Toán học thật kỳ diệu: MỘT cộng MỘT bằng bao nhiêu?

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LBC – MỘT CỘNG MỘT BẰNG BAO NHIÊU? là một câu hỏi đơn giản mà một đứa trẻ mẫu giáo lớn cũng có thể trả lời đúng. Ấy thế mà những câu trả lời của câu hỏi này lại là cơ sở của nhiều ngành Toán học như Tin học, Điều khiển học…

Câu hỏi này cũng gắn với một kỷ niệm đẹp về những ngày tôi còn đứng trên bục giảng của trường Đại học Sư phạm. Nhân ngày Nhà giáo Việt Nam, xin đăng lại để các bạn bè chia sẻ niềm vui đối với người thầy giáo Toán. Kỷ niệm đó chính là nội dung câu chuyện sau:

Một chiều đầu hạ, đang ngồi đọc sách trên phòng làm việc tầng hai, bỗng nghe tiếng chuông reo, ra lan can ngó xuống, tôi thấy một thanh niên cao ráo, ăn mặc lịch sự, cầm một bó hoa tươi đứng trước cửa nhà. Vừa nhìn thấy tôi, người thanh niên nhanh miệng:
Em chào thầy ạ!

Tôi mời khách vào nhà. Anh thanh niên tặng tôi bó hoa rồi nói:
Thưa thầy! Hôm nay em đến thăm và cám ơn thầy, vì thầy đã giúp em xin được việc làm ạ.

Tôi ngạc nhiên:
– Em bảo sao? Tôi giúp em xin được việc làm à?

Anh thanh niên bèn kể lại cho tôi nghe câu chuyện xin việc làm của mình như sau:
Dạ, thưa thầy! Em là Minh, sinh viên cũ của khoa Toán. Thầy có thể không nhớ chúng em, nhưng chúng em không bao giờ quên được thầy, vì thầy đã dạy chúng em rất nhiều môn: Tập hợp, Lô gic, Đại số đại cương, Lý thuyết vành và mô-đun môn, Lý thuyết Galoa…

Em đã tốt nghiệp Đại học cách đây 5 năm nhưng chưa xin được việc ở đâu cả, nộp hồ sơ xin việc rất nhiều nơi nhưng chẳng cơ quan nào nhận.

Sau Tết Nguyên đán, một Công ty liên doanh với nước ngoài thông báo tuyển người lao động có trình độ Đại học, em vội nộp hồ sơ đăng ký dự tuyển. Một tháng sau, Công ty thông báo kết quả sơ tuyển vòng một – vòng xét tuyển hồ sơ. Em có tên trong danh sách 120 người trúng tuyển, được tham gia dự thi 3 môn ở vòng 2: Ngoại ngữ, Tin học và Chuyên ngành. 25 người đã vượt qua kỳ thi, được dự tuyển vòng 3 – vòng phỏng vấn trực tiếp. Công ty sẽ phỏng vấn các ứng viên để tuyển 4 người làm việc tại Công ty.

Hôm tham dự phỏng vấn, em là người thứ 15. Các ứng viên vào phỏng vấn trước em, ra khỏi phòng họ đều nói nhiều câu hỏi hóc búa lắm, không biết trả lời đúng hay sai. Đến lượt em, ngồi đối diện trước 5 vị giám khảo, rất hồi hộp và lo lắng. Mỗi vị giám khảo hỏi một câu, em đều trả lời được. Giám đốc là người cuối cùng phỏng vấn em, lo quá, vì nếu không trả lời được câu hỏi của Giám đốc thì em sẽ đánh mất cơ hội xin việc làm.

Giám đốc nói:
Anh hãy bình tĩnh suy nghĩ trả lời câu hỏi sau: Theo anh, một cộng một bằng bao nhiêu?

Em thoáng nghĩ, câu hỏi này quen quá, hình như mình đã được gặp ở đâu rồi thì phải? Ngồi lặng im một lúc, bỗng nhiên em nhớ lại bài giảng của thầy ngày nào.

Cách đây 9 năm, khi học năm thứ nhất Đại học, vào buổi học thứ 2 môn Lô gic, thầy dạy bài “Các phép toán mệnh đề”. Sau khi ôn lại các kiến thức đã học, thầy bắt đầu bài giảng bằng câu hỏi:
Một cộng một bằng bao nhiêu?

phep-cong-mot-cong-mot
1 + 1 = ?

Cả lớp ai cũng ngạc nhiên vì câu hỏi quá dễ. Đoán được suy nghĩ của chúng em, thầy cười và nói:
Em nào trả lời đúng câu hỏi này, tôi sẽ thưởng một cuốn tài liệu học môn Lô gic.

Cả lớp nhao nhao giơ tay. Thầy nhìn cả lớp rồi nói:
Trật tự, tôi rất mừng vì cả lớp ai cũng muốn được trả lời câu hỏi này và tôi biết các bạn sẽ có cùng một kết quả. Do vậy, để tiết kiệm thời gian, tôi mời Lớp trưởng trả lời thay cho cả lớp, nếu đúng, tôi thưởng cho các em có mặt trong buổi học hôm nay mỗi em một cuốn tài liệu như đã hứa. Các em có đồng ý không?

Cả lớp đồng thanh:
Chúng em đồng ý.

Thầy nói:
Được, Lớp trưởng, em hãy trả lời câu hỏi thay các bạn!

Lớp trưởng đứng dậy và trả lời:
Thưa thầy! 1+1 = 2 ạ.

Thầy cười:
Còn ai có câu trả lời khác không?

Cả lớp ngồi im. Thầy giáo tiếp:
– Im lặng là sự đồng ý, như vậy, mọi người đều có cùng một câu trả lời là 1+1 = 2. Bây giờ chúng ta xét một vài ví dụ, qua đó các em sẽ tự khẳng định được câu trả lời của mình đúng hay không nhé!

Ví dụ 1: Tôi có một giọt nước trên mặt kính, nhỏ thêm một giọt nước vào giọt nước đó. Hỏi có mấy giọt nước?

Cả lớp đồng thanh:
Một giọt nước ạ.

Thầy khẳng định:
– Đúng! Như vậy, các em đã kết luận: Một giọt nước cộng một giọt nước bằng một giọt nước, hay 1+1 = 1. Trong sự im lặng, có tiếng thốt lên: Hay nhỉ!

Thầy nói tiếp:
Chúng ta xét ví dụ 2.

Trong sách Toán lớp 1 (Chương trình thực nghiệm của GS. TSKH Hồ Ngọc Đại) có trình bày phép cộng trong hệ đếm cơ số 2. Thông thường, chúng ta dùng hệ đếm cơ số 10, trong hệ đếm này, 10 ký số: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 được dùng để ghi các số; 1 chục là 10; 10 chục là 100; 10 trăm là 1000… Trong hệ đếm cơ số 2, ta dùng các ký số: 0,1 để ghi các số; 1 chục (chỉ có 2 đơn vị) là 10; 2 chục là 100; 2 trăm là 1000… Phép cộng trong hệ đếm cơ số 2 được thực hiện như sau: 1 + 1 = 10; 10 + 10 = 100… Vậy ta có 1 + 1 = 10. Các em thấy thế nào?

Cả lớp im lặng. Thầy lại nói tiếp:
Ví dụ cuối cùng.

Trên một tấm Pano quảng cáo ở một thành phố nọ, để tuyên truyền chính sách dân số “Mỗi gia đỉnh chỉ nên có 2 con” của Đảng và Nhà nước, nhà quảng cáo đã vẽ một cặp vợ chồng tay dắt hai đứa con. Phía dưới bức tranh có ghi chú: 1 vợ + 1 chồng = 2 con. Nếu chúng ta thừa nhận cách viết của nhà quảng cáo trên là hợp lệ thì chúng ta có: Vế trái là cặp vợ chồng, vế phải là số con của cặp vợ chồng đó và 1 vợ + 1 chồng = 2 con, ta viết gọn là 1 + 1 = 2. Từ lập luận trên ta sẽ có các kết quả sau:
a) Ta có 1 + 1 = 0, nếu cặp vợ chồng không có con (vô sinh).
b) Ta có 1 + 1 = 1, nếu cặp vợ chồng có 1 con.
c) Ta có 1 + 1 = 2, nếu cặp vợ chồng có 2 con.

d) Ta có 1 + 1 = k (k nguyên, k ≥ 0), nếu cặp vợ chồng có k con.

Từ các ví dụ trên, ta có thể trả lời:
1 + 1 = 0; 1 + 1 = 1; 1 + 1 = 2; 1 + 1 = 3; …1 + 1 = 10; v.v…
Như vậy, một cộng một bằng bao nhiêu là do mục đích, yêu cầu và điều kiện mà chúng ta thực hiện phép toán cộng đó.

Tóm lại: 1 + 1 = 2 không phải lúc nào cũng đúng, câu trả lời này chỉ đúng trong toán phổ thông (Toán học sơ cấp), còn những trường hợp khác các em chưa hề gặp bao giờ. Sang năm thứ 2, sau khi học xong môn Đại số đại cương, các em sẽ rất ngạc nhiên, vì toàn bộ kiến thức môn Đại số mà các em đã học từ lớp 1 đến lớp 12 ở trường phổ thông thực chất là các tri thức toán có được từ việc thực hiện các phép toán cộng, trừ, nhân, chia, lũy thừa… trên tập số thực R hoặc tập các số phức C. Đó chỉ là những mô hình cụ thể về cấu trúc trường – một trong những cấu trúc cơ bản của Toán học. Thế mới biết, kiến thức môn Toán mà các em đã được học chỉ là một phần vô cùng nhỏ bé trong biển cả mênh mông các kiến thức toán học của nhân loại.

Các em nên nhớ, để trả lời một vấn đề (một câu hỏi), phải suy nghĩ kỹ, phân tích hoàn cảnh, điều kiện, thời gian, địa điểm và mục đích mà vấn đề đó đặt ra, từ đó mới có câu trả lời thích hợp. Để trả lời được những câu hỏi dạng như “một cộng một bằng bao nhiêu?” và để học tốt các môn Toán, chúng ta phải rèn luyện được các phẩm chất như: Không bảo thủ, không sợ khó, không sợ khổ, sợ thiếu thời gian, sợ thiếu tài liệu và sợ thiếu ngoại ngữ.

Tính bảo thủ sẽ làm cho con người trì trệ, không dám thừa nhận những điều mới mẻ, trái ngược với những điều đã biết.

Ngày xưa, con người quan niệm Mặt trời quay quanh Trái đất, mãi đến năm 1534, nhà toán học, thiên văn học người Ba Lan Copernicus (Cô-péc-ních) 1473 – 1543 là người đầu tiên trong tác phẩm vĩ đại của mình “Thuyết thiên thể vận hành” đã chứng minh rằng: Không phải Mặt trời quay quanh Trái đất mà ngược lại, Trái đất quay quanh Mặt trời. Nhà triết học, thiên văn học người ý G.Galilei (Ga-li-lê) 1564 – 1642 vì bảo vệ chân lý “Trái đất quay quanh Mặt trời” (của Cô-pec-ních), đi ngược lại với quy luật trong Kinh thánh dạy là “Mặt trời quay quanh Trái đất”, ông đã bị Tòa án Thiên Chúa giáo kết án tử hình, thiêu trên giàn lửa.

Khoảng 3 thế kỷ sau sự kiện Ga-li-lê, ở Nga xuất hiện một nhà toán học được mệnh danh là Cô-péc-ních toán học, đó chính là nhà toán học vĩ đại Lobachepvsky (Lô-ba-sép-sky) 1792 – 1856, giáo sư toán học tại trường Đại học Kazan. Xuất phát từ sự phủ nhận tiên đề song song “Từ một điểm ở ngoài đường thẳng kẻ được một và chỉ một đường thẳng song song với đường thẳng đã cho” của Hình học Euclid (Ơ-clít) – Hình học mà chúng ta đã được học ở bậc phổ thông, ông đã đưa ra một tiên đề ngược lại: “Qua một điểm ngoài đường thẳng kẻ được vô số các đường thẳng song song với đường thẳng đã cho” thay thế cho tiên đề song song trong hình học Ơ-clít và bằng suy luận lô gic, Lô-ba-sép-sky đã tạo ra một loại Hình học mới gọi là Hình học Lobachepvsky hay Hình học Hypebolic. Những phát minh vĩ đại của nhà toán học Lô-ba-sép-sky không những không được xã hội thời đó thừa nhận mà ông còn bị coi là một kẻ tâm thần và bị đuổi khỏi trường Đại học.

Vài chục năm sau ông mất, khi trình độ khoa học thế giới phát triển thì các công trình nghiên cứu của ông mới được xã hội công nhận. Nhà toán học Lô-ba-sép-sky là người đặt nền móng cho Hình học phi Ơ-clít (gồm Hình học Hypebolic và Hình học Eliptic). Chúng ta có thể hiểu một cách đơn giản sự khác biệt Hình học Ơ-clít và Hình học phi Ơ-clít như sau:

Trong Hình học Ơ-clít, tổng các góc trong một tam giác bằng 180 độ. Nhưng ngược lại, trong Hình học phi Ơ-clít thì tổng các góc trong của một tam giác khác 180 độ. Chẳng hạn, trong Hình học Lobachepvsky, tổng các góc trong của một tam giác nhỏ hơn 180 độ, còn trong Hình học cầu (hay Hình hoc Eliptic) thì tổng các góc trong của một tam giác lớn hơn 180 độ. Hình học phi Ơ-clít là cơ sở toán học cho lý thuyết tương đối của Albert Einsein (Anh-sơ-tanh), cơ học lượng tử và vật lý thiên văn…

Qua câu chuyện trên cho ta thấy: Đừng ngộ nhận về vốn hiểu biết của mình, phải khiêm tốn, học hỏi, tiếp thu những tri thức mới của nhân loại. Trong một xã hội, những người đứng đầu mà bảo thủ, trì trệ thì xã hội đó sẽ kìm hãm sự phát triển của khoa học và công nghệ, đội ngũ trí thức không những bị coi rẻ, không thể phát huy hết tài năng sáng tạo của mình, mà thậm chí còn bị thiệt mạng…

Ký ức về bài giảng của thầy đã khiến em bừng tỉnh. Em đã vận dụng để trả lời câu hỏi của Giám đốc Công ty như sau:
Thưa Giám đốc! Một cộng một bằng bao nhiêu là do yêu cầu, mục đích và điều kiện của công việc cụ thể cho phép ta thực hiện phép cộng đó ạ.

Chẳng hạn, một Công ty sản xuất các sản phẩm để kinh doanh. Giả sử để có một sản phẩm, Công ty đó phải chi 1 đơn vị (tiền) để mua vật liệu và 1 đơn vị (tiền) để sản xuất ra sản phẩm. Nếu sản phẩm đó có giá bán là x đơn vị ( tiền), ta sẽ có một hạch toán đơn giản sau:
Công thức tính lỗ, lãi một sản phẩm của Công ty là: 1 + 1 = x
Nếu x = 0 (tức là sản phẩm không bán được), thì 1 + 1 = 0.
Nếu x = 1 (tức là sản phẩm bán được 1 đơn vị tiền), thì 1 + 1 = 1.
Nếu x = 2 (tức là sản phẩm bán được 2 đơn vị tiền), thì 1 + 1 = 2.
Nếu x = 3 (tức là sản phẩm bán được 3 đơn vị tiền), thì 1 + 1 = 3.
Khi giá trị x < 3, Công ty sản xuất sản phẩm này không có lãi.
Nếu giá trị x > 2, Công ty sản xuất sản phẩm này có lãi.
Như vậy, để kinh doanh có hiệu quả, Công ty phải dựa vào kết quả tổng hợp bán hàng của các đại lý bán hàng trong từng tháng, từng quý để biết được sản phẩm nào Công ty sản xuất ra sẽ bán được nhiều, với giá trị x > 2. Từ đó, Công ty sẽ lên kế hoạch đầu tư sản xuất các mặt hàng phù hợp với thị trường và có lãi xuất cao. Với tư duy này, Công ty kinh doanh sẽ có hiệu quả và ngày càng phát triển mạnh. Thưa giám đốc, câu trả lời của em là như vậy.

Sau khi em trình bày xong, ngài Giám đốc nói:
Tốt, câu trả lời quá xuất sắc, Công ty đồng ý nhận anh. Thứ 2 tuần sau, anh đến Phòng Tổ chức nhận quyết định làm việc với mức lương tháng: 10 triệu đồng. Anh đồng ý chứ?

Dạ, em cám ơn Hội đồng tuyển dụng và Giám đốc nhiều ạ.

Đấy, câu chuyện thầy đã giúp em xin được việc làm là như vậy, thầy ạ…

Tiễn học trò ra về, lòng tôi lâng lâng một niềm xúc cảm đặc biệt. Thấy vô cùng hạnh phúc và tự hào bởi mình là một thầy giáo Toán. Toán học thật kỳ diệu phải không các bạn? Cuộc sống như những bài toán với đầy ẩn số. Toán học giúp con người khám phá những ẩn số đó của cuộc sống và đem lại những đáp án tuyệt vời không chỉ với quá khứ, hiện tại mà còn hướng tới tương lai.

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