Web de Paco Gómez

Contents

 

1 Division

We will start by reviewing the concept of division. Division consists of forming groups. In a division you are given two terms, the dividend and the divisor. Thus, dividing the dividend by the divisor is to see how many groups of the size of the divisor can be formed with the dividend. Division is a way of grouping objects. For example, if I have 12 as dividend and 3 as divisor, the division 12 ÷ 3 is equivalent to find how many groups of size 3 I can form with 12 objects (see Figure 1). In this case, the answer is 4. The number of groups is called the quotient. This kind of division is called exact division. 

 

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Sometimes, after forming all the groups we see that there are elements left over. Those elements are called the remainder. When a division has no remainder is called even division or exact division. Otherwise, it is called division with remainder. In the previous example, 12 ÷ 3 is an exact division. We can write

If we want to divide 17 by 5 we find it is a division with remainder.

Here 2 is the remainder and 5 is the quotient. In this case we can write

Performing this division as formation of groups,  we have the following:

 

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Groups have been formed by rows. The number of columns is the size of each group.

Let's think a little bit more abstract. Let a be the dividend, b the divisor, q the quotient, and r the remainder of a division. The relation among those numbers through division is given by the following equation

 

There is a point to make here. When dividing two numbers, we form as many groups as we can. That implies that the remainder is always less than the divisor. In terms of our equation, that means 0 ≤ r < b.

 

2 Exercises

1. Classify the following divisions as exact or with remainder: 1 ÷ 1, 3 ÷ 2, 121 ÷ 11, 0 ÷ 7, 1089 ÷ 33, 1405782039 ÷ 2, 1405782039 ÷ 3.
2. Perform the division 20 ÷ 4 by forming groups.
3. Perform the division 21 ÷ 4 by forming groups. Specify the dividend, divisor, quotient and remainder in terms of the groups formed.
4. We are given 19 objects. We want to divide 19 by 3. After performing the division we obtain 5 groups of 3 elements each plus a remainder of 4. Is this division correct?

 

3 Highest Common Factors

The highest common factor is a concept easy to understand. Given two numbers, each has a set of factors, also called divisors. The highest common factor (hcf from now on) is just the highest common factor found in both numbers. The hcf always exists because 1 is a factor of any number.

Here we have an example of the hcf. Take numbers 45 and 75. The factors of those numbers are:

• Factors of 45: {1,3,5,9,15,45}.
• Factors of 75: {1,3,5,15,25,75}.

We easily see that the set of common factors is {1,3,5,15}, the highest being 15. Therefore, the hcf(45,75) = 15.

This method of calculating the hcf consists of enumerating all factors of both numbers. This is a good way of practising division because you have to perform many divisions to find those factors. If the numbers are low, it will be all right. If they are high, it will become cumbersome.

Here there is a question to make: How do we find the list of factors of a number? First, we have to find its prime factors. By the way, what is a prime factor? A prime number, I may recall you, is a number only divisible by 1 and itself. Examples of prime numbers are 3, 5, 7, or 749027409284729381. Every number is the product of a series of prime factors. Normally, you find the prime factors of a number by trial and error. You check if your number is divisible by 2; if not, you carry on with 3, the next prime number, and so on. When a division is exact, you obtain a smaller number, and you continue trying with more, bigger prime numbers. For the number of our example, their decomposition in prime factors is

 

In view of that decomposition it is easy to realize that the hcd is formed by the common factors raised to the least exponent. In our case the common factors are 3 and 5, and the least exponents are 1 in all cases. Therefore, hcd(45,75) = 3 ⋅ 5 = 15. This idea relates the hcd with the decomposition in prime factors. However, it is not a good method for calculating the hcd in general. Again, when the numbers are high, finding the prime factorization is a very hard process to go through.

 

4 Exercises

1. Calculate the hcf of the following pair of numbers by finding their prime factors: 33 and 11; 144 and 60; 121 and 275; 1089 and 924.
2. With the ideas presented above, how would you calculate the hcf of three numbers? Use your ideas to calculate the hcf(35,49,77).

 

5 Euclid’s Algorithm

How can we calculate the hcd in a better way? Euclid, a Greek mathematician who lived 400 years B.C., invented a way to do it. His idea is called Euclid's algorithm. Euclid wrote down the equation of the division:

Every common factor of a and b -Euclid observed- must be a common factor of the remainder r. Let d be such a common factor. Then:

If numbers   and   are natural numbers, so will   be. In particular, if d is the hcd of both numbers, that property will still be true. Therefore, hcd(a, b)=hcd(b, r). However, we should not stop here. We can carry on applying this idea until the remainder is 0. At that moment, the last quotient obtained is the hcd (or, equivalently, the last non-null remainder).

Let’s see how this algorithm¹ works. Take a= 75 and b = 45, as before. We start off by dividing both numbers:

Here the remainder is 30. Is the remainder equal to 0? No. In that case, we continue doing divisions:

Again, is the remainder null? No. We proceed to the next division.

Is the remainder null now? Yes. We are done. The hcd is the quotient of this division, namely, 15.

Again, this process can be thought through formation of groups. Let’s illustrate this by calculating the hcf of 16 and 12; see Figure 3. We start by writing 12 ones followed by 4 zeros. This is equivalent to performing the first successive division 16 by 12.  The remainder is 4. Then, we make groups of 4 elements each and put them below the ones. Therefore, we have made 3 groups and no element has been left. Now, the division is over. Since the groups were formed by putting elements one below another, the hcf is the number of columns in the final box, that is, 4.

 

 

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6 Exercises

1. Calculate the hcf of the following pair of numbers by using Euclid’s algorithm: 33 and 11; 144 and 60; 121 and 275; 1089 and 924. Compare how long it took you to the method of finding the primer factors. Which one is the fastest?
2. Calculate the hcf of 24 and 18 by applying Euclid’s algorithm and formation of groups.

 

7 Music

We go down to the basics of music language. In order to do so, we first become familiar with concepts such as notes, rests, and rhythm. Therefore, we start by considering as a fount for a  time span - that is, a fixed amount of time.  It can be represented by a rectangle.

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That time span can be divided into equal parts, as usually done in Western music. Those parts are called pulses.

 

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Each little box in the Figure is a pulse. Pulses have the same length in time.

A rhythm is formed by combination of notes and rests played on the pulses.

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The boxes with a dot are notes, the empty boxes are rests. Rests are not played, but have to be counted.

Here we are going to study a particular type of rhythm: timelines. A timeline is a rhythm design played throughout a piece with no variations. It is the leading temporal and structural reference within a piece. In the Figure below there are several timelines. Notice that these timelines have different length in terms of number of pulses.

 

Figure 7:

Timelines of different lengths.

 

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8 Exercises

1. Play the timelines given in Figure 7 four times each. Use clapping hands.  Increase speed/tempo gradually, but keep a steady beat all along your practising.
2. Compose your own timelines using time spans of 10 and 14 pulses. Practise them as suggested above.
3. Compose your own timelines. Practise them as suggested above.

 

9 Matherhythm

Let’s now play with maths and music together. Let’s fix a 12-pulse time span. If I want to play a 12-note rhythm, then I have to divide 12 (number of pulses) by 1:

and the resulting rhythm would be

Here an X denotes a note. This is very easy since it is clear that every note must go on one pulse.

If I want to play 4 notes, then I have to divide 12 by 3:

and the resulting rhythm would be

Here a "." (a dot) denotes a rest. If we want to obtain this rhythm through division, all we have to do is divide 12 by 3 with ones (notes) and zeros (rests) as follows.

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Again, since the groups were formed vertically, we produce the rhythm by reading the groups by columns. The resulting rhythm is [X . . X . . X . . X . .]

If we want to play 6 notes, then we have to divide 12 by 2:

and the resulting rhythm would be

If we want to play 8 notes, then we have to divide 12 by...

Oops! It turns out that 8 is greater than 6. No divisor of 12 can be greater than 6.

We’ll use the so-called evenness principle, which reads as follows:

Let’s see how this principle works. If we have this rhythm:

it is clear that its notes are not evenly distributed. However, if we move a few notes here and there, then this new rhythm

does have its notes evenly distributed. How did we do it? By calculating the hcf with Euclid’s algorithm and formation of groups.

 

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The resulting rhythm here is [X . X X . X X . X X . X ], which is not the rhythm we obtained first above. This interesting fact has to do with rotation of rhythms. The evenness of a rhythm depends on the distances between its notes. Therefore, the rotation of an even rhythm is still even. In our case, both rhythms are the same up to a rotation.

Since the hcf of 12 and 8 is 4, our rhythm is formed by a small rhythmic cell, [. X X], repeated 4 times.

Finally, notice that when the number of notes is a divisor of the number of pulses, applying the evenness principle is the same as performing division. In that sense the evenness principle is a generalization of division.

If we want to play 7 notes... we already know we have to apply the evenness principle again. The resulting rhythm would be

Let’s see how we obtained this rhythm.

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This rhythm, as already mentioned, is a rotation of [X . X . X X . X . X . X]. In particular, it is a rotation by 3 positions to the right.

Rhythms produced by using the evenness principle are called Euclidean rhythmsor even rhythms.

Finally, here we have the main four timelines we obtained by considering the divisors of 12 and applying the evenness principle.

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Although these rhythms admit a mathematical explanation, they actually exist. They come from an ancient, rich rhythmic tradition, the Ghana tradition. In particular these rhythms are the timelines of a piece of music called gamamla. They are played on a double bell called gankogui (see Figure 12).

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The gankogui produces two sounds, a low-pitched sound and a high-pitched one. The notes of the timelines are distributed among the two bells. Written in musical notation, the final score for the gamamla is given below; notes below the line mean low bell, whereas notes above the line means high bell.

 

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10 Exercises

1. Explore and practise the rotations of the 6-note and 8-note timelines.
2. Explore and practise the rotations of the 7-note timeline [x . x . x x . x . x . x]. Which are the most interesting ones? Do you have an explanation?
3. Create your own 4 timelines following the ideas in the text. Use a 16-pulse timeline. Play the piece by clapping your hands. Make the necessary adjustments in terms of rotations of rhythms.

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