7 + (-5) + (-2)\). Then, you add up the num­ber of cows (13) and anti-cows (7), and an­nihilate them to get the fi­nal an­swer of \(6\).

Here’s an­other ex­er­cise for you: How would you calcu­late \(13 + 8 - 5 + 6 + 4 - 12 - 9 + 1\)?

First, rewrite each sub­trac­tion op­er­a­tion as ad­di­tion of a nega­tive num­ber:
$$13 + 8 + (-5) + 6 + 4 + (-12) + (-9) + 1$$

Then re­ar­range the equa­tion to group all the pos­i­tive and nega­tive num­bers (cows and anti-cows) to­gether:

$$13 + 8 + 6 + 4 + 1 + (-5) + (-12) + (-9)$$

Then add up all the pos­i­tive and nega­tive num­bers sep­a­rately:

$$(13 + 8 + 6 + 4 + 1) + ((-5) + (-12) + (-9)) = 32 + (-26)$$

And fi­nally, sub­tract it all at once:

$$32 + (-26) = 32 - 26 = 6$$

And voilà, you have six cows. <div><div>

What about \(8 - 6 + 4 - 13 + 7 - 5 - 9 + 12\)?

Same as be­fore, we rewrite, re­group, add up, and sub­tract:

$$8 + (-6) + 4 + (- 13) + 7 + (- 5) + (- 9) + 12 \\ \Downarrow$$

$$8 + 4 + 7 + 12 + (-6) + (-13) + (-5) + (-9) \\ \Downarrow$$

$$(8 + 4 + 7 + 12) + ((-6) + (-13) + (-5) + (-9)) = 31 + (-33)$$

$$31 + (-33) = 31 - 33$$

But hold on — now you have more anti-cows than cows, so you can’t sub­tract di­rectly! Not to worry — sim­ply flip the sub­trac­tion prob­lem around, and sub­tract the cows from the anti-cows, know­ing that the fi­nal re­sult will be nega­tive in­stead.

$$31 - 33 = -(33 - 31) = -2$$
<div><div>

The set of integers

Back to defin­ing the in­te­gers. The set of in­te­gers is sim­ply the set of nat­u­ral num­bers and their ad­di­tive in­verses — i.e. the set of num­bers of cows you can have if you can have anti-cows as well. This set branches out to in­finity on both sides, and we usu­ally write it as \(\{ \ldots, -2, -1, 0, 1, 2, \ldots \}\).

cow mul­ti­pli­ca­tion with herds and anti-herds

check read­abil­ity scores to make sure it’s in grade 7-8 range

Parents:

  • Integer
    • Mathematics

      Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.