## The binomial theorem

Let $k\in {ℤ}_{\ge 0}$. Define $k$ factorial by $0!=1 and k!=k· (k-1)⋯ 3·2·1, if k∈ℤ>0 .$ Let $n,k\in {ℤ}_{\ge 0}$ with $k\le n$. Define $(nk) = n! k! (n-k)! .$

(Binomial theorem) Let $n,k\in {ℤ}_{\ge 0}$ with $k\le n$.

1. Let $S$ be a set of cardinality $n$. Then $\left(\genfrac{}{}{0}{}{n}{k}\right)$ is the number of subsets of $S$ with cardinality $k$.
2. $\left(\genfrac{}{}{0}{}{n}{k}\right)$ is the coefficient of ${x}^{k}{y}^{n-k}$ in ${\left(x+y\right)}^{n}$.
3. $\left(\genfrac{}{}{0}{}{n}{n}\right)=1$, $\left(\genfrac{}{}{0}{}{n}{0}\right)=1$, and if $1\le k\le n-1$ then $(nk) =( n-1 k-1 ) + ( n-1k ).$

This theorem says that the table of numbers $(00) (10) (11) (20) (21) (22) (30) (31) (32) (33) (40) (41) (42) (43) (44) (50) (51) (52) (53) (54) (55) ⋰ ⋮ ⋱$ are the numbers in Pascal's triangle $1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 ⋰ ⋮ ⋱$ and that $(x+y)0 = 1, (x+y)1 = x+y, (x+y)2 = x2+xy+ y2, (x+y)3 = x3+ 3x2y+ 3xy2+ y3, (x+y)4 = x4+ 4x3y+ 6x2y2+ 4xy3+ y4, (x+y)5 = x5+ 5x4y+ 10x3y2+ 10x2y3+ 5xy4+ y5, ⋮⋮$

## The exponential function

The exponential function is the element ${e}^{x}$ of $ℚ\left[\left[x\right]\right]$ given by $ex= ∑ k∈ ℤ≥0 xkk! =1+x+ x22! + x33! +⋯.$

As an element of $ℚ\left[\left[x,y\right]\right]$, $ex+y =exey.$

HW: Show that ${e}^{0}=1$.

HW: Show that ${e}^{-x}=\frac{1}{{e}^{x}}$.

The logarithm is $log(1+x) = ∑ k∈ℤ>0 (-1) k-1 xkk.$

Let $G= { p(x)∈ 𝔽[[x]] | p(0)=1} and 𝔤= { p(x)∈ 𝔽[[x]] | p(0)=0 }.$

1. $\mathrm{log}\left(1+\left({e}^{x}-1\right)\right)={e}^{\mathrm{log}\left(1+x\right)}-1=x$.
2. $G$ is an abelian group under multiplication, $𝔤$ is a commutative group under addition and $G ⟶ 𝔤 p ↦ ep-1$ is an isomorphism of groups.

## Notes and References

The binomial theorem and 'Pascals triangle' are useful computational tools for multiplying out algebraic expressions. The exponential function "is the most important function in mathematics" [Ru, Prologue]. The theorems showing that the exponential function is a homomorphism and that the formal inverse to the exponential function is log are found in [Bou, Alg. Ch. IV § 4 no. 10].

## References

[Bou] N. Bourbaki, Algèbre, Chapitre ?: ??????????? MR?????.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.