Scientific notation

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scientific notation, also called exponential notation, is a way of organizing or writing numbers based on powers of 10, which is especially useful for representation of very large or small values, as well as for calculation with them. This is particularly true in physics and chemistry where these values are frequent, so this notation is suitable for clearly showing significant figures and allowing immediate comparisons of magnitude. For example, in approximate values:

electron mass
0.000 000 000 000 000 000 000 000 000 911 = 9.11 × 10^-31 kg
Avogadro constant (material quantity: mol)
602 000 000 000 000 000 000 = 6.02 × 10²³ Elementary entities
greater observable distance of the universe:
740 000 000 000 000 000 000 000 m = 7.4 × 10²⁶ m
proton mass:
0.000 000 000 000 000 000 000 000 000 001 67 kg = 1.67 × 10^-27 kg

The exponent indicates the places that the comma must be moved to change from scientific notation to decimal notation: to the right if it is positive and to the left if it is negative. When it comes to converting a number to scientific notation, the process is reversed.

Mantissa and order of magnitude

The scientific notation of a number is the form ${displaystyle m times 10^{n}}$ where:

The "mantisa" m is a decimal number whose entire part has a single figure other than zero.
The "order of magnitude" ${displaystyle 10^{n}}$ It's an entire exponent power.

Like this:

520 = 5.2 × 10²
-45.9 = -4.59 × 10¹
575 230 000 000 = 5.7523 × 10¹⁴
0.0523 = 5.23 x 10^-2
0.000 000 000 000 000 690 3 = 6.903 × 10⁻¹⁶

History

Archimedes, the father of scientific notation.

The first attempt to represent numbers that were too large was undertaken by the Greek mathematician and philosopher Archimedes, and described in his work The Sand Counter, in the 3rd century BC. C. He developed a numerical representation system to estimate an upper limit for the number of grains of sand needed to fill the universe. To do this he had to estimate the size of the universe according to the model in force at the time and, furthermore, invent a way of expressing very large numbers. The number estimated by him was 10⁶³ grams.

There are those who think, King Gelon, that the number of grains of sand is infinite. And when I mention sand, I am not only referring to the one that exists in Syracuse and the rest of Sicily, but also the one found in other areas, whether inhabited or uninhabited. Once again, there are those who, without considering it infinite, think that no number was yet appointed to be large enough to exceed its multiplicity. And it is clear that those who have this opinion, if they imagined a sandy mass of the size of the Earth mass, including in this all seas and depressions of the Earth filled to a height equal to the highest of the mountains, it would be much to recognize that any number can be expressed in such a way as to overcome the multiplicity of sand there existing. But I will try to show through geometric demonstrations that you will be able to accompany that, two numbers named by me and that consist in the work I sent to Zeuxipo, some exceed, not only the number of mass of sand equal in magnitude to that of the Earth filled in a way described above, but also the mass equal in magnitude to that of the universe.
Arena counter (Arquímedes), pg. 1

It was through scientific notation that the model for representing real numbers by floating point was conceived. This idea was proposed by Leonardo Torres y Quevedo (1914), Konrad Zuse (1936) and George Robert Stibitz (1939). The floating-point encoding of today's computers is basically base-2 scientific notation.

Programming with the use of numbers in scientific notation enshrined a representation without superscripts, in which the letter e (or E) is the mantissa of the mantissa exponent. Thus, 1.785 × 10⁵ and 2.36 × 10⁻¹⁴ are represented, respectively, by 1.785E5 and 2.36E-14 (as most programming languages do). are based on English, commas are replaced by periods).

Types of scientific notation

In standard scientific notation, the exponent e is chosen such that the absolute value of m remains at least one but less than ten (1 ≤ | m | <10). For example, 350 is written as 3.5 × 10². This form allows a simple comparison of two numbers of the same sign in m, as the exponent and indicates the number of the order of greatness. In standard notation the exponent e is negative for an absolute number with value between 0 and 1 (for example, less than half is -5 × 10⁻¹ ). The 10 and the exponent are generally omitted when the exponent is 0.

In many areas, scientific notation is normalized in this way, except for intermediate calculations, or when a non-standard form, such as engineering notation, is desired. Scientific (normalized) notation is often called exponential notation - although the latter term is more general and also applies when m is not restricted to the interval 1 to 10 (as in engineering notation, for example) and for bases other than 10 (as in 315 × 2²⁰).

E notation

Example: a calculator shows the Avogadro Constant in notation E

Most calculators and computer programs are programmed to display excessively large or small numbers in scientific notation. Despite this, they are generally not capable of illustrating "the traditional way" exponents of powers, such as 10⁷ (the same goes for mathematical subscripts). In these cases, they resort to an alternative format for graphical representation of powers: the notation E, where the letter E, followed by a number, literally represents "multiplied by ten to the power of" (that is, "× 10ⁿ"). In other words, if we take two real numbers m and n, the representation "mEn" would mean exactly m × 10ⁿ.

The character e has nothing to do with the mathematical constant e (confusion is not possible when using the uppercase letter E); and although it represents an exponent, the notation is generally referred to as e (scientific) notation or E (scientific) notation, rather than exponential (scientific) notation (although the latter can also occur).

Examples

In the programming language FORTRAN 6.0221415E23 is equivalent to 6.022 141 5×1023.
The programming language ALGOL 60 uses a subscript ten instead of the letter E, for example

6.0221415₁₀23. ALGOL 68 also allows lowercase E, for example 6.0221415e+23.

The programming language ALGOL 68 has the option of 4 characters in (eE).). Examples:

6.0221415e23, 6.0221415E23, 6.022141523 or 6.0221415⏨23.

The programming language Simula requires the use of & (or & fake for long), for example:

6.0221415&23 (or 6.0221415&&23).

In multiparadigm programming language such as Python it is not relevant to use capital or lowercase for character E or e, having the same meaning.
```
6.0221415e23 = 6.0221415E23
```

Engineering notation

Engineering notation differs from standard scientific notation in that the exponent e is restricted to multiples of 3. Consequently, the absolute value of m is in the interval 1 ≤ |m| <1000, instead of 1 ≤ |m| < 10. Although conceptually similar, engineering notation is rarely called scientific notation.

Numbers in this form are easy to read, using magnitude prefixes like mega (m = 6), kilo (m = 3), milli (m = −3), micro (m = −6) or nano (m = −9). For example, 12.5×10⁻⁹ m can be read as "twelve point five nanometers" or written as 12.5 nm.

Motivation

Example of scientific notation, Avogadro's constant.

Scientific notation is a very convenient way to write small or large numbers and do calculations with them. It also quickly conveys two properties of a measurement that are useful to scientists, significant figures and order of magnitude. Writing in scientific notation allows a person to drop leading or trailing zeros from significant figures. This is very useful for very large or very small measurements in astronomy and in the study of molecules. The following examples can demonstrate this.

Examples

The mass of an electron is approximately 0.000 000 000 000 000 000 000 000 000 000 910 938 22 kg. In scientific notation, this is written 9.109 382 2×10^-31kg.
The mass of the earth is about 5 973 600 000 000 000 000 000 000 kg. In scientific notation, this value is represented by 5.9736x10²⁴ kg.
The Earth's circumference is approximately 40,000 m. In scientific notation is 4×10⁷ m. In engineering notation, it's 40×10⁶ m. In the style of SI representation, it can be written 40 mm (40 meter).

Significant figure

One advantage of scientific notation is that it reduces the ambiguity of the number of significant digits. All digits in standard scientific notation are significant by convention. But, in decimal notation any zero or a series of zeros next to the decimal point are ambiguous, and may or may not indicate significant numbers (as they must be underlined to make explicit that they are significant zeros). In decimal notation, the zeros next to the decimal point are not necessarily a significant number. That is, they may be there just to show where the decimal point is. However, in scientific notation this ambiguity is resolved, because the zeros shown are considered significant by convention. For example, using scientific notation, the speed of light in SI units is 2.99792458×10⁸ m/s and the eminence is 2.54×10⁻² m; both numbers are exact, by definition, the units "inches" per centimeter and m in terms of the speed of light. In these cases, all figures are significant. A single zero or any number of zeros can be added to the right hand side to display more significant digits, or a single zero with a slash across the top can be added to display infinitely many significant digits (as well as in decimal notation).

Ambiguous last digit in scientific notation

It is customary in scientific measurements to record all significant digits of the measurements, and to assume an additional digit, if there were certain information at all available for the observer to make an assumption. The resulting number is considered more valuable than it would be without that extra piece of data, and is considered a significant figure, since it contains some information that leads to greater precision in the measurements and in the aggregation of the measurements (add or multiply them).

Through additional annotations, additional information about accuracy can be conveyed. In some cases, it can be useful to know that it is the last significant algorithm. For example, the accepted value of the elementary unit charge can be validly expressed as 1.602176487(40)×10⁻¹⁹ C, and whose figures appear in parentheses at the end of the value, indicate their uncertainty, specifically it is expressed as 0.000000040×10⁻¹⁹ C, and is a shortcut to the abbreviation of (1.602176487 ± 0.000000040)×10⁻¹⁹ C.

Order of magnitude

Scientific notation allows a quick comparison between various homogeneous quantities. For example:

Proton Mass: 1.6726×10⁻²⁷ kg
Electron Mass: 9.109 382 2×10^-31 kg

To compare them with a sufficient approximation, the quotient between the powers of ten is enough:

{displaystyle {10^{-27}}:{10^{-31}}={10^{-27-(-31)}}={10^{4}}}

That is, the proton is about four orders of magnitude (about 10,000 times) more massive than the electron.

Description

The mass of the Milky Way is 1 × 10⁴¹ kg.

In a broad sense, scientific notation would allow an infinite number of representations of each value. For example, for 739,000,000,000 we would have, among many others, the following:

739×10⁹ = 73.9×10¹⁰ = 7.39×10¹¹

However, the mantissa or coefficient is required to be greater than or equal to 1 and less than 10. Thus we have a unique form of scientific notation:

739 000 000 000 = 7.39×10¹¹

How to Transform

To transform any number into scientific notation, we must move the comma obeying the principle of equilibrium.^{[citation required]}

Let's take the example below:

{displaystyle {253756.42}}

Scientific notation requires that the mantissa (coefficient) be between 1 and 10 in absolute value. In this situation, the appropriate value would be 2.5375642 (note that the sequence of numbers is the same, only the position of the comma changes). For the exponent, pay attention to the equilibrium principle: "Each decimal that decreases the mantissa value increases the exponent by one unit, and vice versa."

In this case, the exponent is 5.

Look at the transformation step by step:

${displaystyle {253756.42}}$
${displaystyle {25375.642 times 10^{1}}}$
${displaystyle {2537.5642 times 10^{2}}}$
${displaystyle {253.75642 times 10^{3}}}$
${displaystyle {25.375642 times 10^{4}}}$
${displaystyle {2.5375642 times 10^{5}}}$

1 mol of molecules has 6.02 × 10²³ molecules.

Another example, with values below 1:

0.0000000475
0.000000475 × 10⁻¹
0.00000475 × 10⁻²
0.0000475 × 10 ⁻³
0.000475 × 10⁻⁴
0.00475 × 10⁻⁵
0.0475 × 10⁻⁶
0.475 × 10⁻⁷
4.75 × 10⁻⁸

Use of spaces

In scientific notation, E notation, and engineering notation, the space (which, in text format, can be represented by either a normal wide space or a thin space), only allowed before and after ×, in front of E or e can be omitted, although it is less common to do so before the character alphabetical.

Mathematical operations with scientific notation

Addition and Subtraction

The human brain has about 1 × 10¹¹ neurons.

To add or subtract two numbers in scientific notation, the exponents need to be the same. That is, one of the values must be transformed so that its exponent is equal to that of the other. The result will probably not be in the required form (mantissa with a single digit other than 0 to the left of the comma) so it will be converted later.

Examples:

${displaystyle {4.2cdot 10^{7}}+{3,5cdot 10^{5}}={420cdot 10^{5}}+{3.5cdot 10^{5}}={423.5cdot 10^{5}}}$ = ${displaystyle {4.235cdot 10^{7}}}$

${displaystyle {6.32cdot 10^{9}}-{6.25cdot 10^{9}}={0.07cdot 10^{9}}}$ = ${7cdot 10^{7}}$

Multiplication

Multiply the mantissas and add the exponents of each value.

Example:

${displaystyle {(6,5cdot 10^{8})}cdot {(3.2cdot 10^{5})}={(6,5cdot 3.2)cdot 10^{8+5}}={20.8cdot 10^{13}}}$ = ${displaystyle {2.08cdot 10^{14}}}$

${displaystyle {(4cdot 10^{6})}cdot {(1.6cdot 10^{-15})}={(4cdot 1.6cdot 10^{6+(-15)})}={6.4cdot 10^{-9}}}$

Division

Divide the mantissas and subtract the exponents of each value:

Examples:

${(8cdot 10^{17})}: {(2cdot 10^{9})} = {(8/2)cdot 10^{17-9}} = {4cdot 10^{8}}$

${displaystyle {(2.4cdot 10^{-7})}:{(6.2cdot 10^{-11})}={(2.4/6.2)cdot 10^{-7-(-11)}}={0.3871}cdot 10^{4}}$ = ${displaystyle {3.871}cdot 10^{3}}$

Exponentiation or Potentiation

The mantissa is raised to the outer exponent and the base ten exponent is multiplied by the outer exponent.

${displaystyle {(2cdot 10^{6})^{4}}={(2^{4})cdot 10^{6.4}}={16}cdot 10^{24}=1.6cdot 10^{25}}$

Filing

Before performing the filing, it is necessary to transform an exponent to a multiple of the index. After this is done, the result is the root of the mantissa multiplied by ten to the power of the ratio of the exponent to the index of the radical.

${displaystyle {sqrt {1.6cdot 10^{27}}}={sqrt {16cdot 10^{26}}}={sqrt {16}}cdot 10^{26/2}=4cdot 10^{13}}$

${displaystyle {sqrt[{5}]{6.7cdot 10^{17}}}={sqrt[{5}]{670cdot 10^{15}}}={sqrt[{5}]{670}}cdot 10^{15/5}approx 3.674cdot 10^{3}}$

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