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Created it, 05/10/15
Update it, 05/12/24
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MATHEMATICS “2nd PART”
We now approach the second part of our lesson.
Previously, we advise you, progressively your reading, to solve the exercises quoted in examples. Each finished exercise, compare your result with that which is given to you. A few days later, start again this time without looking at the development a priori, but always compare the results.
We repeat it, in mathematics, it is necessary to practice and repeat to include/understand and retain.
Let us continue. If we consider, as we said, that the four elementary operations (addition, subtraction, multiplication, division) are known of the reader, it seems useful to us to speak about “rise to a power” and about its reverse, i.e. of “the extraction of a root”. The six arithmetic operations will be then known.
Rise with a power is a particular form of the multiplication.
Definition: One calls power of a number produces it of several equal numbers between them.
For example, the number 100, which one can obtain by multiplying 10 by 10 is a power of 10. Number 1 000, that one can obtain by multiplying 10 by 10 and the result still by 10, is another power of 10. The number the 16, that one can obtain by multiplying 2 by 2, result by 2 and the new result still by 2, is a power of 2. What is written : 10 . 10 = 100, 10 . 10 . 10 = 1000, 2 . 2 . 2 . 2 = 16.
However, in the ordinary mathematical expressions, to indicate these calculations, one has recourse to a convention which has the advantage of being very concise and to highlight the properties of the powers.
As the multiplication factors which are used for calculation of a power are all equal, one only once writes the number being used as factor then one writes beside this one, in top and on the right, the number which represents the quantity of factors. While thus proceeding, rise with the power in the preceding examples will be written in the following way :
The number which represents the equal factors calls the base (numbers 10, 10 and 2 are the bases of the respective powers taken into account). The other number, which indicates the quantity of these same factors is called the exhibitor (numbers 2, 3 and 4, of the examples given are exhibitors).
The unit bases and exposing form the power.
When the exhibitor is 2, the power is called the square or power two. When the exhibitor is 3, the power is called the cube or power three. With exhibitors 4, 5, 6… one respectively says power four, power five, power six, etc…
In connection with these powers respectively, the operations being used to calculate the values of them are called rise with the square, or with the power two, rise with the cube, or with the power three, rise with the power four, the power five, the power six, etc… Let us note that this terminology employed in the language running is not rigorous because one should say : 10 exhibitor 2, 10 exhibitor 3, 2 exhibitor 4 instead of 10 power 2, 10 power 3, 2 power 4, etc…
He can be very useful to write a number in the form of rise to a power to obtain shortened arithmetic expressions, in particular when they are formed very large numbers of few figures and follow-ups of much of zeros, or in calculations of very small numbers, formed of decimal digits preceded by many of zeros.
To fix us the ideas, let us consider the case of a very large number : hundred billion.
In the ordinary form, this number is written with eleven zeros after figure 1, that is to say 100 000 000 000.
Obviously, it is a little tiresome to write such a large number; however, if it is considered that :
100 000 000 000 is equal 10 . 10 . 10 . 10 . 10 . 10 . 10 . 10 . 10 . 10 . 10
and that it is thus a power of 10, we will just like be able after having counted the number of factors (which is 11, the number of zeros), to write in a more concise way 1011, instead of aligning a long file of zeros.
In a similar way, by considering that the number :
1 200 000 is equal to 12 . 10 . 10 . 10 . 10 . 10,
One will be able to transform of rise with a power the part of the multiplication which follows the 12. The result will be :
The expression of the numbers in the form of rise to a power, which in a very obvious way is very useful to shorten the writing, can also involve a notable simplification of the arithmetic operations, when one must carry out calculations with two or several powers.
Briefly let us see the principal rules which calculations of the powers obey.
Regulate 1 : The product of two or several having the same base, is equal to a power having the same base and an exhibitor equal to the sum of the exhibitors.
Examples :
Regulate 2 : The division of two powers having the same base is equal to a power having the same base and an exhibitor equal to the difference of the exhibitors.
Example :
In the applications of this rule, one can encounter three particular cases which we will examine.
CASE 1 : the difference of the exhibitors is equal to 1.
Example :
The exhibitor of the power which constitutes the result of the operation is equal to 1. In this particular case, one thus has 31 = 3, as previously one had 51 = 5. Therefore, exposing it 1 means that there is only one single factor.
CASE 2 : the difference of the exhibitors is equal to zero.
Example :
The value which it is necessary to allot to power 110 is thus 1.
Same manner, one finds for example :
It is interesting to notice that the powers having an exhibitor equal to zero always have the same value : they are equal to 1, one will thus have:
20 = 1 ; 250 = 1 ; 10 0000 = 1 and so on for very other bases having for exhibitor zero.
CASE 3 : the difference of the exhibitors is equal to a negative number.
Example :
What can be written :
72 / 75 = 7 . 7 / 7 . 7 . 7 . 7 . 7 = 7 . 7 . 1 / 7 . 7 . 7 . 7 . 7 = 1 / 7 . 7 . 7 = 1 / 73 = 1 / 343
The value of the power having a negative exhibitor, 7-3 is thus equal to 1 / 343, that is to say 1 / 73
Consequently, any power having a negative exhibitor is equal to a fraction having for numerator 1 and denominator this same power whose exhibitor was made positive.
One can thus make positive all the negative exhibitors, for example :
Or
Regulate 3 : - The product of two or several powers having the same exhibitor is equal to a of the same power exhibitor having for base produces it bases.
Regulate 4 : - The division of two powers having the same exhibitor is equal to a power whose exhibitor is the same one and whose base is equal to the quotient of the two bases.
Examples :
Regulate 5 : The power of the power of a number is the power of this number whose exhibitor is the product of the two exhibitors.
Example :
Although the roots are the subject of the following paragraph, we give the following rule :
Regulate 6 : The root of a power is equal to the base of this power having for exhibitor the quotient of the first exhibitor by the root
Examples :
We give hereafter a summary table of the operations on the powers. We have, by this occasion, an application of literal calculation.
4. - ROOTSThe extraction of root is the opposite operation of rise to the power. Like there are powers two, three, four… There are roots second or square, third or cubic, fourth… In practice, it is the square root which one generally meets. It is thus it which will be more particularly treated in the following lines.
Definition : The square root of a number A is the number B which, multiplied twice by itself, will be equal to A.
Examples :
4.
2. - CUBE ROOTDefinition : The cubic root of a number A is the number B which, multiplied three times by itself, will be equal to A.
Examples :
Definition : The root fourth (fifth…) a number A is the number B which, multiplied 4 times (five times…) by itself, will be equal to A.
Examples :
We now will study the way of extracting a square root. This operation requires a very detailed explanation. But when you assimilate the procedure well, you will see that the extraction of a square root does not take much more time than a division.
We will illustrate our explanations for an example; Find the root square of 266,0161 :
Process :
1 - One writes the number under the radical (sign of the operation consisting in extracting a root).
As one said, the index 2, which should distinguish the square root of the roots with superscript (3, 4, 5…) is omitted.
2 - One separates the number given “in sections” of two digits on the basis of the line towards the left, or, if it acts as in our example of a decimal number, while leaving on the right and on the left the comma.
It can happen that the last group on the left consists of only one figure, like the 2 of our example. The same thing could arrive at the last group on the right of the comma; in this case, it is always necessary to add one zero, so that after the comma all the groups are made of two digits.
3 - Let us examine the first groups on the left, namely the 2. One calculates mentally which is the integer which, high squared, makes it possible to obtain the closest, equal or lower result, with the number formed by the first group of figures.
In our case, this integer is 1, since the square of 1, i.e. 1 x 1 is equal to 1 (while the following integer, is 2, high squared gives the result 4, which is larger than 2). One writes the number found in the space reserved for the root.
If the number formed by the figures of the first group were 21, the sought integer would be four, because the square 4 x 4 = 16 is lower than 21, while the square of the following integer, is 5 x 5 = 25, is higher than 21. On the other hand, if the number of the first group were 25, the sought integer would be 5, because 5 x 5 = 25.
4 - One defers the square of the integer found previously under the figure of the first group and one carries out the subtraction. Since the square of 1 is equal to 1, under figure 2 one writes 1 and one calculates the difference.
5 - Beside the found difference, one defers the figures of the second group, 66, and one separates the last figure by a point.
6 - One doubles the number present in the space reserved for the root (1 x 2 = 2) and one writes under the horizontal line the product obtained.
7 - One mentally calculates the quotient between number 16, obtained by the separation of the last figure of number 166, and numbers it 2, obtained by the doubling of the figure present, in the space reserved for the roots.
Then one writes the quotient beside the 2, under the space of the roots. Lastly, one multiplies the result number by this same quotient. Since 16 / 2 = 8, beside the 2 one writes this number and one multiplies the resulting number (28) by 8.
8 - One compares product 224 with number 166.
The product is larger than 166. One thus repeats the preceding operations by decreasing the quotient of a unit, i.e., by using number 7 instead of number 8. However, even with 7, the product (189) is larger than 166. One thus repeats the same operations with the number lower than 7 is with 6.
The product is lower than 166. One thus stops this series of operations by tracing a horizontal line under the last multiplication.
9 - One writes the product 156 pennies number 166 and one carries out the subtraction. Moreover, beside the difference one defers the third group of figures, namely 01, by separating the last figure by a point.
10 - One defers 6 (namely the number who allowed us to obtain product 156 smaller than 166) in space reserved to the root, beside number 1.
The two figures form number 16. Number 16 must be doubled and the new product (16 x 2 = 32) must be deferred under the preceding multiplications.
After figure 6, in the space reserved for the root, one puts the comma, since the operation is finished for the whole part of number 266, 0161.
11 - Now, by once again repeating the process started as in point 7 (see higher), one seeks the quotient of division 100 / 32, by deferring only the whole part of the quotient. Then, one writes beside 32 the number which represents the whole part of the quotient, namely 3 (100 / 32 = 3,…), and one carries out the multiplication by 3.
Since the product obtained (969) is lower than 1 001, one immediately carries out the subtraction between the two numbers. Moreover, beside the difference one defers the last group of figures, 61, by separating by a point the last figure.
12 - One defers number 3 (namely the number which enabled us to obtain product 969, smaller than 1 001) in the space reserved for the root, immediately after the comma. Then, one doubles number 163, obtained by adding 3 following 16 and by omitting the comma, and one writes the product (163 x 2 = 326) under the last multiplication.
13 - Arrive at this point, examine number 326 obtained by separating the last figure from 3 261 and other number 326, obtained by doubling 16,3 and by removing the comma.
The division of these two numbers 326 / 326 is equal to 1. Consequently, while following the process describes as in points 7 and 11, one adds 1 to 326 and one multiplies by 1 the number thus formed. The result of this multiplication is withdrawn of 3 261. Since the product is him also equal to 3 261, the difference will be equal to zero.
The number 1 is written, with which one carried out the preceding operations, beside the last figure of the number 16,3 in the space reserved for the root.
Calculation is finished (OUF!). Result 16,31 is the exact square root of number 266,0161 since the remainder is equal to zero.
To check this result, it is enough to calculate the square of the root.
In our case, by carrying out the multiplication 16,31 x 16,31, one obtain 266,0161 who is precisely the number of which we calculated the root. We can thus conclude that the result does not contain an error.
To exert us, let us carry out, without describing of them all the operations, the calculation of the square root of number 179.
The result obtained is 13, but since the operation has a remainder, 10, number 13 is not the exact square root of 179.
In cases of this kind, one can continue the operation to find other decimals which, added to the whole root, form a number as close as possible to the exact value of the root.
For this purpose, one adds the comma after the last figure of the number of which one seeks the root. Between in addition to, after the comma one adds a number of couples of zero equal to the number of decimals which one wishes to calculate for the root.
For example, let us suppose that we want to calculate the square root of 179, with two decimals. In this case, the comma being placed after the last figure, one adds two couples of zeros. Then one continues calculations in the usual way.
Result 13,37 constitutes the approximate value, calculated until the second decimal, of the square root of 179. To carry out the checking of this calculation, one proceeds as in the preceding case, i.e. one multiplies 13,37 x 13,37.
The result of this operation is 178,7569 ; as it is seen, it is not equal to 179, since number 13,37 is only the approximate value of the root. To supplement the checking, we should add the value of the remainder:
The result of this last operation, while proving to be equal to the number given at the beginning of calculations, confirms the exactitude of all the operations.
Another example with another provision: Calculate the square root of 0,00027, we write :
As this example shows it, the whole part of the number is zero, therefore the whole part of the root is 0. The first groups comma of the number on the right is consisted two zeros, therefore, the first decimal digit of the root is zero. The second groups comma on the right is consisted 02, which comes down to saying 2. The greatest square contained into 2 is 1 ; its root is 1, and this one will be the second decimal digit of the sought root.
The extraction of the root continues normally. As one saw, the extraction of the root is a somewhat tiresome operation.
In addition, even rise with the square, which is however very simple, can make tiresome a calculation consisted many other operations ; one finds now in the trade at a reasonable or ridiculous price, scientific computer machines allowing to immediately find the root square or the square of a number.
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