T 1,T 2andT 3.Andofoursewith these fewexamplesweanseeashortutforndingothertriangularnumbers.

Épreuve de setion européenne

Amagial multiplyingmethod

I well remember the wonder that I experiened when I learned about logarithms. The thing that

mostaughtmyattentionwasthat yououldmultiplytwonumbers throughanot-so-simpleaddition

proess...Ofourse, this isjust a taste of amuh largerand grander topi in Mathematis,but it is

meant to give a onnetion to what I want to show you this time, namely how you an multiply by

adding,using amuhsimpler,arithmeti basis.

Beforeweanproeed, wehavetointroduetriangularNumbers.10isatriangularnumber,beause

10thingsanbearrangedin atriangulararraylikethis:

⋆

⋆ ⋆

⋆ ⋆ ⋆

⋆ ⋆ ⋆ ⋆

Fromthissortofpitureitiseasytoformanddeterminemanyothertriangularnumbers.

Hereweseethat1,3,and6aretherstthreetriangularnumbers

T ₁

,

T ₂

and

T ₃

.Andofoursewith these fewexamplesweanseeashortutforndingothertriangularnumbers.

T 4 = 10 = 1 + 2 + 3 + 4 T 3 = 6 = 1 + 2 + 3 T 2 = 3 = 1 + 2

T 1 = 1 = 1

Now we're ready to show you the magial way to multiply without multiplying anything. I all it

theMagialMultiplying Method (orthe3M way).First,I mustonfesswewillsubtratalittletoo.

Seond,youwillneedatableontainingthevaluesoftriangularnumbers.

Let'stakeasanexample

15 × 9

:

1. Takethelargerfator15andnd

T ₁₅

inthetablementionedabove.

2. Subtrat

1

from

9

,thesmallerfator,getting

8

.Find

T ₈

inthetable.

3. Subtratthetwofators,

15 − 9

;that's

6

.Find

T ₆

.

4. AddtheresultsofSteps1&2,thensubtrattheresultfrom Step3.That's yourprodut!

Well,Ineversaiditwasgoingto beeasier,shorteroranythinglikethat.

AdaptedfromTerryTrotterwebsitewww.trottermath.net

Questions

1. Therstparagraphofthetextreferstoafamouspropertyoflogarithms.Whihone?

2. a. Express

T _{n +1}

in termsof

T _n

and

n

.

b. Usea. togiveinatable thevaluesof

T n

^,for

n

from1to15.

3. a. Chekthealulation

15 × 9

in thetext.

b. Compute

13 × 12

using the3M way.

4. Translateintoanalgebraiformulathe3M

a × b

,where

a

and

b

aretwounequalnaturalnumbers.

5. a. Provethatforanynaturalnumber

n

,

T _n = ^{n ( n +1)}

2

^.

b. Hene,provetheformulagiveninquestion4togettheprodut

a × b

,

a

beinggreaterthan

b

. . Whathappensif

a = b

?Adaptthemethodtosquarenaturalnumbers.