Basic theoretical information
Some basic information on basic math operations
Rules for multiplying and dividing negative and positive numbers:
- When multiplying or dividing two positive numbers, the result is a positive number.
- When multiplying or dividing two negative numbers, the result is a positive number.
- When multiplying or dividing one positive and another negative number (in any sequence), the result is a negative number.
The main property of a fraction: the numerator and denominator of the fraction can be multiplied or divided by the same number, unequal to zero, and the fraction value will not change. If we divide the numerator and denominator by a number, then this procedure is called a fraction reduction. Multiplying the numerator and denominator by the same number is usually used to reduce several fractions to the same (common) denominator. Note that in the record of an ordinary fraction (ie, in a fraction with a line): the numerator is at the top, and the denominator is at the bottom.
The smallest (best) common denominator of fractions is the smallest of the numbers, which is divided into all denominators of the original fractions.
When performing addition or subtraction of fractions with the same denominators , it is necessary to perform the addition or subtraction of the numerators of these fractions, and write the result of this operation in the numerator, and the denominator to rewrite the original. If you need to add or subtract fractions with different denominators , you must first bring them to a common denominator, and then add the fractions with the same denominator.
To convert a fraction with an integer part into an improper fraction, you can use the following rule: multiply the integer part by the denominator and add the numerator to this work. The result is recorded in the numerator of the wrong fraction, and the denominator is left unchanged.
For the inverse transformation of an irregular fraction into the correct one with the integer part, do the following: First, divide the numerator by the denominator. If you divide a larger number by a smaller number, you get an integer (integer part) and a remainder. The whole part is recorded before the fraction, the remainder of the division is recorded in the numerator, and the denominator is not changed.
To multiply fractions, the following rule is applied (the product of the numerators is recorded in the numerator, and the denominators – in the denominator), and both fractions must be reduced to the wrong form, i.e. they should not be allocated the whole part.
Division of fractions is performed by replacing division by multiplication. Namely: the fraction to which they divide (the second fraction) is inverted, changing the numerator and denominator in places, and instead of the division sign the multiplication sign is put. Then multiply as usual. Fractions again should be without a whole part. This rule can be written as a formula:
When dividing a fraction by a number , it is necessary to present the number as a fraction with a denominator of 1, and then perform the usual division of fractions using the previous property. This rule can also be represented as a formula:
If the division sign (two points) is replaced by another fraction line, then in order to perform the fraction division by number operation, you just need to perform a reverse replacement, and then proceed as usual:
When dividing a number by a fraction, it is necessary to act in a similar way, i.e. replace the number with a fraction with a unit denominator, and then perform standard actions in the usual way. We write the rules for dividing a number by a fraction in the form of formulas:
In order to multiply a number by an amount in brackets or vice versa, it is necessary to multiply this number by each term in brackets and add the results. This rule is valid for any number of terms in brackets. In the form of formulas, this rule can be written as follows:
In order to multiply a bracket into a bracket , each item from the first bracket should be multiplied by each item from the second bracket and the results should be added. This rule also holds for any number of terms in brackets. We write in the form of a formula an example of such an operation:
If when multiplying a bracket by a number or multiplying a bracket by a bracket in one of the brackets there are drawbacks, then you just need to consider each number together with the sign that stands before it, and carefully perform the multiplication and further summation by all the rules.
Performing normal calculations with lots of actions:
- first perform operations in brackets;
- then count the works and / or divisions;
- then add up or subtract;
- and lastly, if it was a multi-story fraction, divide the already completely simplified numerator into a completely simplified denominator too.
Moreover, performing first operations in brackets also observe the same sequence, first works or divisions inside brackets, then summation or subtraction in brackets, and if there is another bracket inside the bracket, then the actions in it are performed first of all.
A monomial is the product of some negative or positive number by one or several variables in different degrees. A polynomial is the sum (or difference) of monomials.
Such terms in the polynomial are such monomial terms in which the combination of variables and their degrees are completely repeated, and the numbers of these monomials may be different. Thus, such terms of a polynomial are such monomials that can be folded, usually this needs to be done, and the procedure for adding all types of such terms is called adduction of such terms.
The solution of the simplest linear equation is as follows:
Algorithm for solving linear equations:
- Expand all brackets.
- All the terms with a variable are moved to the left of the equal sign, and all the terms without a variable to the right of the equal sign, do not forget to change the characters before the terms during the transfer.
- Bring all such terms to the left and right. We obtain an equation of the form: ax = b .
- Find the answer by division, like: x = b / a .
When solving linear inequalities, there is only one big thing: it is necessary to change the inequality sign when dividing (or multiplying) inequality by a negative number. To change the sign of inequality means to change the sign “less” to sign “more” or vice versa. At the same time, plus and minus signs, bypassing previously studied mathematical rules, should not be changed anywhere. If we divide or multiply the inequality by a positive number, the sign of the inequality does not need to be changed. The rest of the solution of linear inequalities is completely identical to the solution of linear equations.
The main property of proportion:
Abbreviated Multiplication Formulas
When performing various algebraic transformations, it is often convenient to use abbreviated multiplication formulas. Often these formulas are used not so much to shorten the multiplication process, but on the contrary, rather to see from the result that it can be represented as a product of some multipliers. Thus, these formulas need to be able to apply not only from left to right, but also from right to left. We list the basic formulas of abbreviated multiplication. Amount square:
The difference of squares:
Sum of cubes:
The latter two formulas are also often conveniently used in the form:
The quadratic equation and the formula for the expansion of the square trinomial factors
Let the quadratic equation be:
Then the discriminant is found by the formula:
If D > 0, then the quadratic equation has two roots, which are found by the formulas :
If D = 0, then the quadratic equation has one root (its multiplicity: 2), which is sought by the formula :
If D <0, then the quadratic equation has no roots. In the case when a quadratic equation has two roots, the corresponding quadratic term can be factorized by the following formula :
If the quadratic equation has one root, then the decomposition of the corresponding square trinomial into factors is given by the following formula (note that the bracket is squared):
Only if the quadratic equation has two roots (that is, the discriminant is strictly greater than zero) the Viet theorem holds . According to the Vieta Theorem, the sum of the roots of a quadratic equation is:
The product of the roots of a quadratic equation can be calculated by the formula:
The graph of a parabola is given by a quadratic function:
A quadratic function, like any other function, intersects the OX axis at the points that are its roots: ( x 1 ; 0) and ( x 2 ; 0). If there are no roots, then the quadratic function means that the OX axis does not intersect, if the root is one, then the quadratic function at this point ( x 0 ; 0) only touches the OX axis , but does not intersect it. A quadratic function always intersects the OY axis at the point with the coordinates: (0; c ). The graph of a quadratic function (parabola) may look as follows (in the figure, examples that far from exhaust all possible types of parabolas):
- if the coefficient a > 0, in the function y = ax 2 + bx + c , then the branches of the parabola are directed upwards;
- if a <0, then the branches of the parabola are directed downwards.
The coordinates of the vertex of the parabola can be calculated by the following formulas. X vertices of a parabola (or the point at which the square trinomial reaches its highest or lowest value):
Parabola vertex game or maximum, if parabola branches are down ( a <0), or minimum, if parabola branches are up ( a > 0), the value of the square triple:
Basic properties of degrees
The formal definition of a natural degree can be given using the following entry:
Mathematical degrees have several important properties, we list them. When multiplying degrees with the same bases, the exponents are added:
When dividing powers with the same bases, the divisor degree is subtracted from the exponent of the dividend:
When a degree is raised to a degree, the exponents are multiplied together:
If numbers are multiplied with the same degree, but with different bases, then you can multiply the numbers first and then build the product to this power. The reverse procedure is also possible, if there is a product in the degree, then each of the multiplicated ones can be raised to this degree separately and the results multiplied:
Also, if numbers are divided with the same degree, but with different bases, then you can first divide the numbers, and then raise the quotient to this power (the inverse procedure is also possible):
Some simple degrees properties:
- Any number in zero degree gives one.
- Any number in the first degree is equal to itself.
- A unit to any degree is equal to one.
- Zero in any positive ( n > 0) degree is equal to zero. Remember: zero can not be raised to a negative or zero degree.
The main property of the negative degree is written as follows:
Basic properties of mathematical roots
The mathematical root can be represented in the form of an ordinary degree, and then use all the properties of the degrees given above. To represent the mathematical root in the form of a degree, use the following formula:
The square root is the second degree mathematical root:
A square root can only be extracted from a non-negative number. The value of the square root is also always non-negative:
For the square root, there are two important properties that are important not to confuse:
If there are several factors under the root, then the root can be extracted from each of them separately. It is important to understand that each of these factors separately (and not just their product) must be non-negative:
In addition, it should be noted that if you use a record with a mathematical root icon, then the exponent of this root can only be an integer, and this number must be greater than or equal to two: