Each of us from the school bench (or rather from the 1st grade elementary school) should be familiar with such simple mathematical symbols as greater sign And less sign, as well as the equals sign.

However, if it is rather difficult to confuse something with the latter, then about how and in what direction are the signs more and less written (less sign And sign over, as they are sometimes called) many immediately after the same school bench and forget, because. they are rarely used by us in everyday life.

But almost everyone sooner or later still has to face them, and to "remember" in which direction the character they need is written is obtained only by turning to their favorite search engine for help. So why not answer this question in detail, at the same time telling our site visitors how to remember correct writing these signs for the future?

It is about how the greater-than sign and the less-than sign are spelled that we want to remind you in this short note. It will also not be superfluous to say that how to type greater than or equal signs on keyboard And less or equal, because this question also quite often causes difficulties for users who encounter such a task very rarely.

Let's get straight to the point. If you are not very interested in remembering all this for the future and it’s easier next time to “google” again, and now you just need an answer to the question “in which direction to write the sign”, then we have prepared a short answer for you - signs more and less are written like this, as shown in the image below.

And now we will tell a little more about how to understand this and remember it for the future.

In general, the logic of understanding is very simple - which side (larger or smaller) the sign in the direction of writing looks to the left - such is the sign. Accordingly, the sign more to the left looks with a wide side - a larger one.

An example of using the greater than sign:

50>10 - number 50 more number 10;
student attendance in this semester was >90% of classes.

How to write a less than sign, perhaps, is not worth explaining again. It is exactly the same as the greater than sign. If the sign looks to the left with a narrow side - a smaller one, then the sign is smaller in front of you.
An example of using the less than sign:

100<500 - число 100 меньше числа пятьсот;
came to the meeting<50% депутатов.

As you can see, everything is quite logical and simple, so now you should not have questions about which way to write the greater than sign and the less than sign in the future.

Greater than or equal/less than or equal sign

If you have already remembered how the sign you need is written, then it will not be difficult for you to add one dash to it from below, so you will get a sign "less or equal" or sign "more or equal".

However, regarding these signs, some have another question - how to type such an icon on a computer keyboard? As a result, most simply put two signs in a row, for example, "greater than or equal to" denoting as ">=" , which, in principle, is often quite acceptable, but can be made more beautiful and more correct.

In fact, in order to type these characters, there are special characters that can be entered on any keyboard. Agree, the signs "≤" And "≥" look much better.

Greater than or equal sign on keyboard

In order to write "greater than or equal to" on the keyboard with one character, you don't even need to go into the table of special characters - just put a greater than sign while holding down the key "alt". Thus, the keyboard shortcut (entered in the English layout) will be as follows.

Or you can just copy the icon from this article if you need to use it once. Here he is, please.

≥

Less than or equal sign on keyboard

As you probably already guessed, you can write "less than or equal" on the keyboard by analogy with the greater than sign - just put the less than sign while holding down the key "alt". The keyboard shortcut to be entered in the English layout will be as follows.

Or just copy it from this page, if it's easier for you, here it is.

≤

As you can see, the rule for writing greater than and less than signs is quite easy to remember, and in order to type the greater than or equal and less than or equal icons on the keyboard, you just need to press an additional key - everything is simple.

How to determine which fraction is smaller. Fraction Comparison

The physiology of women is individual, therefore, for some, size is still important, for some, the most sensitive area is located deep in the vagina, so that only the owner of a not very small penis can fully please them.

But how can a woman find out how suitable this particular man is for her sexually? Best way this one, of course, to try it “in action”, but women have their own signs that allow outward signs find out the size manhood. Some take it all as a joke, but there are those who take these signs very seriously.

Some facts about the size

In most men, the length of the penis reaches 12 to 18 cm during an erection. In an unexcited, sluggish, state, the size of the male penis also varies, but in size in calm state it is impossible to judge what scale it can reach when excited. If we turn to the results of studies, it turns out that in most cases, the larger the penis in sluggish state, the fewer times it will increase, being excited. From the above, one conclusion can be drawn, not all that glitters is gold.

It is well known that external factors significantly affect the size of male dignity. So, for example, the male member decreases in size and fits closer to the body under the influence of cold air, ice water and at strong excitement, Thus, male body shows defensive reaction, and this applies not only to the penis, but also to the scrotum. But under the influence of favorable external conditions, such as warm water and the general relaxed state of the body, the size of manhood increases.

So how do you know the size of a man's penis based on his external data?

Each man's penis parameters are as individual as his face. What nature has awarded, so they exist. If, nevertheless, for a woman, size is not the last thing, she should pay attention to some details.

For starters, growth. As the well-known saying goes, “went to the root”, or, more simply, it is believed that the lower the height of men, the longer their penis. It turns out that the higher the man, the shorter his tool? Frankly, a controversial statement, but there are those who continue to convince the public that he is right. Based on the same saying, we can conclude that lean men have a thicker and longer penis than large and pumped ones.

Scientists who were able to prove that sexual activity can be determined by measuring the ratio of thigh length to leg length came to the aid of women interested in this issue. In other words, the longer the thigh, the higher sexual activity person.

It is also considered that fuller lips men, the greater his dignity. Among other signs, it is often cited that the thickness and length of the penis can be recognized by the length and width of the male foot, as well as by the shape thumb hands, you can find out the shape of the penis. It is also believed that men with long nose, the length of the penis also does not disappoint.

Not so long ago, Korean scientists managed to find a way to find out the size of manhood. For most people in general and men in particular, ring finger arms are longer than the index, so that's what more difference between the large and index parade ground in a man, the longer his penis. Scientists claim that this way is the most reliable, because it was obtained through research. It was previously proven that the size of a person's fingers are formed at the embryonic stage, the length of the ring finger, as well as the size of the male penis, are formed at the same stage under the influence of hormones, or rather, testosterone. Those. how more quantity of this hormone, the longer the ring finger, and accordingly the penis.

Of two fractions with the same denominator, the one with the larger numerator is the larger, and the one with the smaller numerator is the smaller.. In fact, after all, the denominator shows how many parts one whole value was divided into, and the numerator shows how many such parts were taken.

It turns out that each whole circle was divided by the same number 5 , but they took a different number of parts: they took more - a large fraction and it turned out.

Of two fractions with the same numerator, the one with the smaller denominator is the larger, and the one with the larger denominator is the smaller. Well, in fact, if we divide one circle into 8 parts and the other 5 parts and take one part from each of the circles. Which part will be bigger?

Of course, from a circle divided by 5 parts! Now imagine that they shared not circles, but cakes. Which piece would you prefer, more precisely, which share: the fifth or the eighth?

To compare fractions with different numerators and different denominators, you need to reduce the fractions to the lowest common denominator, and then compare the fractions with the same denominators.

Examples. Compare ordinary fractions:

Let's bring these fractions to the smallest common denominator. NOZ(4 ; 6)=12. We find additional factors for each of the fractions. For the 1st fraction, an additional multiplier 3 (12: 4=3 ). For the 2nd fraction, an additional multiplier 2 (12: 6=2 ). Now we compare the numerators of the two resulting fractions with the same denominators. Since the numerator of the first fraction is less than the numerator of the second fraction ( 9<10) , then the first fraction itself is less than the second fraction.

We continue to study fractions. Today we will talk about their comparison. The topic is interesting and useful. It will allow the beginner to feel like a scientist in a white coat.

The essence of comparing fractions is to find out which of the two fractions is greater or less.

To answer the question which of the two fractions is greater or less, use such as more (>) or less (<).

Mathematicians have already taken care of ready-made rules that allow you to immediately answer the question of which fraction is larger and which is smaller. These rules can be safely applied.

We will look at all these rules and try to figure out why this happens.

Lesson content

Comparing fractions with the same denominators

The fractions to be compared come across different. The most successful case is when fractions have the same denominators, but different numerators. In this case, the following rule applies:

Of two fractions with the same denominator, the larger fraction is the one with the larger numerator. And accordingly, the smaller fraction will be, in which the numerator is smaller.

For example, let's compare fractions and and answer which of these fractions is greater. Here the denominators are the same, but the numerators are different. A fraction has a larger numerator than a fraction. So the fraction is greater than . So we answer. Reply using the more icon (>)

This example can be easily understood if we think about pizzas that are divided into four parts. more pizzas than pizzas:

Everyone will agree that the first pizza is bigger than the second one.

Comparing fractions with the same numerator

The next case we can get into is when the numerators of the fractions are the same, but the denominators are different. For such cases, the following rule is provided:

Of two fractions with the same numerator, the fraction with the smaller denominator is larger. The fraction with the larger denominator is therefore smaller.

For example, let's compare fractions and . These fractions have the same numerator. A fraction has a smaller denominator than a fraction. So the fraction is greater than the fraction. So we answer:

This example can be easily understood if we think about pizzas that are divided into three and four parts. more pizzas than pizzas:

Everyone agrees that the first pizza is bigger than the second.

Comparing fractions with different numerators and different denominators

It often happens that you have to compare fractions with different numerators and different denominators.

For example, compare fractions and . To answer the question which of these fractions is greater or less, you need to bring them to the same (common) denominator. Then it will be easy to determine which fraction is greater or less.

Let's bring the fractions to the same (common) denominator. Find (LCM) the denominators of both fractions. The LCM of the denominators of the fractions and that number is 6.

Now we find additional factors for each fraction. Divide the LCM by the denominator of the first fraction. LCM is the number 6, and the denominator of the first fraction is the number 2. Divide 6 by 2, we get an additional factor of 3. We write it over the first fraction:

Now let's find the second additional factor. Divide the LCM by the denominator of the second fraction. LCM is the number 6, and the denominator of the second fraction is the number 3. Divide 6 by 3, we get an additional factor of 2. We write it over the second fraction:

Multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to compare such fractions. Of two fractions with the same denominators, the larger fraction is the one with the larger numerator:

The rule is the rule, and we will try to figure out why more than . To do this, select the integer part in the fraction. There is no need to select anything in the fraction, since this fraction is already correct.

After selecting the integer part in the fraction, we get the following expression:

Now you can easily understand why more than . Let's draw these fractions in the form of pizzas:

2 whole pizzas and pizzas, more than pizzas.

Subtraction of mixed numbers. Difficult cases.

subtracting mixed numbers Sometimes you may find that things don't go as smoothly as you'd like. It often happens that when solving an example, the answer is not what it should be.

When subtracting numbers, the minuend must be greater than the subtrahend. Only in this case will a normal response be received.

For example, 10−8=2

10 - reduced

8 - subtracted

2 - difference

The minus 10 is greater than the subtracted 8, so we got the normal answer 2.

Now let's see what happens if the minuend is less than the subtrahend. Example 5−7=−2

5 - reduced

7 - subtracted

−2 is the difference

In this case, we go beyond the numbers we are used to and find ourselves in the world of negative numbers, where it is too early for us to walk, and even dangerous. To work with negative numbers, we need an appropriate mathematical background, which we have not yet received.

If, when solving examples for subtraction, you find that the minuend is less than the subtrahend, then you can skip such an example for now. It is permissible to work with negative numbers only after studying them.

The situation is the same with fractions. The minuend must be greater than the subtrahend. Only in this case it will be possible to get a normal answer. And in order to understand whether the reduced fraction is greater than the subtracted one, you need to be able to compare these fractions.

For example, let's solve an example.

This is a subtraction example. To solve it, you need to check whether the reduced fraction is greater than the subtracted one. more than

so we can safely return to the example and solve it:

Now let's solve this example

Check if the reduced fraction is greater than the subtracted one. We find that it is less:

In this case, it is more reasonable to stop and not continue further calculation. We will return to this example when we study negative numbers.

It is also desirable to check mixed numbers before subtracting. For example, let's find the value of the expression .

First, check whether the reduced mixed number is greater than the subtracted one. To do this, we translate mixed numbers into improper fractions:

We got fractions with different numerators and different denominators. To compare such fractions, you need to bring them to the same (common) denominator. We will not describe in detail how to do this. If you're having trouble, be sure to repeat.

After reducing the fractions to the same denominator, we get the following expression:

Now we need to compare fractions and . These are fractions with the same denominators. Of two fractions with the same denominator, the larger fraction is the one with the larger numerator.

A fraction has a larger numerator than a fraction. So the fraction is greater than the fraction.

This means that the minuend is greater than the subtrahend.

So we can go back to our example and boldly solve it:

Example 3 Find the value of an expression

Check if the minuend is greater than the subtrahend.

Convert mixed numbers to improper fractions:

We got fractions with different numerators and different denominators. We bring these fractions to the same (common) denominator.

Comparison of fractions, oh yes, this insidious topic awaits young mathematicians already in the 5th grade and is considered simple ... at first glance. It's easy to compare fractions with the same denominators. For example, what do you think which fraction is larger and which fraction is smaller? Or maybe they are even ... equal?

By skimming through the example, you can probably guess why the right fraction is the largest.
And as you already understood, it was about fractions with the same denominators.
Well, everything is simple here. A person whom fate has not yet brought together with fractions, and he can offhand determine which fraction is smaller and which is larger. And if he answers correctly, the teacher will try to puzzle him with a similar example. Oh come on! It's quite easy! He will exclaim, putting so many feelings and emotions into the very word “easy” that it will immediately reach the teacher - it's time to complicate the task for the impudent one.

As a result, our slightly dumbfounded insolent will feverishly think about which fraction is larger and which is smaller, without understanding the fraction comparison algorithm itself. And if this text is exactly about you, I recommend that you first study the theory and examples and the scheme by which the fraction comparison calculator works, and only after that, take on the calculator itself.

Eh, probably, the first part of my article scared you a little. Relax. In fact, comparing fractions, even with different denominators, is easier than a steamed turnip. The main thing is to take it seriously and competently.
I will hasten to assure you right away that our mathematical shot has nothing to do with gun or drum shot. In our case, common fraction is a rational number that consists of two or three fragmented parts.

Surely there are still quite green beginners who do not know what an ordinary fraction looks like. Don't know what a numerator is? What is a denominator? What is a whole part? And how to compare such fractions, even if they have the same common denominator. To get started, take a look at the image below:

Now then, do you understand what "fragmented" parts I wrote about? The number above the bar is the numerator. The number below the line is the denominator. The number that distinguished big size located on left side, is called the integer part. However, in this article, we will not go in cycles in definitions, but will immediately move on to comparisons. So how do you compare fractions?
To compare two fractions with the same denominators, you need to compare their numerators. In this case, the largest fraction is the one with the largest numerator. But this rule only works when both fractions lie in the positive or negative area. If it turns out that one fraction is positive and the other is negative, forget about the numerators and denominators, a negative fraction is always smaller.