# Mathedu

Learn math by practice

# Re-engineering Math

## Adding and Subtracting a Percentage

As we saw in a previous post, adding a percentage is a multiplication.

For instance, adding 20% to a tax free price, to obtain the VAT included price, is multiplying by (1+20/100)=1.20=1.2.

But what I would like you to know today, is that subtracting a percentage is NOT a division, but a multiplication as well!

For instance, a coupon of 30% on a price of \$100 is subtracting 30% of \$100, that is (30/10)*\$100=\$30. The reduced price is \$100-\$30=\$70, that is 0.70 times \$100.

Another way to...

## Compare Integers and Decimal Numbers.

The decimal notation is very useful to compare numbers, as soon as we are able to order the 10 digits from 0 to 9: 0<1<2<3<4<5<6<7<8<9.

To know which number is the greatest between 2 natural integers a≥0 and b≥0, we first compare the number of digits of a and b. If a has more digits than b, then a>b, and reversely. For instance 96<456 and 16>0.

If a and b have the same number of digits, then we compare the digits that are the most to the left. If they are different, then the greatest...

## Greatest Common Divisor and the Fractions

In arithmetics, the numbers by which we may divide an integer, their 'divisors', are quite important.

For instance, if you find a number that divides both the numerator and the denominator of a fraction, for instnace 2 for the fraction 6/4, then you may simplify the fraction: 6/4=3/2, because
6=2x3 and 4=2x2.

The integers that divide both the integer p and the integer q are called the 'common divisors' of p and q.

The positive common divisors of 2 integers p and q are all lower than both...

## The Measurement of Time II: the time on the clock

The day is the smallest time unit that has an astronomic reference: the time by which the earth spins on itself by one turnaround.

But it is not the smallest duration we may evaluate: our clocks show us the hours and minutes, and even the seconds.

These are obtained from the day duration by successive divisions:

• The hour is the 24th part of the day, that is that 1 day lasts 24 hours of same duration.
• The minute is the 60th part of the hour: an hour lasts 60 minutes
• The second is the...

## Find Prime Numbers with Scilab®: Programing of the Sieve of Erastothenes

The Sieve of Erasthotenes is a very old algorithm to find by elimination the prime numbers up to a given integer N.

The manual process is so:

1. write down the numbers from 2 to N (say 100)
2. stripe the multiples of 2, except 2
3. stripe the multiples of 3 not yet striped, except 3
4. iterate the process for the next unstriped number k: stripe the multiples of k not yet striped, except k.
5. continue until you get no more unstriped multiples before N

The unstriped remaining numbers are the numbers...

## The square root of 2 is a non rational number

The rational numbers are the results of exact divisions of integers.

They are represented as fractions or, more exactly, by equivalence classes of fractions.

Namely, each rational number has an infinity of representations as fractions, deduced from a "canonical" one, with mutually prime numbers. The canonical representation of a rational number has a numerator and a denominator that do not share any common divisor. The other representations are so that their numerators and denominators are...

## Music and Math: how is the Major C scale built?

The Greeks discovered the link between the musical intervals and the length of a string. The simpler example is that half the string sounds at the octave of the whole string. We know now that this corresponds to twice the frequency of the vibration.

Thus, 2 sounds differing by a 2 ratio on their vibrations have an interval of an octave. This is why the octave sounds so well to our ears, and was chosen as the periodicity of the scale.

The next such "harmonic" is when we divide the string in...

## Sharing a Pizza in 6 with trigonometry!

You may share a pizza in 6 equal portions with the help of trigonometry laws!

The method is so:

• share the pizza in 2
• share the diameter in 2 radius
• share one radius in 2
• elevate the perpendicular to the radius up the the circle
• cut from the center to the obtained point.

You obtain a portion that is exactly the 6th of the pizza!

Why is it so?

It is because of trigonometry!

Indeed, if you cut a circle into 6 equal portions from the center, the angles of the portions are 60° each. And...

## Infinite in Theory, Finite in Practice

When we start to count on our fingers and continue by heart, we only have a small part of all the possible "natural integers": these are all the possible numbers we may count with.

We know that we may count until very very big numbers, the numbers we may imagine, and even further… These very big numbers may be the size of a Galaxy, the size of the known universe, the number of cells in a human body, in all the human bodies, and so on.

The mathematicians construct the set of all "natural...

## Why real numbers are generally not real

The numbers defined in mathematics are in the increasing complexity order:

- The natural integers, from 0 to infinity, that we use to count

- The relative integers, that are the natural integers plus the negative integers, that we use for instance for accountancy "credit" or "debit" notions

- The rational numbers, that we use to share a quantity: they are the result of divisions

- The real numbers, that are all the possible finite or infinite combination of digits, with or without a...