MATHS

Text tasks

Basic theoretical information

Types of numbers

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Natural numbers are numbers used to count objects. Natural numbers are positive integers 1, 2, 3, 4, 5 … It is important that the number 0 is not natural. The set of integers is denoted N .

Integers are numbers that can be obtained from the natural by addition, subtraction and multiplication. The set of integers is denoted Z . The set of integers is greater than the set of natural numbers. This means that if a number is natural, then it is guaranteed to be integer, but not vice versa. The integer is not necessarily natural.

Rational numbers are numbers that are represented by an ordinary fraction of the form m / n , where integers are in the numerator and denominator. The set of rational numbers denoted Q . The set of rational numbers is greater than the set of integers. This means that if a number is an integer, then it is guaranteed to be rational, but not vice versa. A rational number is not necessarily an integer.

Irrational numbers are numbers that cannot be represented as an ordinary fraction of the form m / n , where integers are in the numerator and denominator. Irrational numbers are, for example, the roots of their “bad” numbers. The set of irrational numbers is denoted by j .

Real numbers are all rational and irrational numbers. It denotes the set of real numbers R . The set of real numbers is greater than the set of natural, integer, rational and irrational numbers.

Simple numbers are natural numbers, which are divided completely into only 1 and into themselves. The sequence of prime numbers starts like this: 2, 3, 5, 7, 11, 13 … It is important that the number 1 is not simple. Among the primes there is a single even number, 2. All other primes are odd.

Composite numbers are natural numbers that are not prime. It is important that the number 1 is not composite. Any composite number can be decomposed into a product of prime numbers, and the only way. In this expansion, the obtained prime numbers are called prime factors.

Co-prime numbers are integers that have no common integer divisors, except for 1 and –1. For example, the numbers 14 and 25 are mutually simple, and the numbers 15 and 25 are not, since they have a common divisor 5.

Opposite numbers are numbers that give a total of 0. For example, the opposite numbers are 5 and –5.

Inverse numbers are numbers that give in product 1. For example, inverse numbers are 5 and 1/5.

To write down a number in the standard form means to move the comma so that there is only one digit in front of it (not zero) and multiply it by 10 to some extent. The formula of the number, written in the standard form :

The formula of the number written in the standard form

Divisibility of numbers, GCD and NOC

Signs of divisibility of numbers:

  • The number is divisible by 2 if it is even. For example, the number 1884 is divisible by 2, since it is even.
  • A number is divisible by 3 if the sum of its digits is divisible by 3. For example, the number 16482 is divisible by 3, since the sum of its digits 1 + 6 + 4 + 8 + 2 = 21 is divisible by 3.
  • A number is divisible by 4, if a number composed of its last two digits is divisible by 4. For example, the number 42852 is divisible by 4, since the number 52 is divisible by 4.
  • A number is divisible by 5 if it ends in 5 or 0. For example, the number 53165 is divisible by 5, as it ends in 5.
  • A number is divisible by 6 if it is divisible by 2 and 3 at the same time, that is, it is even, and the sum of its digits is divided by 3. For example, the number 27366 is divisible by 6, because it is even, and the sum of its digits is 2 + 7 + 3 + 6 + 6 = 24 divided by 3.
  • The number is divisible by 7, if the triple number of tens combined with the number of units is divided by 7. For example, the number 1001 is divided by 7, because it contains 100 tens, and the number 3×100 + 1 = 301 is divided by 7. Why? Because there are 30 dozens in it, and the number 3×30 + 1 = 91 is divisible by 7. Why? Because there are 9 dozens in it, and the number 3×9 + 1 = 28 is divided by 7.
  • A number is divisible by 8 if a number made up of its last three digits is divisible by 8. For example, the number 851208 is divisible by 8, since the number 208 is divisible by 8.
  • A number is divisible by 9 if the sum of its digits is divisible by 9. For example, the number 19765242 is divisible by 9, since the sum of its digits 1 + 9 + 7 + 6 + 5 + 2 + 4 + 2 = 36 is divisible by 9.
  • A number is divisible by 10 if it ends in 0. For example, the number 13287654810 is divisible by 10.
  • The number is divisible by 11, if the sum of its digits standing in even places is equal to the sum of digits standing in odd places, or these sums differ by 11, 22, 33 … For example, the number 81752 divides into 11, since the sum of its digits standing on odd places, it is equal to 8 + 7 + 2 = 17, and the sum of the figures standing in even places is 1 + 5 = 6. Numbers 17 and 6 differ by 11.

The smallest common multiple (LCM) of two integers is the smallest positive integer, which is divided without a balance into both these numbers. Denoted by LCM ( m ; n ). For example, LCM (16; 20) = 80.

The smallest common multiple (NOC) of several integers is the smallest positive integer that is divided without a remainder by all these numbers.

The greatest common divisor (GCD) of two integers is the largest of their common divisors. It is denoted by GCD ( m ; n ). For example, GCD (16; 20) = 4.

To find the NOC and GCD of two numbers, it is necessary to decompose each of the numbers into prime factors. Next, find duplicate factors. Then the LCM will be equal to the product of all prime factors of each of the numbers taken in the number in which they are found in the expansion of each number the greatest number of times. GCD will be equal to the product of all prime dividers that are common to each of the numbers.

Rounding Rules

The basic rule of rounding.  When rounding, you must write the number as a decimal and round with the necessary accuracy, observing the rule: if the first number to be rounded is equal to or greater than 5, then the number is rounded up. If the first number to be rounded is less than 5, then the number is rounded down.

Additional rounding methods:

  • If it is necessary to round a number with an excess, then it is rounded up, regardless of the basic rule.
  • If you want to round a number with a disadvantage, it is rounded down, regardless of the basic rule.

Textual tasks for movement and work

The most important quantities describing motion are the path L , the velocity v, and the time t . They are related to each other by the formula:

Formula Communication path, time and speed

When using this formula, do not forget to reduce all values ​​to the same type of units. For example, if the speed in a task is measured in km / h, then the path should be measured in kilometers, and time – in hours. When translating you, of course, remember that 1 hour is 60 minutes or 3600 seconds. And in one meter 10 decimeters, 100 centimeters or 1000 millimeters. At one kilometer of 1000 meters.

The following observations should be taken into account in the movement tasks:

  • First, if the bodies meet, then we must guess that at the time of the meeting they are at one point in space.
  • Secondly, if the bodies began to move at the same time, then the times of their movements are the same. And if one of the bodies began to move at, say, 1 hour after the second, then the time of movement of the second body is t , and the time of movement of the first body is ( t –1), since it began to move later, therefore, it moves for a shorter time .

The most important quantities describing the performance are work A , performance P, and time t . They are related to each other by the formula:

Formula Relationship of Work, Time, and Performance

When using this formula, do not forget to reduce all values ​​to the same type of units. For example, if the machine productivity in a task is measured in details per hour, then the work should be measured in details, and time – in hours. If productivity is measured in kilograms per day, then the work must be converted to kilograms, and time – in days.

In joint work tasks, the following points should be considered:

  • First, if workers do the work at the same time, their productivity is summed up, and 
    the work done will add up to the total work.
  • Secondly, if workers began to work simultaneously, then the times of their work are the same. And if one of the workers started working for, say, 1 hour after the second, then the second employee’s work time is t , and the first employee’s work time is ( t –1), since he started working later, therefore, moves for a shorter time .
  • Often, if the amount of work is not explicitly specified, you can take it as a conventional unit.

Interest

Percentage is one hundredth of the number. For example, 1 is 1% of 100. It is often convenient in tasks to express the values ​​given in percent as percentages. So, for example, 20% of a certain value is twenty one-hundredths, or two-tenths, i.e. 0.2 of the same value. Or 73% is 0.73. And so on. It is also important to understand, for example, that if a certain value is reduced by 24%, then this means that 76% or 0.76 of the initial value remains from it. If the value is increased, say by 32%, this means that in the end it was 132% or 1.32 of the initial value.

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