C (complex numbers) is a field, just like R (real numbers). It's an algebraically closed field, unlike R, so in some sense it's actually the most natural set to call "numbers".
Well I was answering the question about why is C considered numbers - it's a field, hence elements of C behave exactly like numbers. Moreover, it's an 'algebraic closure' of R, hence a more natural choice for "all numbers", and it's a maximal one at that (i.e. quaternions and such are no longer fields and don't really behave like numbers, while C still does).
Edit: FWIW, I consider the terminology of 'real' vs 'imaginary' completely stupid and misleading. This terminology didn't really make it to other languages, e.g. in Russian it's 'material' vs 'complex' numbers, but they don't use the term 'imaginary'.
Analytically differentiable (being able to derive a formula for the derivative) is not the same thing at all as analytic functions of complex analysis (which are infinitely differentiable, etc). If anything, the term "analytic function" seems more of a misnomer to me, but "analytically differentiable" is used in its proper sense in the article.
> It takes about 2 years to save up $100K on a typical Bay Area engineer's salary
Have you actually done this? Know anyone who has?
This seems completely implausible, outside of some extreme frugal lifestyle cases (I'm talking "eating garbage dumped by grocery stores" level frugal)...
I actually saved up $100K in about 18 months, working at a large Bay Area tech company and living in Mountain View. I don't feel I made significant compromises to do so, either: I had roommates for 6 of those 18 months but lived in a 1BR afterwards, I had friends (many of whom I still keep up with), I owned a car and got out on weekends. I did cook the majority of my (non-employer-provided) meals at home, and my friends tended to like relatively cheap activities like LAN parties, board games, and hiking rather than expensive meals out. I didn't go on any major vacations, but my family lives back East, and so that includes rather expensive airfare at Christmas, Thanksgiving, and 4th of July to see them.
Paycheck calculator shows me that $120K [1] in CA leaves you with $6,220 after-tax monthly income. If you really made about as much, and managed to save ~$5.5K a month, then you spent only ~$660 a month, which is just impossible, sorry. Even if you're sharing a 1BR with several roommates, which, by the way, is not really an option for many.
[1] Googling "average software engineer salary california" shows figures ranging from 107k to 135k, 120k seems to be more or less the median of Google's figures.
It was 2009-2010, so rents were lower, but my paycheck was lower as well (I started at only $100K/year base, compared to the ... well, much higher amount a few years later). After taxes and 401k, I netted $5400/month. 18 months encompassed 2 bonus cycles, so about $40K of that was bonuses, after tax. My rent was $900/month with roommates, and then $1400/month in the 1BR, and total monthly expenditures were about $2K. I was packing away about $3.5K/month; 18 months of that is $63K, +$40K from bonuses = just over $100K.
You must admit that $40k bonus after tax is a huge, atypical amount of money. It makes your effective yearly income to be about $156k.
It reminds me of the mrmoneymustache.com guy, saying "anyone can retire early! Just look at me, all I had to do was get $1M in stock from Cisco!"
No disrespect intended for you; I'm sure compared to some of your friends you lived frugally. The point I'm making is that to duplicate your living standard today, very few comparable developers would be able approach your level of savings.
It's over 2 years. $20K or so after tax per year. A bit over $30K pre-tax.
Not sure what typical bonuses are these days, but I don't think that's out of line for a big tech company. Certainly stock grants can be much more than that, but it usually takes more than 18 months for them to amount to anything.
Thanks for sharing numbers. Wow, that's an amazing bonus, and you should feel fortunate! This is coming from someone who does work for a big tech company. I've never seen anything remotely close to that.
I will unequivocally state that a total comp of $150k+ for a new grad is at the very least, the top 5% of software development salaries for that level of experience.
was the op a new grad? sorry, I missed that. yes, $150k for a ncg in the first year is high. but honestly not that high, and for a 2nd year... doable from my experience [1].
Yes, doable at Google, a company known for being at the top of the market in compensation.
The whole point of the original statement was "I made $X and saved $Y, it's not that hard." But when you examine the X and the Y, they are not representative at all!
It has similar merit to a lottery winner saying, "why can't you win the lottery too? All I did was buy a lottery ticket."
Google pays substantially more than that - matching that number does not put you at the top of the market. Companies paying that much include many of the largest tech employers in the area. Compensation for job switchers has been increasing rapidly since the no-poaching ring was busted, but you have to know to ask for it. Lots of people are anchored from that period and don't know what they are missing.
Well, good for you. I don't think your situation was "average" really, but kudos on the savings :) Coming from a guy in NYC whose monthly expenditures are through the roof...
- rents $2000K/mo 1BR apartment (single but wants some comfort in the Peninsula, no hellish commutes from East Bay)
- transportation $200/mo
- free lunch (and possibly dinner) at the company - eating out and groceries $300/mo
- entertainment $200/mo
That's $2700/mo. Let's round it up to $3000/mo because you're a well paid software engineer and can spend $300/mo on top of what I listed whenever you feel like it.
So far, $36k expenses + $50k savings and you still have $8k left to go on ski trips to Tahoe, Outside Lands and other non-frugal lifestyle expenses.
A decent 1BR on the peninsula (in San Mateo, Redwood City, Mountain View, etc.) is likely closer to >$2,500 a month plus $200/month for utilities/renters insurance. For transportation, I pay $200/month for a 15 year old Civic with liability insurance. If you have a moderately nice car that isn't paid off, you are going to pay >$200/month (especially if you commute daily). If you cook moderately nice meals and drink some alcohol, that food budget is likely going to be >$400/month. Plus going out to happy hour once a week, which isn't that uncommon, can easily cost you an extra $30-50 a week (or more). A good number of people have college debt as well, so add a monthly payment for that. I agree with you that saving is definitely possible when single and making >$100,000. I just think you low balled most of your numbers to prove your point.
If we're talking about low balled figures, I guess we need to consider the income side of the equation as well. The original question depends a good deal on what kind of job you're doing in the Bay Area. If it's someone coming to the Bay Area to work for Google/Facebook or similar, the figure I used is way too low, for example.
"A decent 1BR" - It all depends on the definition of decent. I pay $2k for a 1BR walking distance to Redwood City downtown/Caltrain.
But forget about renting an apartment, if you want to save and you're single, having roommates allows you to rent in the Peninsula for much less than $2k/mo.
Also, I added $300 discretionary spending which lets you to cover some of the things you described, and you're left with $8k at the end of the year.
Can you easily save $50k while spending $3k on rent+utilities to live in a fancy apartment in the Peninsula ? Maybe not, but you certainly don't need to be "eating garbage dumped by grocery stores".
> Maybe not, but you certainly don't need to be "eating garbage dumped by grocery stores".
Nice strawman :) I was specifically talking about the situation conditional on saving the 100k in 18 months, my comment was a response to the comment above it.
Separate point: your figures seem very low. I thought SF is roughly equivalent to NYC in terms of cost of living. In which case 300/mo is a zero off as an estimate for all food expenses.
Sure, I did this (and more if you count my 401k). I don't think I was that frugal: I certainly didn't want for anything, and ate my fair share of sushi.
Like the article mentions, saving a lot is easy if you're fine with a roommate and living in a non-luxury apartment, but if your housing standards are higher than that then it's tougher.
This is, of course, at the compensation levels talked about in the article. Some companies (early stage start-ups, banks) pay much less.
I went from a net worth of <$50k at age 20 to >$150k at 22, while never being particularly frugal at all. Heck, I didn't even have roommates. My monthly spending averaged $4k/m.
NYC, not SF, but the difference isn't huge.
Most people are atrocious with their money by either overspending or under-optimizing.
OK, it all depends on what the cash flow looks like :) You're leaving out that one very important piece of the puzzle...
Also, the average apartment in Manhattan (if you do rent without roommates) is pretty close to your cited figure for total monthly expenses. Which means you either lucked out, or seriously compromised on your living arrangements.
> OK, it all depends on what the cash flow looks like
I was making $150k/yr, which is pretty standard for a developer in NYC.
> Also, the average apartment in Manhattan (if you do rent without roommates) is pretty close to your cited figure for total monthly expenses.
I take it you don't live in NYC. The average is a useless metric because it's distorted by millionaires and billionaires. Just go on Craiglist and you'll find plenty of suitable apartments in the $2-3k range.
Nobody I know pays more than $3k/m for their apartment. Heck, my friend just rented a beautiful luxury single-bedroom for $2.8k.
In my case, I chose to go for a non-luxury apartment for $1.8k/m. This was definitely a deal, but even though it wasn't luxury there was plenty of space and I even had a back yard.
> Nobody I know pays more than $3k/m for their apartment.
Few people I know pay under. We live in parallel dimensions I guess.
> In my case, I chose to go for a non-luxury apartment for $1.8k/m.
I don't know what NYC you live in, but this is non-existent AFAIK. I know someone who went all the way to Rego Park (this is really far from Manhattan) to rent a 1BR for $2.5k.
I did it, paying $3400/month rent & supporting a family. Salary took care of living + unexpected expenses, and stock grants/purchase plans took care of the saving + knowing where to keep the money (thanks, years of scraping by on non-SF-ludicrous pay packets).
edit I don't know what a 401k is good for (I already have two retirement saving schemes in other countries), mostly did staycations since SF is such a tourist destination.
Well, understanding calculus and being able to compute indefinite integrals or solve differential equations analytically on paper are very different things.
Definitely, this is arguably the most important feature.