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April 10

What's the present value of a million a year (inflation adjusted) till you die?

US dollars. Depends on how long you'll live of course. Sagittarian Milky Way (talk) 04:22, 10 April 2017 (UTC)[reply]

If you live for n years then (assuming an annual lump sum) the amount is

\$1,000,000\sum _{i=1}^{n}{\frac {1}{1+I(i)}}

where

I

is the net inflation rate since today. You need a model of the expected inflation to say more than this, and I'm not an economist so I won't speculate on that. @Sagittarian Milky Way: An exercise for you, to encourage you to think critically: suppose this is not a single annual lump sum but instead, an infinitesimal amount of money is given to you every instant (i.e. a million a year is the value of the derivative of the total amount of money given to you with respect to time). What is the formula instead? (this assumes knowledge of basic integral calculus).--Jasper Deng (talk) 06:45, 10 April 2017 (UTC)[reply]

If each payment is inflation adjusted then the present value of each payment is $1m (by definition), so the present value of the annuity is $1m x years you can expect to live. In actuarial notation this is $1m x

e_{x}

where

e_{x}

is the curtate expected lifetime at age x. Gandalf61 (talk) 08:57, 10 April 2017 (UTC)[reply]

Correction: The present value is the sum of the values at each time discounted back to the present by the rate of interest. Since the payments are expressed in real (inflation adjusted) terms, the real interest rate r (roughly the interest rate minus the inflation rate) is used. So the present value is

\sum _{i=0}^{n}\$1,000,000\times {\frac {1}{(1+r)^{i}}}.

Loraof (talk) 13:40, 10 April 2017 (UTC)[reply]

The upper limit of the sum should be n−1 in order to make n payments. Then the sum

\sum _{i=0}^{n-1}(1+r)^{-i}

is reduced to

(1-(1+r)^{-n})(1-(1+r)^{-1})^{-1}

Bo Jacoby (talk) 04:44, 12 April 2017 (UTC).[reply]

Good software for doing math symbolically?

What would be a good (and preferably free) tool for doing math symbolically? I want to be able to develop proofs and derivations, in an exploratory way, on a computer. (I'm not trying to typeset a solution that I've already worked out. Rather, I'm trying to work out a solution on a computer.) I want to be able to manipulate equations algebraically, say by moving a term from one side to another, or by selectively substituting a sub-expression with another expression. I want to be able to tell the tool what manipulation to do, and have the tool take care of the details for me. Any suggestions? --134.242.92.97 (talk) 17:46, 10 April 2017 (UTC)[reply]

The most famous one is probably Mathematica, but that's not free (or even cheap); WolframAlpha is an online, free, dumbed-down version that is good enough for dirty work such as simplification of trigonometric fractions. As totally free software goes, I heard good things about Maxima but never tested it myself. (EDIT: also, I just discovered from perusing WP's page that SageMath seems to be another option.) Tigraan^{Click here to contact me} 18:36, 10 April 2017 (UTC)[reply]

Thanks. With Maxima and SageMath, how do you do something like "substituting (1) into the l.h.s. of (2)" or "multiplying both sides of (1) by the denominator of the l.h.s."? --134.242.92.97 (talk) 19:28, 10 April 2017 (UTC)[reply]

In Mathematica, there is a special syntax for rules and replacement (see documentation for commands Rule and Replace). The wolfram language is a cellular automaton based on tree rewriting rules. An open source implementation of the language is "mathics", which is surprisingly good, but can be tricky to install because of Python versioning. Sławomir Biały (talk) 11:17, 11 April 2017 (UTC)[reply]

It sounds from your description that Mathematica isn't exactly what you're looking for. Suppose you have an equation - Mathematica is designed to take the equation and solve it for you; it's not designed to ask you what manipulations you want to do to it on your way to solving it. Of course, you can do it if you insist, as Sławomir indicated - you can do pretty much anything - but at least from a UI perspective, it's not optimized for what you are describing.

I suspect the other tools in the category are similar, though there could be special-purpose tools for what you want.

Of course, this could be an XY problem, and what Mathematica is doing might be what you really want but didn't realize.

Myself I'm a long-time hardcore Mathematica fan, but my enthusiasm has cooled somewhat after some unsatisfactory engagements with their support. -- Meni Rosenfeld (talk) 20:30, 11 April 2017 (UTC)[reply]

There also is Maple (software), and a whole category of Computer algebra systems. If you really want to develop formal proofs, a proof assistant might be the way to go - the big ones are Isabelle (proof assistant), Coq, HOL (proof assistant). --Stephan Schulz (talk) 10:14, 11 April 2017 (UTC)[reply]