# Mathedu

Learn math by practice

# Re-engineering Math

## Why is a kilobyte 1,024 bytes rather than 1,000?

In the measurement units system, the prefix "kilo" is to multiply by 1,000, "mega" to multiply by 1,000,000 end "giga" to multiply by 1,000,000,000.

These are powers of 10:

• 1,000 is 10 to the power 3,
• 1,000,000 is 10 to the power 6
• and 1,000,000,000 is 10 to the power 9.

Also, 1,000,000 = 1,000 x 1,000 and 1,000,000,000 = 1,000 x 1,000,000, so that a mega is 1,000 kilo and a giga is 1,000 mega.

But in usual computer science, it is not so: a kilobyte is 1,024 bytes rather than 1,000...

## The Cross Product Law as a Linear Equation

Proportionality calculations lead to the use of the so-called 'rule of 3', also nammed the 'cross product law'.

Let's study it with an example.

If it lasts 15 minutes to bake 500 g of rastbeef, how much time shalll I bake my 950 g rastbeef?

Let's call x the time in minutes that we are looking for.

Then the time per gram is either 15 min over 500 g, or x min over 950 g:

x               15
------    =    ------    (1)
950            500

If we multiply...

## 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...

The Euclidian Division is the way integers divide together without introducing the fractions.

It deals with problems of sharing quantities of objects that can not be individually shared.

For instance, if we want to share 13 marbles between 3 children equally, then each child will receive 4 marbles, and it will remain 1 marble unshared.

This is written 13 ÷ 3 = 4, remains 1, that is equivalent to 13 = 4 x 3 + 1.

The definition of the Euclidian division is so: a ÷ b, with a and b...

## 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...

## Helix in architecture: Spiral Stairs

"Spiral" Stairs are stairs that go up together with tunling around a central axis.

They follow a mathematical curve called "helix".

But what is the equation of an helix?

Let's suppose that the vertical axis "z" is the central axis around wich the helix turns, oriented to the top, and that the first horizontal axis "x" is along the basis of the stairs.

Let the parameter a be the horizontal angle of a dot on the curve with the "x" axis.  Then the horizontal projection of the dot has...

## Exponentiation and the exponential increasing

The exponentiation is the action of multiplying a number by itself a given number of times.

For instance, 3 to the power 2 is 3x3=9, 10 to the power 3 is 10x10x10=1,000.

A sequence starting wit u(0)=C and continuing with the iterative formula u(n+1)=a x u(n) has for value u(n) equal to C multiplied by a to the power n.

Such a sequence is called a geometrical sequence of initial point C and "reason" a>0.

There are 3 cases for the behavior of the sequence:

• if a>1, the sequence goes to...

## 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...

## The Measurement of Time I: the dates in the calendar

As we saw in a previous post, the metric system for the measurement of lengths is a decimal system, and the time measurement system is not.

First, the basic time units are NOT the hour or the second, but the day, the month and the year:

• The day is the time by which the earth spins on its axis completely.
It used to be measured by the time between 2 instants at which the sun is at its highest position in the sky.
• The year is the time by which the earth turns around the sun by a full...