1
Scientific Method: General Concepts
1.1 Introduction
In this chapter, we will examine the scientific method and its concepts. These concepts must be well understood and are fundamental to achieve a solid foundation in electrical engineering.
1.2 Powers of 10
In science it is common to represent numbers like 100, 1000, and 10000, as 102, 103, and 104, respectively, in what is called powers of ten.
Powers of 10 show how many times 10 has to be multiplied by itself to obtain the final value.
In other words,
is equal to 10 multiplied by itself 4 times.
However, if we want to represent very small numbers like 0.1, 0.01, and 0.001, we have to use negative exponents, as 10−1, 10−2, 10−3, respectively.
This negative exponent represents how many times the number must be divided by 10.
Numbers like
10−1, 10−2, 10−3 represent
1/10, 1/100, 1/100, respectively.
1.3 Roots
Roots are the inverse operation of powers.
Roots are expressed in scientific notation as a fractional exponent:
1.4 Scientific Notation as a Tool
In science, we constantly have to deal with numbers that are very hard to write in natural form, for example, 0.000000000003434033323. Imagine how hard it would be to compare this number with 0.0000000000000212817716 and know which one is bigger.
To solve this problem and make it easy to compare and write huge or minuscule numbers, a methodology called scientific notation was born.
1.4.1 Very Large Numbers
To convert a large number like 324484738, for example, to scientific notation, we divide this number by 10, several times, until we get a nonzero number with just one digit in the integer part.
Example:
| 324484738 | ÷ | 10 | = | 32448473.8 |
| 32448473.8 | ÷ | 10 | = | 3244847.4 |
| 3244847.38 | ÷ | 10 | = | 324484.74 |
| 324484.738 | ÷ | 10 | = | 32448.474 |
| 32448.4738 | ÷ | 10 | = | 3244.8474 |
| 3244.84738 | ÷ | 10 | = | 324.48474 |
| 324.484738 | ÷ | 10 | = | 32.448474 |
| 32.4484738 | ÷ | 10 | = | 3.2448474 |
These numbers show that we have to divide the initial number 8 times by 10 to get a single digit alone in the integer part, like 3.24484738.
For that reason, the given number in scientific notation is represented as 3.24484738 × 108.
1.4.2 Very Small Numbers
To convert a very small number to scientific notation, we must multiply the number by 10 several times, until we get a single‐digit number in the integer part. The number of times the multiplication happens will be the exponent.
In order to convert, for example, 0.00156 to scientific notation, it would be necessary to multiply it 3 times by 10 to obtain a single nonzero‐digit number:
| 0.00156 | × | 10 | = | 0.0156 |
| 0.0156 | × | 10 | = | 0.156 |
| 0.156 | × | 10 | = | 1.56 |
For that reason, 0.00156 is represented in scientific notation as 1.56 × 10−3.
Like we said before, negative exponents mean division. So, 1.56 × 10−3 can be also written as
That is exactly 0.00156.
1.4.3 Operations with Powers of 10
Operations like multiplication, division, etc. can be done with numbers represented in powers of 10.
This is how they are done.
1.4.3.1 Multiplication
To multiply two numbers, for example, 1.56 × 10−3 and 3.33 × 108, we must first multiply the numbers without the powers
and then sum the exponents
The result would be 5.19 × 105.
If the final result is not a number with a single nonzero digit in the integer part, we must make sure it is.
A final result of 24 × 1011 should be corrected to 2.4 × 1012.
1.4.3.2 Division
As explained before, any number raised to a negative exponent is equal to that number divided by the same power:
If we want to divide 3 × 109 by 8 × 102, we can represent that division as
The denominator can be converted to a nominator if we invert the exponent
and we just converted a division into a multiplication.
We can now multiply the numbers without the powers obtaining
and add the powers
and the final result will be 24 × 107, which should be converted to a single‐digit number equal to 2.4 × 108.
1.4.3.3 Adding and Subtracting
To add or subtract numbers represented in powers of 10, we must first make the numbers have the same exponent.
To add, for example, 2.4 × 108 and 3 × 109, we must first make both numbers have the same base. We chose 108. That will make the second number change to 30 × 108. We can now add both numbers:
The final result will be 32.4 × 108, corrected to 3.24 × 109.
1.4.3.4 Raising to a Number
To raise a number expressed in powers of 10 to a given exponent, we must raise the number without the power and then multiply the power by the exponent.
The cube of 2.4 × 108 will be equal to the number alone raised to the cube
and the base multiplied by the exponent
The final result will be 13.824 × 1024, corrected to 1.3824 × 1025.
1.4.3.5 Expressing Roots as Exponents
All roots in the form 
can be expressed as an exponent in the form 
.
For example,
1.4.3.6 Extracting the Root
To extract a root of a number expressed in powers of 10, we have to raise the number alone, without the power, to the correspondent exponent and multiply the power by that exponent.
To calculate the square root of
we first have to write the number in the exponent form
We then extract the root of the number alone
and then multiply the base by the exponent
The result is 1.549 × 104.
If the root is cubical, like
we first extract the cubical root of the number alone
and multiply the base by the equivalent exponent
The result is 1.338 × 103.
A problem arises if we need to extract a root of a number expressed in powers of 10 that is raised to another number, like
This root can be written as
As usual, we could start by extracting the root of the number alone and then multiplying the base by the exponent.
But when we do that, we have an exponent that is 8 divided by 3, resulting in 108/3 or 102.666, which is not a very pretty result.
To improve the result, we have to convert the exponent first to something that can be divided by 3. The closest candidate is 9.
Hence, we convert 2.4 × 108 into 0.24 × 109, that is exactly the same number written differently.
Now we can proceed. We first extract the cubical root of the number alone
and multiply the base by the exponent
The result is 0.6214465 × 103, which must be corrected to 6.214465 × 102.
1.4.4 Computers and Programming
Numbers written in scientific notation inside computer programs and other computer contexts use a specific syntax.
A number like 1.38 × 1025, for example, is normally written as 1.38E25 or 1.38e25.
Numbers with negative exponents have the sign before the number, for example, 1.38e − 25 or −3.43e − 12.
In almost all the cases, computer languages use decimal numbers as fraction number separators.
1.4.5 Engineering Notation
In engineering, scientific notation numbers are expressed with exponents of 10 that are multiples of 3, so they match one of the multiples of the International System (see Appendix A).
Following that rule, a number like 2 × 10−7 must be converted to 0.2 × 10−6 or 200 × 10−9. Normally it is always preferred to have numbers with first digits nonzero. For that reason, the later form will be generally preferred. In this case, the term 10−9 is equivalent to the prefix nano. So, the given number will be written as 200 nano or 200 n.