In order to understand how to bring an arbitrary rational number to a standard form, you need to know what the first significant digit of a number is.
The first significant digit of the number call it the first digit on the left that is nonzero.
2 5, 3, 05, 0, 1 43, 0, 00 1 2. The first significant digit is highlighted in red.
In order to bring the number to standard form, you need:
- Move the comma so that it is immediately after the first significant digit.
- Multiply the resulting number by 10 n, where n is the number, which is determined as follows:
- n> 0, if the comma was shifted to the left (multiplying by 10 n, indicates that the comma should actually be to the right),
- n 0, if the comma was shifted to the right (multiplying by 10 n, indicates that the comma should actually be to the left),
- the absolute value of the number n is equal to the number of digits by which the comma has been shifted.
25 = 2 , 5 ← , = 2,5 ⋅ 10 1
The comma moved to the left by 1 digit. Since the comma is shifted to the left, the degree is positive.
Already reduced to a standard form, you do not need to do anything with it. It can be written as 3.05 ⋅ 10 0, but since 10 0 = 1, we leave the number in its original form.
0,143 = 0, 1 → , 43 = 1,43 ⋅ 10 − 1
The comma shifted to the right by 1 digit. Since the comma is shifted to the right, the degree is negative.
− 0,0012 = − 0, 0 → 0 → 1 → , 2 = − 1,2 ⋅ 10 − 3
The comma moved to the right by three digits. Since the comma is shifted to the right, the degree is negative.
An algebraic expression is a notation made up with a meaning in which numbers can be indicated by letters and numbers, and it can also contain signs of arithmetic operations and brackets.
Any letter denoting a number, and any number represented by numbers, is considered to be also an algebraic expression in algebra.
The algebraic expressions that make up the formulas can be applied to solving particular arithmetic problems if they replace letters with given numbers and perform the indicated actions. The number that will turn out if we take instead of letters any numbers and perform the indicated actions on them is called numerical value algebraic expression. From this it is easy to conclude that the same algebraic expression for different values of the letters included in it can have different numerical values, for example, the expression
at a=2, m=5, b=1, n= 4 is calculated: 2 · 5 + 1 · 4 = 14, and when a=3, m=4, b=5, n= 1 is calculated: 3 · 4 + 5 · 1 = 17, etc., the expression
The product of several factors a, b, c, dis written abcd. If, in addition to letter factors, there is a numerical one (anyway, integer or fractional), then it is usually put in front and called coefficient. In this way,
product of quantities m, n, , p write like this:.
The numbers 4 and are coefficients. Obviously 4abcd = abcd + abcd + abcd + abcd and just as well. So, the coefficient shows how many times the whole algebraic expression or its known part is taken by the term.
If there is no coefficient in an algebraic expression, then it is assumed that it is equal to unity, since a = 1 · a, bc = 1 · bc and so on.
Types of Expressions
An algebraic expression that does not include letter divisors is called whole, otherwise fractional or algebraic fraction.
For example, 7a 2 b, - integer expressions,, - fractional expressions.
Rootless expressions are called rational, and containing roots - irrational or radical. For example, all the expressions above, which are integer or fractional, can also be called rational.
, - irrational or radical expressions.
Definition of a concept
What expressions are called algebraic? This is a mathematical notation made up of numbers, letters and signs of arithmetic operations. The presence of letters is the main difference between numerical and algebraic expressions. Examples:
A letter in algebraic expressions denotes a number. Therefore, it is called a variable - in the first example it is the letter a, in the second - b, and in the third - c. The algebraic expression itself is also called variable expression.
The meaning of algebraic expression Is the number obtained as a result of performing all arithmetic operations that are indicated in this expression. But in order to receive it, letters must be replaced by numbers. Therefore, the examples always indicate which number corresponds to the letter. Consider how to find the value of the expression 8a-14 * (5-a), if a = 3.
We substitute the figure 3. Instead of the letter a, we get the following record: 8 * 3-14 * (5-3).
As in numerical expressions, the solution of an algebraic expression is carried out according to the rules for performing arithmetic operations. Let's solve everything in order.
Thus, the value of the expression 8a-14 * (5-a) with a = 3 is -4.
The value of a variable is called valid if the expression makes sense with it, that is, it is possible to find its solution.
An example of a valid variable for the expression 5: 2a is the number 1. Substituting it into the expression, we get 5: 2 * 1 = 2.5.
If two expressions turn out to be equal for any values of the variables included in their composition, they are called identical.
Identity Expression Example:
4 (a + c) and 4a + 4c.
Whatever values the letters a and c take, the expressions will always be equal. Any expression can be replaced by another, identical to it. This process is called the identity transformation.
Identity Transformation Example.
4 * (5a + 14s) - this expression can be replaced by an identical one, applying the mathematical law of multiplication. To multiply a number by the sum of two numbers, you need to multiply this number by each term and add the results.
Thus, the expression 4 * (5a + 14s) is identical to 20a + 64s.
A number in an algebraic expression before a literal variable is called a coefficient. Coefficient and variable are factors.
Algebraic expressions are used to solve problems and equations.
Consider the problem. Petya came up with a number. In order to be guessed by his classmate Sasha, Petya told him: first I added to the number 7, then subtracted from it 5 and multiplied by 2. As a result, I got the number 28. What number did I guess?
To solve the problem, you need to designate the hidden number with the letter a, and then perform all the specified actions with it.
Now we solve the resulting equation.
Petya made up the number 12.
What did we learn?
Algebraic expression - a record composed of letters, numbers and signs of arithmetic operations. Each expression has a value that is found by performing all arithmetic operations in the expression. A letter in an algebraic expression is called a variable, and the number in front of it is called a coefficient. Algebraic expressions are used to solve problems.