Measure Wiktionary. Measuring quantities

This lesson will not be new for beginners. We have all heard from school such things as centimeter, meter, kilometer. And when it came to mass, they usually said gram, kilogram, ton.

Centimeters, meters and kilometers; grams, kilograms and tons have one common name - units of measurement of physical quantities.

In this lesson we will look at the most popular units of measurement, but we will not delve too deeply into this topic, since units of measurement go into the field of physics. Today we are forced to study some physics because we need it for further study of mathematics.

Units of length

The following units of measurement are used to measure length:

• millimeters;
• centimeters;
• decimeters;
• meters;
• kilometers.

millimeter(mm). Millimeters can even be seen with your own eyes if you take the ruler that we used at school every day

Small lines running one after another are millimeters. More precisely, the distance between these lines is one millimeter (1 mm):

centimeter(cm). On the ruler, each centimeter is marked with a number. For example, our ruler, which was in the first picture, had a length of 15 centimeters. The last centimeter on this ruler is marked with the number 15.

There are 10 millimeters in one centimeter. You can put an equal sign between one centimeter and ten millimeters, since they indicate the same length:

1 cm = 10 mm

You can see this for yourself if you count the number of millimeters in the previous figure. You will find that the number of millimeters (distances between lines) is 10.

The next unit of length is decimeter(dm). There are ten centimeters in one decimeter. An equal sign can be placed between one decimeter and ten centimeters, since they indicate the same length:

1 dm = 10 cm

You can verify this if you count the number of centimeters in the following figure:

You will find that the number of centimeters is 10.

The next unit of measurement is meter(m). There are ten decimeters in one meter. One can put an equal sign between one meter and ten decimeters, since they indicate the same length:

1 m = 10 dm

Unfortunately, the meter cannot be illustrated in the figure because it is quite large. If you want to see the meter live, take a tape measure. Everyone has it in their home. On a tape measure, one meter will be designated as 100 cm. This is because there are ten decimeters in one meter, and one hundred centimeters in ten decimeters:

1 m = 10 dm = 100 cm

100 is obtained by converting one meter to centimeters. This is a separate topic that we will look at a little later. For now, let's move on to the next unit of length, which is called the kilometer.

The kilometer is considered the largest unit of length. There are, of course, other higher units, such as megameter, gigameter, terameter, but we will not consider them, since a kilometer is enough for us to further study mathematics.

There are a thousand meters in one kilometer. You can put an equal sign between one kilometer and a thousand meters, since they indicate the same length:

1 km = 1000 m

Distances between cities and countries are measured in kilometers. For example, the distance from Moscow to St. Petersburg is about 714 kilometers.

International System of Units SI

The International System of Units SI is a certain set of generally accepted physical quantities.

The main purpose of the international system of SI units is to achieve agreements between countries.

We know that the languages ​​and traditions of the countries of the world are different. There's nothing to be done about it. But the laws of mathematics and physics work the same everywhere. If in one country “twice two is four,” then in another country “twice two is four.”

The main problem was that for each physical quantity there are several units of measurement. For example, we have now learned that to measure length there are millimeters, centimeters, decimeters, meters and kilometers. If several scientists speaking different languages ​​gather in one place to solve some problem, then such a large variety of units of length measurement can give rise to contradictions between these scientists.

One scientist will state that in their country length is measured in meters. The second may say that in their country the length is measured in kilometers. The third may offer his own unit of measurement.

Therefore, the international system of SI units was created. SI is an abbreviation for the French phrase Le Système International d’Unités, SI (which translated into Russian means the international system of units SI).

The SI lists the most popular physical quantities and each of them has its own generally accepted unit of measurement. For example, in all countries, when solving problems, it was agreed that length would be measured in meters. Therefore, when solving problems, if the length is given in another unit of measurement (for example, in kilometers), then it must be converted into meters. We'll talk about how to convert one unit of measurement to another a little later. For now, let's draw our international system of SI units.

Our drawing will be a table of physical quantities. We will include each physical quantity studied in our table and indicate the unit of measurement that is accepted in all countries. Now we have studied the units of length and learned that the SI system defines meters to measure length. So our table will look like this:

Mass units

Mass is a quantity indicating the amount of matter in a body. People call body weight weight. Usually when something is weighed they say “It weighs so many kilograms” , although we are not talking about weight, but about the mass of this body.

However, mass and weight are different concepts. Weight is the force with which the body acts on a horizontal support. Weight is measured in newtons. And mass is a quantity that shows the amount of matter in this body.

But there is nothing wrong with calling body weight weight. Even in medicine they say "person's weight" , although we are talking about the mass of a person. The main thing is to be aware that these are different concepts.

The following units of measurement are used to measure mass:

• milligrams;
• grams;
• kilograms;
• centners;
• tons.

The smallest unit of measurement is milligram(mg). You will most likely never use a milligram in practice. They are used by chemists and other scientists who work with small substances. It is enough for you to know that such a unit of measurement of mass exists.

The next unit of measurement is gram(G). It is customary to measure the amount of a particular product in grams when preparing a recipe.

There are a thousand milligrams in one gram. You can put an equal sign between one gram and a thousand milligrams, since they mean the same mass:

1 g = 1000 mg

The next unit of measurement is kilogram(kg). The kilogram is a generally accepted unit of measurement. It measures everything. The kilogram is included in the SI system. Let us also include one more physical quantity in our SI table. We will call it “mass”:

There are a thousand grams in one kilogram. You can put an equal sign between one kilogram and a thousand grams, since they denote the same mass:

1 kg = 1000 g

The next unit of measurement is hundredweight(ts). In centners it is convenient to measure the mass of a crop collected from a small area or the mass of some cargo.

There are one hundred kilograms in one centner. One can put an equal sign between one centner and one hundred kilograms, since they denote the same mass:

1 c = 100 kg

The next unit of measurement is ton(T). Large loads and masses of large bodies are usually measured in tons. For example, the mass of a spaceship or car.

There are one thousand kilograms in one ton. One can put an equal sign between one ton and a thousand kilograms, since they denote the same mass:

1 t = 1000 kg

Time units

There is no need to explain what time we think is. Everyone knows what time is and why it is needed. If we open the discussion to what time is and try to define it, we will begin to delve into philosophy, and we do not need this now. Let's start with the units of time.

The following units of measurement are used to measure time:

• seconds;
• minutes;
• watch;
• day.

The smallest unit of measurement is second(With). There are, of course, smaller units such as milliseconds, microseconds, nanoseconds, but we will not consider them, since at the moment this makes no sense.

Various parameters are measured in seconds. For example, how many seconds does it take for an athlete to run 100 meters? The second is included in the SI international system of units for measuring time and is designated as "s". Let us also include one more physical quantity in our SI table. We will call it “time”:

minute(m). There are 60 seconds in one minute. One minute and sixty seconds can be equated because they represent the same time:

1 m = 60 s

The next unit of measurement is hour(h). There are 60 minutes in one hour. An equal sign can be placed between one hour and sixty minutes, since they represent the same time:

1 hour = 60 m

For example, if we studied this lesson for one hour and we are asked how much time we spent studying it, we can answer in two ways: “we studied the lesson for one hour” or so “we studied the lesson for sixty minutes” . In both cases, we will answer correctly.

The next unit of time is day. There are 24 hours in a day. You can put an equal sign between one day and twenty-four hours, since they mean the same time:

1 day = 24 hours

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Russian system of measures- a system of measures traditionally used in Rus' and the Russian Empire. The Russian system was replaced by the metric system of measures, which was approved for use in Russia (optional) according to the law of June 4, 1899. Below are the measures and their meanings according to the “Regulations on Weights and Measures” (1899), unless indicated other. Earlier values ​​of these units may have differed from those given; so, for example, the code of 1649 established a verst of 1 thousand fathoms, while in the 19th century the verst was 500 fathoms; versts of 656 and 875 fathoms were also used.

Sazhen, or sazhen (sazhen, sazhenka, straight sazhen)- old Russian unit of distance measurement. In the 17th century the main measure was the official fathom (approved in 1649 by the “Cathedral Code”), equal to 2.16 m, and containing three arshins (72 cm) of 16 vershok each. Even in the time of Peter I, Russian measures of length were equalized with English ones. One arshin took the value of 28 English inches, and a fathom - 213.36 cm. Later, on October 11, 1835, according to the instructions of Nicholas I “On the system of Russian weights and measures”, the length of a fathom was confirmed: 1 government fathom was equal to the length of 7 English feet , that is, to the same 2.1336 meters.

Machaya fathom- an old Russian unit of measurement equal to the distance in the span of both hands, at the ends of the middle fingers. 1 fly fathom = 2.5 arshins = 10 spans = 1.76 meters.

Span– Old Russian unit of measurement of length. Since 1835 it has been equal to 7 English inches (17.78 cm). Initially, the span (or small span) was equal to the distance between the ends of the outstretched fingers of the hand - the thumb and index. The “big span” is also known - the distance between the tip of the thumb and middle finger. In addition, the so-called “span with a somersault” (“span with a tumble”) was used - a span with the addition of two or three joints of the index finger, i.e. 5–6 vershoks. At the end of the 19th century it was excluded from the official system of measures, but continued to be used as a folk measure.

The height of humans and large animals was indicated in vershok above two arshins, for small animals - in excess of one arshin. For example, the expression “a man is 12 inches tall” meant that his height is 2 arshins 12 inches, that is, approximately 196 cm.

Shtof, half-shtof, shtof– was used, among other things, when measuring the amount of alcoholic beverages in taverns and taverns. In addition, any bottle of damask volume could be called a half-damask. A shkalik was also a vessel of the appropriate volume in which vodka was served in taverns.

Russian measures of length

Russian measures of area

Russian measures of volume

Russian measures of bulk solids “grain measures”

Russian measures of liquid bodies “wine measures”

Russian weight measures

Russian measures of weight (mass) are apothecary and troy.

Pharmacist's weight is a system of mass measures used when weighing medicines until 1927.

1 pound = 12 ounces = 358.323 g.
1 ounce = 8 drachms = 29.860 g.
1 drachm = 1/8 ounce = 3 scruples = 3.732 g.
1 scruple = 1/3 drachm = 20 grains = 1.244 g.
1 grain = 62.209 mg.

In Rus', the Russian system of measures was used to determine area, length, weight, volume, and distance.

About the system

The measuring system was based on man and his physical capabilities. For example, the measure of weight is how much he can lift on his shoulders.

Throughout history, the system has changed several times. New values ​​were added to it or existing ones underwent changes. For example, a verst, which in the 19th century was equated to 500 fathoms, according to the law of 1649 was twice as long - 1000 fathoms.

In 1899, there was a transition to the metric system of measures.

“You need to drink in moderation”

Quite a common expression. But everyone has their own measure. A glass is enough for some, others will drink until they fall. But the word “measure” has several meanings. One of them is the immediate amount that is optimal for some situation or person. But she has a very specific expression. Then 1 measure is how much? It is comparable to eight garnets or one quadrangle. In the modern sense - 26.24 liters.

Upon learning this, many say that their ancestors loved to drink. But the measure was not intended for wine or liquid. It belonged to the category of loose measures, in particular, grain measures.

How the liquid was measured

The barrel was considered the largest wine measure. How much is this? 1 barrel contained 40 buckets. If we convert this volume into liters, we get a value of about 491.96 liters.

The bucket was another common measuring instrument; it was also a container. It could be made of wood, metal or leather. Traditionally it was shaped like a cylinder with a handle so that it could be lifted and carried. It was believed that a woman could carry two buckets at once. Often a rocker arm was used for convenience. One bucket had a volume of 12 liters, in rare cases - 15 liters.

Another unit of volume in this system was the bottle. Despite the fact that there were two of them - for vodka and for wine, only wine was recognized as measuring. It was equal to approximately 0.75-0.77 liters, which, in turn, was equal to a sixteenth of a bucket or three glasses.

Shkalik, shtof and half-shtof served, among other things, to measure alcohol poured in taverns. First of all, this is a vodka measure. How much is this in liters? One scale contained 61.5 ml. The bottle contained 20 scales and was approximately 1.23 liters. The shalik also meant the container into which vodka was poured in drinking establishments.

How was the length measured?

In Rus', the fathom was most widespread. But it can be called, rather, a general category, since there were a large number of fathoms. Each of them had not only its own name, but also its own size. In terms of the most commonly used units, how many are in the metric system? The longest was the “oblique fathom”. It was defined as the distance from the toe of the foot to the fingers of the opposite hand extended upward. In the metric system, it was equal to 2.48 m. “Makhovaya fathom” was equal to the distance from the fingertips of one hand to the other, which were placed apart, or 1.76 m. “Tsarskaya” was equal to 197.4 m, “breech” - 217 .6 m. At the same time, the fathoms did not correlate with each other in any way and did not have any multiplicity.

Based on the fact that each person has individual physical characteristics, special wooden “warehouses” and ropes were used for construction, dividing land plots and measuring the required amount of goods.

How was weight determined?

The need to establish weights arose due to trade relations. There were several most common categories. One of the well-known ones is a pood, which is 16.38 kg. But the hryvnia was very often used. How much was this in kilograms? It was 410 gr. or 96 spools.

The hryvnia was used to denote two quantities - monetary and weight. It was widely used by traders. The hryvnia was used to measure a wide variety of products. And also for silver and gold. The hryvnia was replaced by the pound.

The hryvnia was usually used in retail trade. For large weight categories used in the wholesale segment, Berkovets was used. Most often used for weighing wax or honey. The name of this weight category has an interesting origin. It comes from the name of the island of Bjork. Merchant ships were leaving in his direction, on which barrels of wax were placed. The maximum weight that a person could roll onto a ship alone is how much? The measure was 163.8 kg. Mentions of Berkovets were found in documents of the 12th century.

The measuring system in Rus' was quite complex. There were many gradations in it. In addition, a unit of measurement with one name could have different expressions in different areas.

Magnitude is something that can be measured. Concepts such as length, area, volume, mass, time, speed, etc. are called quantities. The value is measurement result, it is determined by a number expressed in certain units. The units in which a quantity is measured are called units of measurement.

To designate a quantity, write a number, and next to it the name of the unit in which it was measured. For example, 5 cm, 10 kg, 12 km, 5 min. Each quantity has countless values, for example the length can be equal to: 1 cm, 2 cm, 3 cm, etc.

The same quantity can be expressed in different units, for example kilogram, gram and ton are units of weight. The same quantity in different units is expressed by different numbers. For example, 5 cm = 50 mm (length), 1 hour = 60 minutes (time), 2 kg = 2000 g (weight).

To measure a quantity means to find out how many times it contains another quantity of the same kind, taken as a unit of measurement.

For example, we want to find out the exact length of a room. This means we need to measure this length using another length that is well known to us, for example using a meter. To do this, set aside a meter along the length of the room as many times as possible. If it fits exactly 7 times along the length of the room, then its length is 7 meters.

As a result of measuring the quantity, we obtain or named number, for example 12 meters, or several named numbers, for example 5 meters 7 centimeters, the totality of which is called compound named number.

Measures

In each state, the government has established certain units of measurement for various quantities. An accurately calculated unit of measurement, adopted as a standard, is called standard or exemplary unit. Model units of the meter, kilogram, centimeter, etc. were made, according to which units for everyday use were made. Units that have come into use and are approved by the state are called measures.

The measures are called homogeneous, if they serve to measure quantities of the same kind. So, gram and kilogram are homogeneous measures, since they are used to measure weight.

Units of measurement

Below are units of measurement of various quantities that are often found in mathematics problems:

Weight/mass measures

• 1 ton = 10 quintals
• 1 quintal = 100 kilograms
• 1 kilogram = 1000 grams
• 1 gram = 1000 milligrams

• 1 kilometer = 1000 meters
• 1 meter = 10 decimeters
• 1 decimeter = 10 centimeters
• 1 centimeter = 10 millimeters

• 1 sq. kilometer = 100 hectares
• 1 hectare = 10,000 sq. meters
• 1 sq. meter = 10000 sq. centimeters
• 1 sq. centimeter = 100 square meters millimeters

• 1 cu. meter = 1000 cubic meters decimeters
• 1 cu. decimeter = 1000 cubic meters centimeters
• 1 cu. centimeter = 1000 cubic meters millimeters

Let's consider another quantity like liter. A liter is used to measure the capacity of vessels. A liter is a volume that is equal to one cubic decimeter (1 liter = 1 cubic decimeter).

Measures of time

• 1 century (century) = 100 years
• 1 year = 12 months
• 1 month = 30 days
• 1 week = 7 days
• 1 day = 24 hours
• 1 hour = 60 minutes
• 1 minute = 60 seconds
• 1 second = 1000 milliseconds

In addition, time units such as quarter and decade are used.

• quarter - 3 months
• decade - 10 days

A month is taken to be 30 days unless it is necessary to specify the date and name of the month. January, March, May, July, August, October and December - 31 days. February in a simple year is 28 days, February in a leap year is 29 days. April, June, September, November - 30 days.

A year is (approximately) the time it takes for the Earth to complete one revolution around the Sun. It is customary to count every three consecutive years as 365 days, and the fourth year following them as 366 days. A year containing 366 days is called leap year, and years containing 365 days - simple. One extra day is added to the fourth year for the following reason. The Earth's revolution around the Sun does not contain exactly 365 days, but 365 days and 6 hours (approximately). Thus, a simple year is shorter than a true year by 6 hours, and 4 simple years are shorter than 4 true years by 24 hours, i.e., by one day. Therefore, one day is added to every fourth year (February 29).

You will learn about other types of quantities as you further study various sciences.

Abbreviated names of measures

Abbreviated names of measures are usually written without a dot:

Measuring instruments

Special measuring instruments are used to measure various quantities. Some of them are very simple and designed for simple measurements. Such instruments include a measuring ruler, tape measure, measuring cylinder, etc. Other measuring instruments are more complex. Such devices include stopwatches, thermometers, electronic scales, etc.

Measuring instruments usually have a measuring scale (or scale for short). This means that there are line divisions on the device, and next to each line division the corresponding value of the quantity is written. The distance between the two strokes, next to which the value of the value is written, can be additionally divided into several smaller divisions; these divisions are most often not indicated by numbers.

It is not difficult to determine what value each smallest division corresponds to. So, for example, the figure below shows a measuring ruler:

The numbers 1, 2, 3, 4, etc. indicate the distances between the strokes, which are divided into 10 identical divisions. Therefore, each division (the distance between the nearest strokes) corresponds to 1 mm. This quantity is called at the cost of a scale division measuring device.

Before you start measuring a value, you should determine the scale division value of the instrument you are using.

In order to determine the division price, you must:

1. Find the two closest lines on the scale, next to which the values ​​of the quantity are written.
2. Subtract the smaller number from the larger value and divide the resulting number by the number of divisions between them.

As an example, let’s determine the price of the scale division of the thermometer shown in the figure on the left.

Let's take two lines, near which the numerical values ​​of the measured value (temperature) are plotted.

For example, bars indicating 20 °C and 30 °C. The distance between these strokes is divided into 10 divisions. Thus, the price of each division will be equal to:

(30 °C - 20 °C) : 10 = 1 °C

Therefore, the thermometer shows 47 °C.

Each of us constantly has to measure various quantities in everyday life. For example, in order to arrive at school or work on time, you have to measure the time that will be spent on the road. Meteorologists measure temperature, atmospheric pressure, wind speed, etc. to predict the weather.