Adrian Said:

Answers for If the physiological density is much higher than the arithmetic density, then a country has?

We Answered:

sorry - looks like you should be in geography not physics

Katherine Said:

Can you help me with this arithmetic sequence and series word problem?

We Answered:

Well, we know that they were penalized $5,000 a day for the first 5 days. $5,000 x 5 = $25,000

For each day after, they get penalized $200 more a day than the day before.

If you subtract 25,000 from 65,600, you're left with 40,600.

On the first "late" day, the company gets charged 5200, then 5400, then 5600, then 5800, then 6000, then 6200, then 6400.

If you add these up, it equals the remaniing 40,600, so they were 7 days late.

Ron Said:

how does visualbasic treat mixed modes of arithmetic operation?

We Answered:

you can use brackets. the answer in such a case will be a single precision number.

Terrence Said:

why first order/higher order arithmetic are considered as mathematic logic?

We Answered:

first order logic is a kind of logical system. it is called "first-order" because it uses variables and quantifiers in addition to simple declarative propositions.

first order arithmetic is a first order theory in first order logic.

in other words, first order logic is the language in which the rules of the theory are stated. the rules of the theory are usually called axioms.

Zermelo-Fraenkel Set theory is another first order theory, which deals with sets, instead of arithmetic.

the study of arithmetic as a first order theory has shed light on the foundations of logic itself. to make an analogy with language: we can define a language, and give it rules, and show that these rules make cohesive sense. but the point of language is not in the words itself, but rather the meanings the words express. first order logic, in and of itself, would be uninteresting, unless it there exist interpretations of it that have some value for us.

one such interpretation is in arithmetic, we can make logical statements about numbers that reflect our actual experiences. perhaps the most famous of such statements is the oft-quoted: "2 + 2 = 4". in first-order logic, we can PROVE that, and that indicates that our notions of "2", "4", "+" and "=" which were orginally abstracted from our experiences in counting and tallying, make sense- that is, are logical.

in other words, you can study Russian for years, but if you never plan to speak it, read it, or write it, you will never fully understand it. logic is the same, it is the TOOL by which logical structures are understood.

so, first order arithmetic is not first order logic, but to fully understand first order logic, you should at least study one first order theory. first order theories are what first order logic is used for.

Shelly Said:

Have any of you heard of kurt Godel's proof, that arithmetic is "Incomplete"?

We Answered:

Godel's incompleteness theorm not only speaks to mathematics but to all formal systems. Godel's theorm is not directed at arithmatic per se, but rather at Russell and Whitehead's formalizaion of arithmatic through logic expounded in their magum opus, Principia Mathematica. Godel showed that any system which is sufficiently powerful to handle arithmatic is also necessarily incomplete.

The holy grail of logicians (and presumably mathematicians?) is a formal system that is both consistant and complete, but Godel's theorm is a powerful challenge to that even being possible, that to my knowledge has not been fully answered even today.

This finding has profound ramifications not only for math but for information processing in general, which drives deep into the heart of philosophy and science as well. Perhaps a new radically skeptical argument could be advanced that uses the incompleteness theorm to state that certain knowledge is impossible...

For a thourough, interesing, incicive, and beautiful discussion of Godel's Incompleteness Theorm and its ramifications for both Philosophy and Mathematics I reccomend you read:

Godel, Escher, Bach & I Am a Strange Loop
both by Douglas Hofstadter.

Angela Said:

Can anyone recomend a good book that teaches percents from basic math to the higher level maths?

We Answered:

Most math books don't treat percentages in any special way. A percentage, such as 75%, is really just a way of expressing a certain number, just like a fraction (3/4) or a decimal expansion (0.75).

Lots of math books use percentages when it is convenient to express numbers that way. Otherwise, they'll use fractions or decimal expansions.

You can fully understand percentages at an early stage in your mathematical education; after that, there's really not much more to learn about them.

Yes, they are used extensively in mathematics courses you'll take later on, such as probability and statistics. They're also encountered in calculus and differential equations courses (for instance, in mixture problems).

But there's no difference between a percentage you'd encounter in arithmetic or algebra, and a percentage you'd encounter in calculus. They all mean the same thing, and once you figure out what a percentage represents, you'll never have to worry about percentages again.

In other words, you are on the verge of learning everything there is to know about percentages. When you get percentages figured out, you'll never have to worry about them again--although you will find them useful in lots of situations.

A percentage is only the numerator of a fraction whose denominator is 100. There is nothing more to know about it than that.

For instance, 66/100 = 66%.

Also, 33/50 = 66%, because you can multiply the denominator (50) by 2 to get 50 x 2 = 100, and, multiplying the numerator by 2, you get 33 x 2 = 66%.

Also, 198/300 = 66%, because you can divide the denominator by 3 to get 100, and that makes the numerator 198/3 = 66.

A percentage is simply another way, in addition to fractions and decimal expansions, to express a number.

So I doubt there is any book that follows the use of percentages specifically throughout the mathematics curriculum. The math books I am familiar with simply use percentages numerically, in addition to the other ways of expressing numbers.

Vernon Said:

Why is math so hard for some and so fun or easy for some?

We Answered:

I love math. The only way to understand and like math is to practice, practice, practice and more practice each day.