From the very beginning it should be recalled, so as not to get confused: there are numbers - there are 10. From 0 to 9. There are numbers, and they consist of their numbers. The numbers are infinitely many. Exactly more than the stars in the sky.
A mathematical expression is an instruction written down with the help of mathematical symbols, what actions need to be performed with numbers in order to get a result. Do not "go" to the desired result, as in statistics, but to find out how many of them were exactly. But what and when it was - is no longer included in the sphere of interests of arithmetic. It is important not to make a mistake in the sequence of actions, what is addition or multiplication at first? An expression in a school is sometimes called an example.
Addition and Subtraction
What actions can be performed with numbers? There are two basic ones. This is addition and subtraction. All other actions are built on these two.
The simplest human action: take two piles of stones and mix them into one. This is addition. In order to get the result of such an action, you may not even know what addition is. Just take a bunch of stones from Petya and a bunch of stones from Vasya. Put it all together, count it all over again. The new result of successively counting stones from a new pile is the sum.
In the same way, you may not know what subtraction is, just take and divide the pile of stones into two parts or pick up a certain amount of stones from the pile. So what is called the difference will remain in the heap. You can only pick up what is on the heap. Credit and other economic terms are not considered in this article.
In order not to count the stones every time, because it happens that there are a lot of them and they are heavy, they came up with mathematical actions: addition and subtraction. And for these actions they came up with a computing technique.
The sum of any two numbers is stupidly memorized without any technique. 2 plus 5 equals seven. You can count on counting sticks, stones, fish heads - the result is the same. Put 2 sticks first, then 5, and then count everything together. There is no other way.
Those who are smarter, usually cashiers and students, learn more, not only the sum of two digits, but also the sum of the numbers. But most importantly, they can add numbers in the mind using different techniques. This is called an oral counting skill.
To add numbers consisting of tens, hundreds, thousands and even larger digits, use special techniques - addition by a column or a calculator. With a calculator, you can’t be able to add even numbers, and you don’t need to read further.
Column addition is a method that allows you to add large (multi-digit) numbers by learning only the results of adding the numbers. When adding in a column, the corresponding decimal digits of two numbers (that is, in fact two digits) are sequentially added, if the result of adding two digits exceeds 10, then only the last digit of this sum is taken into account - units of a number, and 1 is added to the sum of the following digits.
Multiplication
Mathematicians like to group similar actions to simplify calculations. So the operation of multiplication is a grouping of the same actions - the addition of the same numbers. Any product N x M - is N operations of addition of numbers M. This is just a form of writing addition of identical terms.
To calculate the product, the same method is used - at first the table of multiplying the numbers by each other is stupidly memorized, and then the bitwise multiplication method is used, which is called "in the column".
What first is multiplication or addition?
Any mathematical expression is actually a record of the counter "from the field" about the results of any actions. Let's say tomato harvesting:
- 5 adult workers collected 500 tomatoes each and complied with the norm.
- 2 schoolchildren did not attend math classes and helped adults: they collected 50 tomatoes each, did not comply with the norm, ate 30 tomatoes, bit and spoiled another 60 tomatoes, 70 tomatoes were removed from the pockets of the assistants. Why they took them with them to the field is incomprehensible.
All the tomatoes were handed over to the clerk, he stacked them in heaps.
We write the result of the "harvest" of the crop in the form of an expression:
- 500 + 500 + 500 + 500 + 500 are a handful of adult workers;
- 50 + 50 is a handful of minors;
- 70 - withdrawn from the pockets of schoolchildren (spoiled and bitten into the offset of the result does not go).
We get an example for the school, a record of the results counter:
500 + 500 +500 +500 +500 + 50 +50 + 70 = ?;
Here you can apply the grouping: 5 piles of 500 tomatoes each - this can be written through the multiplication operation: 5 ∙ 500.
Two heaps of 50 - this can also be written through multiplication.
And one bunch of 70 tomatoes.
5 ∙ 500 + 2 ∙ 50 + 1 ∙ 70 =?
And what to do in the example first - multiplication or addition? So, you can add only tomatoes. You can not add 500 tomatoes and 2 heaps. They do not add up. Therefore, first you always need to bring all records to the basic operations of addition, that is, first of all, calculate all operations of grouping-multiplication. In very simple words - multiplication is performed first, and addition then. If you multiply 5 piles of 500 tomatoes each, you get 2500 tomatoes. And then they can already be stacked with tomatoes from other piles.
2500 + 100 + 70 = 2 670
When a child is studying mathematics, it must be conveyed to him that this is a tool used in everyday life. Mathematical expressions are, in fact (in the simplest version of elementary school), warehouse records about the amount of goods, money (very easily perceived by schoolchildren), and other items.
Accordingly, any work is the sum of the contents of a certain number of identical containers, boxes, piles containing the same number of items. And that at first multiplication, and addition then, that is, at first I began to calculate the total number of objects, and then already add them together.
Division
The division operation is not considered separately, it is the opposite of multiplication. You need to distribute something in the boxes, so that in all the boxes there is the same specified number of items. The most direct analogue in life is packaging.
Parentheses
Of great importance in solving the examples are the brackets. Brackets in arithmetic - a mathematical sign used to regulate the sequence of calculations in an expression (example).
Multiplication and division take precedence higher than addition and subtraction. And parentheses take precedence higher than multiplication and division.
Everything that is written in brackets is calculated first. If the brackets are nested, the expression in parentheses is first evaluated. And this is an immutable rule. As soon as the expression in brackets is calculated, the brackets disappear, and a number appears in their place. Options for disclosing parentheses with unknowns are not considered here. This is done until all of them disappear from the expression.
((25-5): 5 + 2): 3 =?
- It's like candy boxes in a big bag. First you need to open all the boxes and pour them into a big bag: (25 - 5) = 20. Five candies from the box immediately sent the excellent pupil Lyuda, who was sick and does not participate in the celebration. The rest of the candy is in the bag!
- Then tie the candies into bunches of 5 pieces: 20: 5 = 4.
- Then add another 2 bundles of sweets to the bag so that you can divide into three children without a fight. Signs of division by 3 are not considered in this article.
(20: 5 + 2): 3 = (4 +2): 3 = 6: 3 = 2
Total: three children, two bunches of candies (one bundle per hand), 5 candies per bundle.
If you calculate the first brackets in the expression and rewrite everything again, the example will become shorter. The method is not fast, with a large consumption of paper, but surprisingly effective. At the same time, training mindfulness when transcribing. An example is brought to a view when only one question remains, first multiplication or addition without parentheses. That is, to this kind when the brackets are no longer there. But the answer to this question already exists, and there is no point in discussing what comes first - multiplication or addition.
"Cherry on the cake"
And finally. The rules of the Russian language are not applicable to the mathematical expression - read and execute from left to right:
5 - 8 + 4 = 1;
This simple example can bring a child to hysteria or spoil the evening for his mother. Because she will have to explain to her second grader that there are negative numbers. Or destroy the authority of Marya Vanovna, who said that: “We need from left to right and in order.”
"Quite a cherry"
An example walks on the Web, causing difficulties for adult uncles and aunts. He is not quite on the subject under discussion, that at first - multiplication or addition. It seems to be about the fact that you first perform the action in parentheses.
From the rearrangement of the terms, the sum does not change, from the rearrangement of factors, too. You just need to write the expression so that it will not be excruciatingly embarrassing.
6: 2 ∙ (1 + 2) = 6 ∙ ½ ∙ (1 + 2) = 6 ∙ ½ ∙ 3 = 3 ∙ 3 = 9
Now for sure!