Pascal's triangle: Difference between revisions
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For instance, we can use Pascal's triangle to compute the binomial expansion of | For instance, we can use Pascal's triangle to compute the binomial expansion of | ||
<math>(x+y)^4 = 1 \times a^4 + 4 \times a^3b + 6 \times a^2b^2 + 4 \times ab^3 + 1 \times b^4 ~</math> | :<math>(x+y)^4 = 1 \times a^4 + 4 \times a^3b + 6 \times a^2b^2 + 4 \times ab^3 + 1 \times b^4 ~</math> | ||
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Each coefficient in the triangle is the sum of the two coefficients over it<ref>This rule does not apply to the ones bordering the triangle. We just insert them.</ref>. For instance, <math>6 = 3 + 3</math>. The binomial coefficients relate to this construction by Pascal's rule, which states that if | Each coefficient in the triangle is the sum of the two coefficients over it<ref name="not_for_ones">This rule does not apply to the ones bordering the triangle. We just insert them.</ref>. For instance, <math>6 = 3 + 3</math>. The binomial coefficients relate to this construction by Pascal's rule, which states that if | ||
:<math> {n \choose k} = \frac{n!}{k! (n-k)!} </math> | :<math> {n \choose k} = \frac{n!}{k! (n-k)!} </math> | ||
is the ''k''th binomial coefficient in the [[binomial expansion]] of (''x''+''y'')<sup>''n''</sup>, then | is the ''k''th binomial coefficient in the [[binomial expansion]] of (''x''+''y'')<sup>''n''</sup>, then | ||
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== Properties == | == Properties == | ||
In order to ease the understanding of some properties, the triangle should be presented differently : | In order to ease the understanding of some properties, the triangle should be presented differently : | ||
<div style="margin-left: 40px;"> | <div style="margin-left: 40px;"> | ||
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</math> | </math> | ||
</div> | </div> | ||
Each coefficient is the sum of the coefficient exactly over it and the other to its left. For instance, <math>3 + 3 = 6</math>. | Each coefficient is the sum of the coefficient exactly over it and the other to its left. For instance, <math>3 + 3 = 6</math>. | ||
Let's call this rule the "addition rule" | Let's call this rule the "addition rule"<ref name="not_for_ones"/>. | ||
Using this format, it is easy to apply an index to each row : | |||
Revision as of 07:21, 25 October 2007
The Pascal's triangle is a convenient tabular presentation for the binomial coefficients. Already known in the 11th century[1], it was adopted in Western world under this name after Blaise Pascal published his Traité du triangle arithmétique ("Treatise on the Arithmetical Triangle") in 1654.
For instance, we can use Pascal's triangle to compute the binomial expansion of
The coefficients 1, 4, 6, 4, 1 appear directly in the triangle.
Each coefficient in the triangle is the sum of the two coefficients over it[2]. For instance, . The binomial coefficients relate to this construction by Pascal's rule, which states that if
is the kth binomial coefficient in the binomial expansion of (x+y)n, then
for any nonnegative integer n and any integer k between 0 and n.[3]
Those coefficients have applications in algebra and in probabilities. From the triangle, we can equally compute the Fibonacci numbers and create the Sierpinski triangle. After studying it, Isaac Newton expanded it and found new methods to extract the square root and to calculate the natural logarithm of a number.
Properties
In order to ease the understanding of some properties, the triangle should be presented differently :
Each coefficient is the sum of the coefficient exactly over it and the other to its left. For instance, .
Let's call this rule the "addition rule"[2].
Using this format, it is easy to apply an index to each row :
Starting the indices at zero facilitates many calculations.
The sum of any row is , with being the row index : For instance, the sum of row 4 is .
References
- ↑ Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji, School of Mathematics and Statistics, University of St Andrews. Consulted 2005-09-03.
- ↑ 2.0 2.1 This rule does not apply to the ones bordering the triangle. We just insert them.
- ↑ By convention, the binomial coefficient is set to zero if k is either less than zero or greater than n.
