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Matematikte (-) ile (-) nin çarpımı neden (+) olur?

Aslında çok eğlenceli olabilecek matematik bizlere katı formüllerle ve mantığın kolay kabul edemeyeceği ifadelerle öğretilince bir kabus olup çıkıyor. Artının artı ile, eksinin eksi ile çarpım sonucu artı iken artı ile eksinin çarpım sonucu eksi oluyor. Peki bunun mantıki izahı nedir? Yani -5 derece sıcaklıkla -8 derece sıcaklığı çarpınca sonuç +40 derece olup ortalık ısınıyor mu? 

Tabii bu bir  şaka,  şaşırtmaca. Esas bilmemiz gereken (-2)x(-2)=(+4) diye bir eşitlik yazdığımızda, bunun sadece rakamların ve önlerindeki işaretlerin belirlediği mantıksal bir denklem olmadığı, bir beyan, bir ifade olduğudur.  Eğer sayıları bir çizgi üzerinde gösterirsek, ‘-1′ sıfırın eksi tarafındaki ilk sayı olarak düşünülebilir ama eşitlik içinde bu böyle değildir. Çizginin neresinde olursanız olun bir adım geri atmaktır. Yani çizgide ‘+4′ noktasında iseniz ve ona

‘-1′ ilave ederseniz, bir adım geri atarak ‘+3’e gelmiş olursunuz. Toplama ve çıkartmada nispeten kolay olan bu açıklama, iş çarpmaya gelince biraz zorlaşıyor. Örneğin haftanın 5 günü işe otobüs ile gidip geliyorsunuz. Her

sefer bir milyonluk bir biletle yapılıyor. 10 milyon tutarında 10 tane bilet aldınız. Her gün gidiş-geliş kullandıkça iki tanesi eksiliyor. Bunun eşitlikteki yeri ‘-2’dir. Siz bu işi 5 gün süresince yani 5 kere yaparsanız (-2)x(+5)=(-10) olur ki biletler

biter. Diyelim ki bayram tatilinin iki günü o haftanın perşembe ve cuma günlerine denk geldi ve tatil. Bu sefer yapmanız gereken hareketi yapmıyorsunuz.  İki günlük 4 bileti kullanmıyorsunuz. Bu hareket yapmanız gerekene göre negatif

yani ters yönde bir harekettir. Her gün bilet almak yerine iki gün süresince hiç bilet kullanmıyorsunuz.  İki kere negatif hareketi ‘-2′ bilet üzerinde yapınca o hafta elinizde (-2)x(-2)=(+4) bilet kalıyor.  Hala biraz karışık değil mi? Bir örnek daha verelim. Bir eşitliğin başına ‘-2′ yazdığınız zaman başlangıçta bu sizin sıfır noktasından iki kere geri sıçrayarak ‘-

2′ noktasına ulaşacağınız anlamına gelir. Ama siz yapacağınız bu hareketin tam tersini yani negatifini iki defa yapıyorsunuz. Sıfırdan ‘-2’ye sıçrama hareketini iki kere ters yönde (-2) yapıyorsunuz ve sonunda ‘+4′ noktasına ulaşıyorsunuz. Ters bir kararın tersini yapınca doğruyu buluyorsunuz yani.

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