Rational numbers

The quotient a: b, b ≠ 0 (a, b ∈ Z), is considered a number, which is called a fraction and is denoted by

The set of all fractions is called the set of rational numbers and is denoted by Q


1. Equality of fractions

Fractions are equal if and only if a ∙ d = b ∙ c


2. Expanding and shortening fractions

To expand a fraction a / b means to multiply both the numerator and the denominator by the same number k ≠ 0, ie.

To shorten a fraction means to divide both the numerator and the denominator by one
the same number k ≠ 0, ie.

If the numerator and denominator of a given fraction are mutually prime numbers, then it is called an indefinite fraction.


3. Fraction operations

3.1. Addition and subtraction of fractions

Fractions with equal denominators are added according to the rule:

If the fractions we need to add have different denominators, then they will first expand to have the same denominator, and then add.

For the fraction the opposite fraction is


3.2. Multiplication of fractions

Fractions are multiplied by the rule:


3.3. Fractionation

The fraction b / a is called the reciprocal value of the fraction a / b

(Reciprocal fraction)

Two fractions can be divided according to the following rule:


3.4. Double fractions

The fraction of the shape:

is called a double fraction, and in doing so:


4. Decimal numbers

A fraction with denominator 10, 100,1000, …, (decimal unit) is called a decimal or decimal fraction.

The decimal fraction written without a denominator is called a decimal number,

Example 1.

Decimal numbers in which a group of digits, from somewhere, begins to be repeated, are called periodic decimal numbers. The number that is repeated, written in parentheses, is called a period.
Each periodic decimal number represents a fraction, t, is a rational number.

Example 2. Let us represent the number 1, (25) in a fraction.
1, (25) =?
x = 1, (25) = 1.252525 …
100x = 125, (25) = 125.252525 …,
———————————
100x-x = 124
99x = 124

x = 124/99


4.1 Rounding decimal numbers

Rule:

– if the first digit of the numbers we omit is 0,1,2,3 or 4, then the digit before it does not change
– if the first digit of the numbers we omit is 6,7,8 or 9 then the digit before it increases by 1
– if the first digit of the numbers we omit is 5, then the digit in front of it increases by 1
– if the first digit we omit is 5 and after that there are no other digits then the rule of even digits applies
a) the digit in front of it is even then the pattern does not change: 1.825 ≈ 1.82
b) if the digit in front of it is odd then it increases by 1 and becomes an even example: 6,555 ≈ 6,56

Example:
We will circle the given decimal number 2,34562398. This number has 8 decimal places.

The underlined digits are the digits that we will omit. The number will be rounded up:
Rounded to one decimal place will be:
2.34562398 ≈ 2.3
Rounded to two decimal places will be:
2.34562398 2.35
Rounded to three decimal places will be:
2.34562398 2.346
Rounded to four decimal places will be:
2.34562398 2.3456
Rounded to five decimal places will be:
2.34562398 ≈ 2.34562 etc.

Example
: The given decimal number 2,34562398 will be rounded to the nearest whole number.
To say which integer we will have if we round this number, it means that we have to omit all the digits from the decimal part, so we will have: 2,34562398 2

Example:
Round to the nearest whole number
The 2.4 ≈ 2 digit 4 is omitted, it does not change the digit in front of it
76.2 ≈ 76 digit 2 is omitted and it does not change the digit in front of it
7.8 ≈ 8 we omit the digit 8 and it increases the digit in front of it by one

Exampl
e: 3.42 ≈ 3.4 rounded to one decimal 4
7.3948 ≈ 47.39 rounded to two decimal places 0
.047 ≈ 0.05 rounded to the nearest 0.01
0.07649 ≈ 0.076 rounded to the nearest 0.001

– Rational numbers –



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