ทำไม 0.999… ถึงเท่ากับ 1 พิสูจน์แบบละเอียดพร้อมสมการทีละขั้นตอนสำหรับนักเรียน ม.ต้น และ ม.ปลาย

ทำไม 0.999… ถึงเท่ากับ 1: พิสูจน์แบบละเอียด

ก่อนที่เราจะไปดูวิธีพิสูจน์ พี่กฤษณ์อยากให้น้องๆ ลองคิดดูว่า ปกติแล้วเวลาที่เราเห็นตัวเลขทศนิยมไม่รู้จบแบบมีทศนิยมซ้ำ เราก็สามารถแปลงให้เป็นรูปเศษส่วนได้ใช่ไหมครับ? อย่างเช่น $0.333… = frac{1}{3}$ หรือ $0.666… = frac{2}{3}$ แล้วถ้าเป็น $0.999…$ ล่ะครับ มันจะเท่ากับเท่าไหร่? เรามาดูการพิสูจน์ง่ายๆ กันก่อนครับ

วิธีที่ 1: การใช้พีชคณิตเบื้องต้น (สำหรับน้องๆ ม.ต้น และ ม.ปลาย)

วิธีนี้เป็นวิธีที่นิยมที่สุดและเข้าใจง่ายที่สุดเลยครับ เราจะใช้หลักการแก้สมการเข้ามาช่วยครับ

1. กำหนดให้ตัวแปรเท่ากับค่าที่เราสงสัย:

   เรากำหนดให้ $x$ มีค่าเท่ากับ $0.999…$ ครับ

   $x = 0.999…$

   หรือจะเขียนแบบให้เป็นสมการที่ 1 ก็ได้ครับ

   $(1)quad x = 0.999…$

2. คูณสมการด้วย 10:

   เพื่อให้ทศนิยมเลื่อนไปหนึ่งตำแหน่ง เราจะนำ 10 ไปคูณทั้งสองข้างของสมการที่ 1 ครับ

   $10 times x = 10 times 0.999…$

   ผลลัพธ์ที่ได้คือ

   $10x=9.999\dots$

   เราให้เป็นสมการที่ 2 ครับ

   $\left(2\right)10x=9.999\dots$

3. นำสมการมาลบกัน:

   ตอนนี้เรามีสองสมการที่ดูคล้ายกันมากเลยใช่ไหมครับ ลองนำสมการที่ 2 ลบด้วยสมการที่ 1 ดูครับ

   $(10x – x) = (9.999… – 0.999…)$

   มาดูแต่ละข้างกันครับ:

   • ข้างซ้าย: $10x – x = 9x$
   • ข้างขวา: $9.999… – 0.999… = 9$ (เพราะทศนิยมที่ซ้ำกันจะหักล้างกันไปหมด เหลือแค่ตัวเลขหน้าทศนิยมครับ)

   ดังนั้นเราจะได้สมการใหม่เป็น:

   $9x = 9$

4. แก้สมการหาค่า x:

   จากสมการ $9x = 9$ เราสามารถหาค่า $x$ ได้โดยการนำ 9 ไปหารทั้งสองข้างครับ

   $frac{9x}{9} = frac{9}{9}$

   และแล้วเราก็ได้ผลลัพธ์ออกมาครับ

   $x = 1$

จากที่เรากำหนดให้ $x = 0.999…$ ในตอนแรก และตอนนี้เราได้พิสูจน์แล้วว่า $x=1$ แสดงว่า $0.999…$ มีค่าเท่ากับ $1$ จริงๆ ครับ!

วิธีที่ 2: การแปลงทศนิยมซ้ำเป็นเศษส่วน (สำหรับน้องๆ ม.ต้น และ ม.ปลาย)

วิธีนี้จะใช้ความรู้เรื่องการแปลงทศนิยมซ้ำเป็นเศษส่วน ซึ่งน้องๆ ม.ต้น อาจจะเคยเรียนมาบ้างแล้วครับ

1. หลักการแปลงทศนิยมซ้ำ:

   ถ้าเรามีทศนิยมซ้ำ เช่น $0.abar{b}c$ ที่มีตัวเลขซ้ำแค่ตัวเดียวอย่าง $0.dbar{d}bar{d}…$ เราสามารถแปลงเป็นเศษส่วนได้โดยเอาตัวเลขที่ซ้ำนั้นเป็นตัวเศษ และมี 9 เป็นตัวส่วนตามจำนวนหลักที่ซ้ำครับ

   ตัวอย่างที่น้องๆ คุ้นเคยกันดีคือ:

   $0.333… = frac{3}{9} = frac{1}{3}$
   $0.666… = frac{6}{9} = frac{2}{3}$

2. นำหลักการนี้มาใช้กับ $0.999…$:

   ในกรณีของ $0.999…$ ตัวเลขที่ซ้ำคือ 9 และมีเพียงหลักเดียวที่ซ้ำครับ ดังนั้น เราจะเขียนในรูปเศษส่วนได้เป็น:

   $0.999… = frac{9}{9}$

3. หาค่าของเศษส่วน:

   เมื่อเรานำ 9 หารด้วย 9 ผลลัพธ์ที่ได้คือ:

   $frac{9}{9} = 1$

ดังนั้น จากวิธีนี้ก็ยืนยันอีกครั้งว่า $0.999\dots$ เท่ากับ $1$ ครับ

วิธีที่ 3: การใช้อนุกรมเรขาคณิตอนันต์ (สำหรับน้องๆ ม.ปลาย)

วิธีนี้น้องๆ ม.ปลาย ที่เรียนเรื่องลำดับและอนุกรมมาแล้วจะเข้าใจได้ดีเลยครับ เราสามารถมอง $0.999…$ เป็นผลบวกของอนุกรมเรขาคณิตอนันต์ได้ครับ

1. เขียน $0.999…$ ในรูปอนุกรม:

   เราสามารถแตก $0.999…$ ออกมาเป็นผลบวกของแต่ละหลักได้ดังนี้ครับ

   $0.999… = 0.9 + 0.09 + 0.009 + 0.0009 + ldots$

2. ระบุพจน์แรกและอัตราส่วนร่วม:

   อนุกรมนี้คืออนุกรมเรขาคณิตอนันต์ครับ โดยมี:

   • พจน์แรก ($a$) คือ $0.9$
   • อัตราส่วนร่วม ($r$) คือ การนำพจน์ที่สองหารด้วยพจน์แรก หรือพจน์ที่สามหารด้วยพจน์ที่สองครับ

   $r = frac{0.09}{0.9} = 0.1$

   หรือเขียนในรูปเศษส่วนได้ว่า $a = frac{9}{10}$ และ $r = frac{1}{10}$ ครับ

3. ใช้สูตรผลบวกอนุกรมเรขาคณิตอนันต์:

   สำหรับอนุกรมเรขาคณิตอนันต์ที่มีค่าสัมบูรณ์ของอัตราส่วนร่วม $|r| < 1$ (ในกรณีนี้ $0.1 < 1$ จึงเข้าเงื่อนไขครับ) ผลบวกอนันต์ $S$ คำนวณได้จากสูตร:

   $S = frac{a}{1-r}$

4. แทนค่าและหาผลลัพธ์:

   แทนค่า $a = 0.9$ และ $r = 0.1$ ลงในสูตรครับ

   $S = frac{0.9}{1-0.1}$

   คำนวณส่วนที่อยู่ข้างล่าง:

   $S = frac{0.9}{0.9}$

   และสุดท้ายเราก็จะได้:

   $S = 1$

เห็นไหมครับว่าไม่ว่าเราจะใช้วิธีไหน ผลลัพธ์ก็ออกมาเหมือนกันเสมอ! $0.999…$ จึงเท่ากับ $1$ จริงๆ ครับ

ทำไมน้องๆ ถึงมักรู้สึกว่ามันแปลก หรือขัดแย้งกับความรู้สึก?

เหตุผลที่หลายคนรู้สึกแปลกๆ กับเรื่องนี้ เพราะเรามักจะคิดว่าต้องมี “ช่องว่าง” เล็กๆ ระหว่าง $0.999…$ กับ $1$ ใช่ไหมครับ? เหมือนว่ามัน “ใกล้เคียง” แต่ไม่ “เท่ากัน” ซะทีเดียว

แต่ในทางคณิตศาสตร์แล้ว ตัวเลขจริง (Real Numbers) มีความหนาแน่นมากครับ การที่ $0.999…$ เท่ากับ $1$ หมายความว่าไม่มีตัวเลขจริงอื่นๆ แทรกอยู่ระหว่าง $0.999…$ และ $1$ เลยครับ ถ้ามีช่องว่างแม้แต่นิดเดียว แสดงว่าสองตัวนี้ไม่เท่ากันครับ

ลองคิดดูว่าถ้า $0.999…$ ไม่เท่ากับ $1$ มันจะต้องมีตัวเลขอะไรบางอย่างที่อยู่ระหว่างกลางใช่ไหมครับ? เช่น <math data-latex="0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999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