Decimal notation

Seven­teen is the num­ber that rep­re­sents as many things as there are x marks at the end of this sen­tence: xxxxxxxxxxxxxxxxx. Writ­ing out num­bers by say­ing “the num­ber rep­re­sent­ing how many things there are in this pile:” gets un­wieldy when the pile gets large. Thus, we rep­re­sent num­bers us­ing the nu­mer­als 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Speci­fi­cally, we write the num­ber rep­re­sent­ing this many things: xxx as “3″, and the num­ber rep­re­sent­ing this many things: xxxxxxxxxxx as 11, and the num­ber sev­en­teen as “17”. This is called “dec­i­mal no­ta­tion,” be­cause there are ten differ­ent sym­bols that we use. Num­bers don’t have to be writ­ten down in dec­i­mal no­ta­tion, it’s also pos­si­ble to write them down in other no­ta­tions such as bi­nary no­ta­tion. Some num­bers can’t even be writ­ten out in dec­i­mal no­ta­tion (in full); con­sider, for ex­am­ple, the num­ber \(e\) which, in dec­i­mal no­ta­tion, starts out with the digits 2.71828… and just keeps go­ing.

How dec­i­mal no­ta­tion works

How do you know that 17 is the num­ber that rep­re­sents the num­ber of xs in this se­quence: xxxxxxxxxxxxxxxxx? In prac­tice, you know this be­cause the rules of dec­i­mal no­ta­tion were in­grained in you in a young child. But do you know those rules ex­plic­itly? Could you write out a se­ries of rules for tak­ing in some in­put sym­bols like ‘2’, ‘4’, and ‘6’ and us­ing those to figure out how many peb­bles to add to a pile?

The an­swer, of course, is this many:

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

But how do we perform that con­ver­sion in gen­eral?

In short, the num­ber 246 rep­re­sents \((2 \cdot 100) + (4 \cdot 10) + (6 \cdot 1),\) so as long as we know how to do ad­di­tion and mul­ti­pli­ca­tion, and as long as we know what the ba­sic nu­mer­als 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 mean, and as long as we know how to get to pow­ers of 10 (1, 10, 100, 1000, …), then we can ex­plic­itly un­der­stand dec­i­mal no­ta­tion.

(What do the ba­sic nu­mer­als 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 mean? By con­ven­tion, they rep­re­sent as many things as are in the fol­low­ing ten se­quences of xs: , x, xx, xxx, xxxx, xxxxx, xxxxxx, xxxxxxx, xxxxxxxx, and xxxxxxxxx, re­spec­tively.)

This ex­pla­na­tion as­sumes that you’re already quite fa­mil­iar with dec­i­mal no­ta­tion. Ex­plain­ing dec­i­mal no­ta­tion from scratch to some­one who doesn’t already know it (which was a task peo­ple ac­tu­ally had to do back when half the world was us­ing Ro­man nu­mer­als, a much less con­ve­nient sys­tem for rep­re­sent­ing num­bers) is a fun task; to see what that looks like, re­fer to Rep­re­sent­ing num­bers from scratch.

Other com­mon notations

The above text made use of unary no­ta­tion, which is a method of rep­re­sent­ing num­bers by mak­ing a num­ber of marks that cor­re­spond to the rep­re­sented num­ber. For ex­am­ple, in unary no­ta­tion, 17 is writ­ten xxxxxxxxxxxxxxxxx (or ||||||||||||||||| or what­ever, the ac­tual marks don’t mat­ter). This is per­haps some­what eas­ier to un­der­stand, but writ­ing large num­bers like 93846793284756 gets rather un­gainly.

His­tor­i­cal no­ta­tions in­clude Ro­man nu­mer­als, which were a pretty bad way to rep­re­sent num­bers. (It took hu­man­ity quite some time to find good tools for rep­re­sent­ing num­bers; the dec­i­mal no­ta­tion that’s been in­grained in your head since early child­hood is the re­sult of many cen­turies worth of effort. It’s much harder to in­vent good rep­re­sen­ta­tions of num­bers when you don’t even have good tools for writ­ing down and rea­son­ing about num­bers. Fur­ther­more, the mod­ern tools for rep­re­sent­ing num­bers aren’t nec­es­sar­ily ideal!)

Com­mon no­ta­tions in mod­ern times (aside from dec­i­mal no­ta­tion) in­clude bi­nary no­ta­tion (of­ten used by com­put­ers), hex­adec­i­mal no­ta­tion (which is a use­ful for­mat for hu­mans read­ing bi­nary no­ta­tion). Bi­nary no­ta­tion and hex­adec­i­mal no­ta­tion are very similar to dec­i­mal no­ta­tion, with the differ­ence that bi­nary uses only two dis­tinct sym­bols (in­stead of ten), and hex­adec­i­mal uses six­teen.

Children:

  • 0.999...=1

    No, it’s not “in­finites­i­mally far” from 1 or any­thing like that. 0.999… and 1 are liter­ally the same num­ber.

Parents:

  • Mathematics

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