An order of magnitude larger than necessary. Standard notation for a number, mantissa of a number, exponent of a number

standard form notations of a number, mantissa of a number, exponent of a number

Positive number written in standard form, has the form

The number m is a natural number or a decimal fraction, satisfies the inequality

and called the mantissa of a number written in standard form.

The number n is an integer (positive, negative, or zero) and is called the order of the number written in standard form.

For example, the number 3251 in standard form is written like this:

Here the number 3.251 is the mantissa and the number 3 is the exponent.

The standard notation for a number is often used in scientific calculations and is very useful for comparing numbers.

In order to compare two numbers written in standard form, you must first compare their orders. The larger number is the one whose order is higher. If the orders of the compared numbers are the same, then you need to compare the mantissas of the numbers. In this case, the larger number will be the one whose mantissa is larger.

For example, if we compare numbers written in standard form with each other

and ,

then obviously the first number more than a second, because it has more order.

If we compare the numbers

then, obviously, the second number is greater than the first, since the orders of these numbers are the same, and the mantissa of the second number is larger.

On our website, you can also familiarize yourself with the educational materials developed by the teachers of the Resolventa training center for preparing for the Unified State Examination and the OGE (GIA) in mathematics.

For schoolchildren who want to prepare well and pass the exam or OGE (GIA) in mathematics, physics or Russian language for a high score The educational center"Resolvent" holds

(Photo: Paul Matthew Photography, Shutterstock)

Often they say “an order of magnitude more”, “an order of magnitude less”, or even “more / less by several orders of magnitude”. It is intuitively clear that “an order of magnitude more” means “much more”, “significantly more” - but I would like to know by how much exactly? If you read this article, you will know for sure.

Any real number... Sorry... Perhaps not everyone remembers what it is. And you know, it doesn't matter. As Uncle Murphy said: "If you do not understand any term in a technical article or documentation, feel free to skip it - the article will completely retain its meaning without this term."

My answer: "Yes, very simple!". Now in every computer science textbook there is such a topic as "Translating numbers from one number system to another", and special attention ...

So let's try again: any number X, other than zero, can be represented as

X \u003d Mantissa * 10 ^ Exponenta,

i.e. "mantissa multiplied by ten to the power of the exponent", where
mantissa is a number, modulo (that is, without a sign), not less than one and less than ten, and
exhibitor – any integer (... -3, -2, -1, 0, +1, +2, +3, ...).
Well, it’s just that these numbers are called so: one is the mantissa, the other is the exponent. There is no need to “hang out” much on this, we go further.

Zero, by the way, cannot be written in this way, because the mantissa, by definition, is not zero, but ten to whichever integer power you raise, you still get a number greater than zero, and the product of two numbers that are not equal to zero is not equal to zero.

For example,
1024 = 1.024 * 10^3
-3.14 = -3.14 * 10^0
1"000"000 = 1 * 10^6

This type of notation is called scientific or standard. It is convenient, for example, because numbers written in such a notation are convenient to compare: if the numbers have the same sign (both positive or both negative), then the exponents are compared first, and only then, if the exponents are equal, are the mantissas compared.

And this is where we come to the answer to the question, what does “an order of magnitude more” mean. Another, more Russian, name of the exhibitor is “order”. The number 256 is a second order number because 256 = 2.56 * 10^2. A million is a number of the sixth order, a billion is the ninth. Actually, 1024 is exactly 4 times more number 256, but if you just need to determine which of them is greater, it is quite sufficient to state that the first is an order of magnitude greater than the second.

Just think, you say, discovered America! And so it is clear: we look at which number is “longer” - then more! In general, yes. Intuitively, this concept was already included in the circle of your concepts, in this article we simply formalized them and gave them b about greater clarity.

A couple more examples:
five billion is three orders of magnitude greater than seven million;
read/write data speed HDD(milliseconds, 10^(-3)) is three orders of magnitude slower than the speed of access to random access memory(microseconds, 10^(-6)).

Here, in the first approximation, and all. Now you can flaunt that term with confidence. Or just use it wisely and appropriately. The latter is perhaps preferable.

Learning the multiplication table is no easy task. It really "grows" into the head only after several years of practical calculations. And at first, children either perform several addition operations in their minds, or utter a mantra like “seven, four ...

Why "in the first approximation"? Hmm... There is a rather well-known joke in the circles of programmers: for a programmer, "an order of magnitude" means "twice". Why at two? We just said that "an order of magnitude" is "ten times"? How can I tell you ... There is one caveat. But that's a topic for another conversation. To find the answer to your question, use the form -

In the order of spheropsidal (pycnidial) fungi, there are currently 750 genera, uniting about 6000 species. Many representatives of this order have pycnidia spores in the form of small, barely visible to the naked eye ... ... Biological Encyclopedia

Desmid algae are characterized by an amazing variety of outlines, beauty of forms. and remarkable cell symmetry. The algae that make up this order have long attracted the attention of not only professional ... ... Biological Encyclopedia

Filamentous, non-branching bright green algae of this order are extremely widespread in fresh water bodies of all continents. Even in the cold streams of Antarctica, they, although not for long (short summer time), delight the eye with their emerald ... ... Biological Encyclopedia

This order includes the vast majority of representatives of the class of protococcal algae. They are commonly referred to as protococcal algae proper. They are characterized by the most fully expressed coccoid body structure, i.e. ... ... Biological Encyclopedia

Order, m. 1. only units. The state of improvement and well-established, systematic, correctness in the arrangement of something, in the course of affairs; opposite mess. "Get your impressions in order." Turgenev. The room is in perfect order. Restore order... Dictionary Ushakov

Husband. a set of objects standing in a row, side by side, in a row, hardly, in a row, not randomly, not randomly, but one after another; row, line, rank, system; each side of the street, a row of houses, forms an order (in St. Petersburg. line). What order to go then? Row … Dahl's Explanatory Dictionary

Ergots form perithecia in well-developed stroma, consisting only of fungal hyphae. Stromas are usually fleshy, light or brightly colored, dark in some representatives of the order. Their form is varied, from prostrate to substratum ... ... Biological Encyclopedia

In Nostocaceae, trichomes are always uniserial, always with heterocysts and often with spores, non-branching or falsely branching. They come with or without vaginas, usually with one trichome in each vagina. The order includes 9 families. ... ... Biological Encyclopedia

A certain connection between the surrounding world and a person, characterized by stability, structural certainty, will follow. the course of development, as well as acquiring meaning for a person and expressed through symbols in the language of culture (in ... ... Encyclopedia of cultural studies

In the broad sense of the word, a harmonious, expected, predictable state or arrangement of something, as well as: the order in physics, the arrangement of atoms, which has some invariance with respect to shift; there is only one order in biology ... ... Wikipedia

This order of marsupial fungi unites several hundred species, most of which develop on plant litter, dried branches and leaves of woody plants, shrubs and shrubs, as well as on herbaceous and higher spore ... ... Biological Encyclopedia

Any real number... Sorry... Perhaps not everyone remembers what it is. And you know, it doesn't matter. As Uncle Murphy said: “If you don’t understand a term in a technical article or documentation, feel free to skip it - the article will completely retain its meaning without this term.”

So let's try again: any number X, other than zero, can be represented as
X \u003d Mantissa * 10 ^ Exponenta, i.e. "mantissa multiplied by ten to the power of the exponent", where
mantissa is a number, modulo (that is, without a sign), not less than one and less than ten, and
exhibitor – any integer (... –3, –2, –1, 0, +1, +2, +3, ...).
Well, it’s just that these numbers are called so: one is the mantissa, the other is the exponent. There is no need to “hang out” much on this, we go further.

For example,
1024 = 1.024 * 10^3
–3.14 = –3.14 * 10^0
1"000"000 = 1 * 10^6

Just think, you say, discovered America! And so it is clear: we look at which number is “longer” - then more! In general, yes. Intuitively, this concept was already included in the circle of your concepts, in this article we simply formalized them and gave them b about greater clarity.

A couple more examples:
five billion is three orders of magnitude greater than seven million;
the speed of reading/writing data to the hard disk (milliseconds, 10^(–3)) is three orders of magnitude slower than the speed of accessing RAM (microseconds, 10^(–6)).

Here, in the first approximation, and all. Now you can flaunt that term with confidence. Or just use it wisely and appropriately. The latter is perhaps preferable.

School of Life

Other related news:

Natural numbers are called numbers that appeared as a result of counting. The numbers one, two, three, four, and so on, are natural numbers. Negative and fractional numbers do not belong to natural numbers. Zero, most often, is not considered a natural number. Natural numbers are numbers

Number transformations are one of the most important operations in mathematics. To solve a particular problem, we may need to represent the number in the desired form. Moreover, the list of tasks is practically unlimited - it can be both a physical task and an arbitrary equation. You will need

The emergence of the concept of a real number is due to practical use mathematics to express using a certain number the value of any magnitude, as well as an internal extension of mathematics. Sponsored by the placement of P&G Articles on the topic "What are real numbers" How to calculate

Natural numbers are numbers that arise when counting, numbering and listing objects. They do not include negative and non-integer numbers, i.e. rational, real and others. P&G Placement Sponsor Related Articles "What is natural number" How to read the periodic table How to translate

The module of the number n is the number of unit segments from the origin to the point n. And it does not matter in which direction this distance will be counted - to the right or left of zero. Sponsored by the placement of P&G Articles on the topic "How to find the modulus of a number" How to extract from the modulus How to calculate the modulus

Raising a number to a power is one of the simplest algebraic operations. In everyday life, erection is rarely used, but in production, when performing calculations, it is almost everywhere, so it is useful to remember how this is done. P&G Placement Sponsor Related Articles "How to Build