The numbers
A number is some essence that denotes the number of some objects. Examples: two apples, five spoons, ten books, one hundred rubles, seven tulips. Every day we turn to numbers, sometimes without even noticing it.
The first time, when people were taught to read and write, the number of objects was depicted using sticks:
One subject was portrayed as |
Two items like | |
Three items like | | |
Four subjects like | | | |
Five items like | | | | |
When people became more literate, they realized that a large number of objects could not be depicted with sticks, and replaced these sticks with numbers.
Today in mathematics, numbers are denoted by numbers. These are the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The numbers scare away the majority of schoolchildren and students, as they are associated with mathematics and "terrible formulas." In fact, there is nothing wrong. Numbers are simply a collection of symbols that are meant to represent numbers. Simply put, to indicate the number of items.
That's all. The introductory lesson on numbers is now complete. We'll learn it much better in the future, but for now, what we've covered in this tutorial will suffice.
Basic operations
The basic operations that are used in mathematics are addition, subtraction, multiplication and division. In addition to these operations, there are also relations operations, such as equal (=), greater than (>), less than (<), greater than or equal to (≥), less than or equal to (≤), not equal to (≠).
In general, operations can be divided into two types:
1. action operations;
2. relationship operations.
Action operations are:
• addition (+)
• subtraction (-)
• multiplication (×)
• division (÷).
Relationship operations are:
• equals (=)
• more (>)
• less (<)
• greater than or equal to (≥)
• less than or equal (≤)
• not equal (≠).
Lesson content
• Relationship operations
• Operation of addition
• Subtraction operation
• Operation of multiplication
• Division operation
• Tasks for independent solution
Relationship operations
Let's start with relation operations. The word "attitude" speaks for itself. Examples from life: something is related to something. Dad is related to mom. This relationship is called marriage:
There are many examples of relationships. We can say that our beautiful world, which is developing harmoniously, also consists of relationships.
If five is greater than three, then we say that "five is greater in relation to three" and write it as 5> 3 (read: five is more than three). The acute angle of the ratio sign should be directed towards the lower number. In our example, the number 3 was less than the number 5, so the acute angle of the ratio sign was directed towards the number 3.
Another example. The number 11 is less than the number 15. This phrase can be written like this:
11 <15
In mathematics, relationships can be used to write laws, formulas, equations, and functions. You can write down that one expression is equal to another, or some action is unacceptable in relation to some object, number, law.
For example, the famous phrase "you cannot divide by zero" is written as follows:
Let's not get ahead of events and get ahead of ourselves. Let's just say that in this expression, any numbers can be used instead of a and b. But then it says that b should not be zero.
The equal sign = stands between the quantities and indicates that these quantities are equal to each other.
For example, “five equals five” is written as 5 = 5. It is clear that two fives are equal. In addition to prime numbers, more complex expressions can be connected with an equal sign, for example: 9 + x + y = 4 + 5 + x + y.
Another example: if one large watermelon weighs 20 kg, and two small watermelons weigh 10 kg each, then an equal sign can be put between a 20 kg watermelon and two 10 kg watermelons. This ratio can be read as follows: "one watermelon weighing 20 kilograms is equal to the weight of two watermelons, each of which weighs 10 kilograms." After all, 20 kg = 10 kg + 10 kg.
The sign is not equal ≠ is placed between values ​​when they are not equal to each other.
For example, 5 ≠ 7. It is clear that five is not equal to seven. More examples: an excellent student is not equal to a poor student, a dog is not equal to a cat, a tangerine is not an orange:
excellent student ≠ poor student
dog ≠ cat
tangerine ≠ orange
You can look around you and find many examples of relationships that can be interpreted mathematically.
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Addition operation
The addition operation is indicated by a plus sign (+) and is used when adding numbers.
The numbers that add up are called terms. The number that results from their addition is called the sum.
For example, let's add the numbers 3 and 2.
Rewrite 3 + 2 = 5
In this example, 3 is the term, 2 is the second term, 5 is the sum.
In the future, you will have to add up quite large numbers. But adding up those big numbers will ultimately boil down to adding up the small ones.
Therefore, you need to learn how to add small numbers in the range from 0 to 9. For example:
2 + 2 = 4
3 + 4 = 7
7 + 2 = 9
0 + 7 = 7
You can practice by writing a few simple examples in your notebook. Believe me, there is nothing shameful in this.
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Subtraction operation
The subtraction operation is denoted by a minus sign (-) and is used when another number is subtracted from one number.
The number from which another number is subtracted is called the subtracted number. The number that is subtracted from the number to be reduced is called the subtracted number. The resulting number is called the difference.
For example, subtract the number 2 from the number 10.
10 - 2 = 8
In this example, 10 is the decrement, 2 is the subtracted, and 8 is the difference.
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Multiplication operation
It is denoted by the multiplication sign (×) and is used when one number is multiplied by another. The word multiplication speaks for itself - a certain number increases by a certain number of times, that is, it multiplies.
For example, writing 4 × 3 means that the four will be increased three times during the multiplication operation.
The number that is being increased is called the multiplicable. The number that shows how many times the multiplier needs to be increased is called the multiplier. The resulting number is called the product.
For example, let's multiply the number 4 by 3.
4 × 3 = 12
In this example, 4 is the multiplier, 3 is the multiplier, 12 is the product.
Notation 4 × 3 can be understood as "repeat the number 4 three times." For example, if we have four candies and we repeat them three times, then we get twelve candies:
In other words, multiplying 4 by 3 can be thought of as the sum of three fours. Schematically, it looks like this:
Multiplication can be understood in another way, namely as taking something a certain number of times. Let's say there are candy in a vase. Let's take four candies once:
4 conf. × 1 = 4 conf.
We will have four candies in our hands.
Let's try to take four candies 2 times:
4 conf × 2 = 8 conf.
We will have eight chocolates in our hands.
Let's try to take four candies zero times, that is, never:
4 × 0 = 0
We will not have sweets on our hands, since we have never taken them. Therefore, multiplying any number by zero gives zero in the answer.
In some books, the multiplier and the multiplier are called by one common word - factors. For example, in the notation 4 × 3, the multiplier is 4, and the factor is 3, but these two numbers can also be called factors. It won't be a mistake.
In the future, we will be multiplying quite large numbers. But multiplying large numbers amounts to multiplying small ones. Therefore, you first need to learn how to multiply small numbers. Fortunately, they have already been multiplied and written into a special table called the multiplication table. If you live in Russia or in the countries of the former USSR, then you probably know this table by heart. If you don't know, be sure to learn!
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Division operation
It is denoted by a division sign (÷ or:) and is used when numbers are divided.
The number that is being divided is called the dividend. The number that indicates how many parts the dividend is divided into is called the divisor. The resulting number is called the quotient.
For example, let's divide the number 10 by 2.
10: ¬ 2 = 5