Make coefficients appear

Highlight a line and read each number. In line 4, the 6 comes from 3 + 3 in the previous line.

Em telas pequenas, deslize o gráfico na horizontal para ler os detalhes.

Compare the sums of lines zero to six: 1, 2, 4, 8, 16, 32, and 64.

From line 3 to the cube of a sum

  1. The line 3 has the coefficients 1, 3, 3, 1.
  2. The exponents of a decrease: 3, 2, 1, 0.
  3. The exponents of b increase: 0, 1, 2, 3.
  4. Thus, (a + b)³ = a³ + 3a²b + 3ab² + b³.

In the previous post, we talked about notable products. Here, we will extend the concept of notable products with Pascal’s triangle. Pascal’s triangle is a triangle of integers, where each number is the sum of the two numbers above it. Pascal’s triangle is a very useful tool for calculating binomial coefficients.

In addition to his masterpiece, Pascal’s Triangle, Blaise Pascal was one of the precursors of current computing, as with the La Pascaline machine he revolutionized the era with his mechanical calculator. We can quickly perform addition and subtraction calculations. He did this as a distraction, as his partner was a tax collector.

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Traite du triangle arithmétique (1654 - 1754)

“All I know is that I’m going to die and that eu ignoro is a death I can’t avoid.”

Blaise Pascal Pascal’s tombstone
Pascal's triangle from line zero to line six.
Each internal value sums the two immediately above.
Ampliar figura para ler os detalhes
Pascal's triangle from line zero to line six.

Here we will build Pascal’s triangle based on the results of several translated and adapted authors, with the aim of understanding the extent of Pascal’s philosophy.

“And the design of the arithmetic triangle is the construction figure…”

$$ \begin{array}{ccccc} 1 & & & & \\ 1 & 1 & & & \\ 1 & 2 & 1 & & \\ 1 & 3 & 3 & 1 & \\ 1 & 4 & 6 & 4 & 1\\ \vdots & \vdots & \vdots & \vdots & \vdots\\ \end{array} $$

How did he arrive at this result?

Pascal's triangle from line zero to line six.
Each internal value sums the two immediately above.
Ampliar figura para ler os detalhes
Pascal's triangle from line zero to line six.
n Coefficients
1 $1 \quad 1$
2 $1 \quad \overbrace{2}^{1+1} \quad 1 $
3 $1 \quad \overbrace{3}^{1+2} \quad \overbrace{3}^{2+1}\quad 1 $
4 $1 \quad \overbrace{4}^{1+3} \quad \overbrace{6}^{3+3} \quad \overbrace{4}^{3+1}\quad 1 $
$\vdots$ $\vdots$

Comparison of remarkable product and Pascal’s triangle

Next, we will observe that notable products is a particular case of Pascal’s triangle.

Notable product Pascal’s triangle Result
$(a+b)$ $1 \quad 1 \quad \quad \quad $ $1a + 1b$
$(a+b)^2$ $1 \quad 2 \quad 1 \quad \quad $ $1a^2+ 2ab + 1b^2$
$(a+b)^3$ $1 \quad 3 \quad 3 \quad 1 \quad $ $1a^3+ 3 a^2b+ 3 ab^2 + 1b^3$
$(a+b)^4$ $1 \quad 4 \quad 6 \quad 4 \quad 1$ $1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b4$

Historical Sources - Pascal’s Triangle

Pomegranate
Blaise Pascal.
Europe
Blaise Pascal.
Arabic
Blaise Pascal.

Application to polynomials

$$f(x)= (1+x)^n$$
$$ \begin{aligned} n = 0 \qquad &\Rightarrow \qquad f(x) = 1 \\ n = 1 \qquad &\Rightarrow \qquad f(x) = 1 + x\\ n = 2 \qquad &\Rightarrow \qquad f(x) = 1 + 2x + x^2\\ n = 3 \qquad &\Rightarrow \qquad f(x) = 1 + 3x + 3x^2 + x^3\\ n = 4 \qquad &\Rightarrow \qquad f(x) = 1 + 4x + 6x^2 + 4x^3 + x^4\\ \qquad\vdots \qquad& \qquad\vdots\qquad\\ \end{aligned} $$
$$ \begin{array}{ccccc} 1 & & & & \\ 1 & 1 & & & \\ 1 & 2 & 1 & & \\ 1 & 3 & 3 & 1 & \\ 1 & 4 & 6 & 4 & 1\\ \vdots & \vdots & \vdots & \vdots & \vdots\\ \end{array} $$

“And the rooster crows for the first time!”

Drunk’s walk

Be it a bar, a drunk, your house, n steps and an intention…

To illustrate this example from Pascal, observe:

$$ \begin{aligned} n = 1 \qquad &\Rightarrow \qquad S + F\\ n = 2 \qquad &\Rightarrow \qquad SS + SF + FS + FF\\ n = 3 \qquad &\Rightarrow SSS + SSf +SFS + FSS + SFF +FSF +FFS +FFF\\ \qquad\vdots \qquad& \qquad\vdots\qquad\\ n = 1 \qquad &\Rightarrow \qquad S + F\\ n = 2 \qquad &\Rightarrow \qquad S^2 + 2SF + F^2\\ n = 3 \qquad &\Rightarrow S^3 + 3S^2f +3SF^2 +F^3\\ \qquad\vdots \qquad& \qquad\vdots\qquad\\ \end{aligned} $$
drunk walk.

“And the rooster crows for the second time!”