3. Pascal’s Triangle

파스칼의 삼각형

Pascal’s Triangle

"이항 곱셈공식의 계수는 외울 필요가 없어요"

" you don’t have to memorize

the coefficients in binomial expansions "

오늘은 프랑스 철학자이자 수학자인 파스칼이 정리해 놓은 아주 유명하고 재미있는 파스칼의 삼각형을 소개합니다.

파스칼의 삼각형은 이항정리의 계수를 알아내거나 삼각수 (triangular number) 등 여러가지의 특이한 숫자들의 규칙을 찾아내는 데에 활용되고 있습니다.

중학수학에서도 두 가지의 경우를 선택하는 경우의 수를 구하거나, 세제곱 이상의 곱셈공식 전개식에서 계수를 알아내는 데에 아주 편리하고 재미있는 내용이므로 원리를 잘 이해해 두고 활용법을 기억해 두기 바랍니다.

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스마트폰에서 수학 수식을 보시려면, 왼쪽 버튼을 누른 후

[데스크톱 보기] 를 설정하세요.

Please select [desktop view] on the mobile

to read math equations

♧   ♧   ♧   ♧   ♧   ♧

곱셈공식에서 배웠던 내용 중에서, 두 개의 항만으로 이루어진 이항의 거듭제곱을 복습삼아 전개해 보도록 할까요?

As we learned earlier in [polynomial expansions], let’s review the square and cube form of binomial expansion which has only two terms.

(a + b)¹ = a + b

(a + b)² = a² + 2ab + b²

(a + b)³ = a³ + 3a²b + 3ab² + b³

만일 4 차 이상의 이항식을 전개하면 그 계수는 어떻게 될까요?

What will be the coefficient in binomial expansions if we increase the power higher than 3?

이럴 때, 파스칼의 삼각형을 이용하면 아주 쉽게 그 계수들을 알아낼 수 있습니다.

Pascal’s triangle is a very convenient tool to find out the coefficient in binomial expansions.

아래 그림에서 보는 것과 같이 윗 줄의 좌우 두 숫자의 합을 적어 놓되, 숫자가 없을 때에는 0 이 있다고 가정하면 됩니다.

As shown below, just add two left and right hand side numbers in the upper row and assume that it’s 0 when there is no number.

0   1   0

v    v

1    1

v    v    v

1    2    1

v    v    v    v

1    3    3    1

v    v    v    v    v

1    4    6    4    1

v    v     v     v     v    v

1    5   10   10    5    1

v    v     v     v     v     v    v

1    6    15   20   15    6    1

︙

파스칼의 삼각형에서 계수들이 좌우대칭의 모습을 보이고 있지요?

Do you see the triangle numbers are symmetrical?



일반적으로 전개식은 (1) 삼각형에서 알아낸 계수를 쓰고 (2) 앞의 문자 a 는 차수를 하나씩 낮추고 (3) 뒤의 문자 b 는 차수를 하나씩 올리면서 곱해주는 방식으로 정리하면 됩니다.

The rules are, in general, (1) After finding out the coefficients from the triangle, (2) decrease the power of the first letter ‘a’ and (3) increase the exponent of ‘b’.

(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

(a + b)⁵ = a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

︙

또한, 파스칼의 삼각형에서는 여러가지 특이한 숫자들의 규칙을 찾아낼 수 있습니다.

In addition, we can find special number patterns in Pascal’s triangle.



정삼각형 도형을 만들기 위해 사용되는 물건의 총수를 나타내는 삼각수 (triangular numbers) 에 대해서 알아 보도록 하지요. 이 수들의 규칙은 뒤에서 배울 계차수열의 하나입니다.

Triangular number counts the number of objects that can form an equilateral triangle. This pattern is also known as quadratic sequence, which we will study later.

                                                                     O
                                              O                  O  O
                           O              O  O             O  O  O
             O         O  O         O  O  O        O  O  O  O     ...
    O    O  O    O  O  O    O  O  O  O    O  O  O  O  O

    a₁       a₂             a₃                  a₄                   a₅ 

아래 그림의 파스칼 삼각형에서 사선의 빨간색으로 표시된 수열이 바로 삼각수 (triangular numbers) 가 되는 것을 발견할 수 있습니다.

In the Pascal’s number table shown above, you can discover that the diagonal red band numbers are triangular numbers.

0   1   0

v    v

1    1

v    v    v

1    2    1

v    v    v    v

1    3    3    1

v     v    v    v     v

1     4    6    4     1

v    v     v    v     v    v

1    5    10   10    5    1

v    v     v     v     v     v    v

1    6    15   20   15    6    1

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또한, 파스칼의 삼각형에서는 여러가지 특이한 숫자들의 규칙을 찾아낼 수 있습니다. 고등학생들도 조금 어려워하는 피보나치 (Fibonacci) 수열을 알아 보도록 할까요?

In addition, we can find special number patterns in Pascal’s triangle. Let’s investigate the famous but quite advanced level sequence – Fibonacci.

1
1	1
1	2	1
1	3	3	1
1	4	6	4	1
1	5	10	10	5	1
1	6	15	20	15	6	1
1	7	21	35	35	21	7	1
1	8	28	56	70	56	28	8	1

                   (source : wikipedia – Pascal’s triangle)

위의 표로 나타낸 파스칼 숫자에서, 사선으로 보이는 같은 색의 셀들을 더해서 차례로 나열하면 피보나치 수열이 됩니다.

In the Pascal’s number table shown above, each sum of the diagonal same colored bands from the top makes Fibonacci sequence.

a₁ = 1

a₂ = 1

a₃ = 1 + 1 = 2

a₄ = 2 + 1 = 3

a₅ = 1 + 3 + 1 = 5

a₆ = 3 + 4 + 1 = 8

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피보나치 수열은 제 3 항부터의 값이 직전에 있는 두 개의 항의 값을 합하여 결정되는 특징을 가지고 있습니다.

Fibonacci sequence has a characteristic rule such that each subsequent number is the sum of previous two and its recurrence relation will be as follows :

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a_n = a_n-1 + a_n-2

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점화식으로 나타낸 다음 실제로 맞는지 확인해 보도록 할까요?

Let’s check whether the numbers are identical or not.

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a_n = a_n-1 + a_n-2

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a₁ = 1

a₂ = 1

a₃ = a₁ + a₂ = 1 + 1 = 2

a₄ = a₂ + a₃ = 1 + 2 = 3

a₅ = a₃ + a₄ = 2 + 3 = 5

a₆ = a₄ + a₅ = 3 + 5 = 8

︙