{"id":349,"date":"2026-07-09T05:49:38","date_gmt":"2026-07-09T05:49:38","guid":{"rendered":"https:\/\/potentsky.com\/math\/?p=349"},"modified":"2026-07-09T05:51:31","modified_gmt":"2026-07-09T05:51:31","slug":"eulers-number-e-why-do-we-raise-it-to-power-x-when-was-calculated-n-tended-to-infinity","status":"publish","type":"post","link":"https:\/\/potentsky.com\/math\/eulers-number-e-why-do-we-raise-it-to-power-x-when-was-calculated-n-tended-to-infinity\/","title":{"rendered":"Euler’s number e, why do we raise it to power x when was calculated n tended to infinity"},"content":{"rendered":"\n

Compounding frequency versus compounding scaling.<\/strong><\/h2>\n\n\n\n

The reason we transition from a limit where \\(n\\) tends to infinity to raising \\(e\\) to the power of \\(x\\) comes down to a beautiful mathematical bridge: compounding frequency versus compounding scaling.<\/strong><\/p>\n\n\n\n

When you calculate \\(e\\) normally, you are looking at what happens to a base unit of \\(1\\) growing at a \\(100\\%\\) interest rate over infinitely small intervals (\\(n \\to \\infty\\)). When you raise it to the power of \\(x\\), you are altering the total time<\/strong> or the internal rate<\/strong> of that growth engine.<\/p>\n\n\n\n

Here is the exact mathematical breakdown of how \\(e^{x}\\) naturally emerges from that infinity limit, structured so you can easily translate it into a high-retention Remotion animation.<\/p>\n\n\n\n

\n\n\n\n

The Mathematical Proof (The “Why”)<\/h3>\n\n\n\n

The Mathematical Proof (The “Why”)<\/p>\n\n\n\n

The standard definition of \\(e\\) when \\(n\\) tends to infinity is:
\\(e=\\lim _{n\\rightarrow \\infty }\\left(1+\\frac{1}{n}\\right)^{n}\\)<\/p>\n\n\n\n

Now, imagine instead of growing at a \\(100\\%\\) rate, your system grows at a rate of \\(x\\) (for example, a \\(200\\%\\) rate, where \\(x = 2\\)). The formula for continuous compounding at a rate of \\(x\\) becomes:
\\(\\text{Growth}=\\lim _{n\\rightarrow \\infty }\\left(1+\\frac{x}{n}\\right)^{n}\\)<\/p>\n\n\n\n

To see why this equals \\(e^{x}\\), we use a clever algebraic substitution. Let\u2019s define a new variable, \\(m\\), such that:
\\(m=\\frac{n}{x}\\quad \\implies \\quad n=mx\\)<\/p>\n\n\n\n

As \\(n\\) approaches infinity (\\(\\infty \\)), \\(m\\) must also approach infinity because \\(x\\) is just a constant number. Now, let’s substitute \\(m\\) back into our growth equation: [1<\/a>, 2<\/a>]<\/p>\n\n\n\n

\\(\\lim _{n\\rightarrow \\infty }\\left(1+\\frac{x}{n}\\right)^{n}=\\lim _{m\\rightarrow \\infty }\\left(1+\\frac{1}{m}\\right)^{mx}\\)<\/p>\n\n\n\n

Using the laws of exponents, we can pull the power of \\(x\\) outside the main brackets:
\\(\\lim _{m\\rightarrow \\infty }\\left[\\left(1+\\frac{1}{m}\\right)^{m}\\right]^{x}\\)<\/p>\n\n\n\n

Look closely at the expression inside the brackets. It is the exact definition of \\(e\\) that we started with, just using the letter \\(m\\) instead of \\(n\\):
\\(\\left[\\lim _{m\\rightarrow \\infty }\\left(1+\\frac{1}{m}\\right)^{m}\\right]^{x}=e^{x}\\)<\/p>\n\n\n\n

The Story Hook:<\/strong> Raising \\(e\\) to the \\(x\\) isn’t an arbitrary rule we invented later. The math proves that changing the growth rate inside the infinity limit is fractionally identical<\/em> to taking the final infinite baseline (\\(e\\)) and scaling it exponentially by \\(x\\).<\/p>\n","protected":false},"excerpt":{"rendered":"

Compounding frequency versus compounding scaling. The reason we transition from a limit where \\(n\\) tends to infinity to raising \\(e\\) to the power of \\(x\\) comes down to a beautiful mathematical bridge: compounding frequency versus compounding scaling. When you calculate \\(e\\) normally, you are looking at what happens to a base unit of \\(1\\) growing […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_themeisle_gutenberg_block_has_review":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-349","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/posts\/349","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/comments?post=349"}],"version-history":[{"count":4,"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/posts\/349\/revisions"}],"predecessor-version":[{"id":353,"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/posts\/349\/revisions\/353"}],"wp:attachment":[{"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/media?parent=349"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/categories?post=349"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/tags?post=349"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}