Chapter Summary (Express)
The preliminary terror, which chokes off most highschoolers from even attempting to learn how to calculate, can be abolished once for all by simply stating what is the meaning—in common-sense terms—of the two principal symbols that are used in calculating.
These dreadful symbols are:
\(d\), which merely means "a little bit of." For example, \(dx\) means a little bit of \(x\), and \(du\) means a little bit of \(u\). Mathematicians prefer to use the phrase "an element of" instead of "a little bit of." However, both terms refer to infinitesimally small quantities.
(1) \(d\) which merely means “a little bit of.”
Thus \(dx\) means a little bit of \(x\); or \(du\) means a little bit of \(u\). Ordinary mathematicians think it more polite to say “an element of,” instead of “a little bit of.” Just as you please. But you will find that these little bits (or elements) may be considered to be indefinitely small.
\(\displaystyle{\int}\), which is a long \(S\), means "the sum of" in calculations. For example, \(\displaystyle{\int dx}\) sums up all the little bits of \(x\), and \(\displaystyle{\int dt}\) totals the little bits of \(t\). Mathematicians call this symbol "the integral of." By considering \(x\) as composed of a lot of little bits, called \(dx\), adding them together yields the sum of all \(dx\)’s, which is the same the whole of \(x\). The term "integral" simply means "the whole." When encountering an expression starting with \(\displaystyle \int\), it instructs you to add up all the little bits represented by subsequent symbols.
(2) \(\displaystyle{\int}\) which is merely a long \(S\), and may be called (if you like) “the sum of.”
Thus \(\displaystyle{\int dx}\) means the sum of all the little bits of \(x\); or \(\displaystyle{\int dt}\) means the sum of all the little bits of \(t\). Ordinary mathematicians call this symbol “the integral of.” Now any fool can see that if \(x\) is considered as made up of a lot of little bits, each of which is called \(dx\), if you add them all up together you get the sum of all the \(dx\)’s, (which is the same thing as the whole of \(x\)). The word “integral” simply means “the whole.” If you think of the duration of time for one hour, you may (if you like) think of it as cut up into \(3600\) little bits called seconds. The whole of the \(3600\) little bits added up together make one hour.
When you see an expression that begins with this terrifying symbol, you will henceforth know that it is put there merely to give you instructions that you are now to perform the operation (if you can) of totalling up all the little bits that are indicated by the symbols that follow.
That’s all.
Full Chapter
The preliminary terror, which chokes off most highschoolers from even attempting to learn how to calculate, can be abolished once for all by simply stating what is the meaning—in common-sense terms—of the two principal symbols that are used in calculating.
These dreadful symbols are:
\(d\) which merely means “a little bit of.”
Thus \(dx\) means a little bit of \(x\); or \(du\) means a little bit of \(u\). Ordinary mathematicians think it more polite to say “an element of,” instead of “a little bit of.” Just as you please. But you will find that these little bits (or elements) may be considered to be indefinitely small.
\(\displaystyle{\int}\) which is merely a long \(S\), and may be called (if you like) “the sum of.”
Thus \(\displaystyle{\int dx}\) means the sum of all the little bits of \(x\); or \(\displaystyle{\int dt}\) means the sum of all the little bits of \(t\). Ordinary mathematicians call this symbol “the integral of.” Now any fool can see that if \(x\) is considered as made up of a lot of little bits, each of which is called \(dx\), if you add them all up together you get the sum of all the \(dx\)’s, (which is the same thing as the whole of \(x\)). The word “integral” simply means “the whole.” If you think of the duration of time for one hour, you may (if you like) think of it as cut up into \(3600\) little bits called seconds. The whole of the \(3600\) little bits added up together make one hour.
When you see an expression that begins with this terrifying symbol, you will henceforth know that it is put there merely to give you instructions that you are now to perform the operation (if you can) of totalling up all the little bits that are indicated by the symbols that follow.
That’s all.
