Savatcha
Savatingiz boʻsh

Miqdori:
0

Umumiy:
0

Do‘kon

Mathematics 6.

İçindekiler
Divisibility
How to solve problems?
Rational numbers I.
Axial symmetry
Rational numbers II.
Fractions (revision)
Operations with fractions (revision)
Negative fractions
Multiplying fractions by fractions
The reciprocals of numbers
Division by a fraction
Properties of operations
Rational numbers
Mixed exercises
Proportion
Percentages
Probability, statistics
Supplementary material
Rational numbers
\latex{\frac{3}{4} }
\latex{-\frac{5}{6} }
\latex{\frac{1}{5} }
\latex{\frac{7}{8} }
\latex{-1 }
\latex{-4 }
\latex{5 }
\latex{0 }
\latex{7 }
\latex{-5 }
\latex{-20 }
\latex{-2 }
Natural numbers
Natural numbers are \latex{0; 1; 2; 3; ....}
\latex{0}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{5}
\latex{6}
\latex{7}
The symbol of the set of natural numbers: \latex{\N}
\latex{\N=\left\{0;1;2;3;4;...\right\} }
\latex{\N}
\latex{0}
\latex{7}
\latex{2}
\latex{1}
\latex{5}
Various operations can be performed with natural numbers.
Example 1
Select the operations whose results are natural numbers.
  1. \latex{452+1328+8472}
  1. \latex{3796-2985}
  1. \latex{689\times1072}
  1. \latex{486-697}
  1. \latex{5954\div13}
  1. \latex{417\div9}
Solution
The results of the addition and the multiplication are natural numbers.
  A)    \latex{452+1328+8472=10,252}
C)    \latex{689\times1072=738,608}
The result of a subtraction is a natural number if the minuend is larger than the subtrahend. B)   \latex{3,796-2,985=811}
If the subtrahend is larger than the minuend, the result is not a natural number.
D)   \latex{486-697=-211}
If the dividend is a multiple of the divisor, the result is a natural number. If it is not, then the result is not a natural number.
E)   \latex{5,954\div13=458}: natural number
F)   \latex{417\div9=46.33}: not a natural number.
The sums and the products of natural numbers are also natural numbers. The results of subtractions and divisions are not always natural numbers.
Integers
If the subtrahend is larger than the minuend, the difference of two natural numbers is a negative number.
\latex{-5}
\latex{-4}
\latex{-3}
\latex{-2}
\latex{-1}
\latex{0}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{5}
\latex{0}
negative integers
natural numbers
positive integers
The symbol of the set of integers: \latex{\Z}
\latex{\Z=\left\{...;-2;-1;0;1;2;3;...\right\} }
\latex{\Z}
\latex{\N}
\latex{5}
\latex{7}
\latex{1}
\latex{0}
\latex{-20}
\latex{-1}
\latex{-2}
\latex{-4}
\latex{-3}
\latex{-5}
\latex{2}
Example 2
In the following operations, some numbers have been covered with a card. At least, how many cards must be removed in each case so you can decide whether the result is an integer or not?
  1. \latex{462-\square 28}
  1. \latex{\square 140+(-963)}
  1. \latex{(-7\square 3)\times 7}
  1. \latex{(-\square 6\square )\div 8}
Solution
The result of the subtraction, the addition and the multiplication is undoubtedly an integer; therefore, in cases A), B), and C), you do not have to remove any cards.
In case D), both cards must be removed because
\latex{-768\div 8=-96}        integer,
\latex{-868\div 8=-18.5}      not an integer.
Can you determine the sign of the result of the addition and the subtraction without removing the cards?
The sums, differences, and products of integers are always integers, but their quotient is not always an integer.
Rational numbers
Numbers that can be expressed as the quotient of two integers are called rational numbers.
negative numbers
positive numbers
\latex{0}
\latex{-1}
\latex{-2}
\latex{-3}
\latex{...}
\latex{1}
\latex{2}
\latex{3}
\latex{...}
\latex{0}
\latex{-\frac{4}{3} }
\latex{\frac{1}{2} }
\latex{-\frac{498}{2342} }
\latex{\frac{1000001}{999999} }
The symbol of the set of rational numbers:  \latex{\Q}
\latex{\Q}
\latex{\Z}
\latex{\N}
\latex{0}
\latex{7}
\latex{2}
\latex{1}
\latex{5}
\latex{-1}
\latex{-4}
\latex{-2}
\latex{-20}
\latex{-5}
\latex{-3}
\latex{-\frac{5}{6} }
\latex{\frac{3}{4} }
\latex{\frac{7}{8} }
All integers are rational numbers: \latex{5=\frac{5}{1};0=\frac{0}{1};-3=\frac{-3}{\;\;1};...}
The sums, differences, products, and quotients of rational numbers are also rational numbers (\latex{ 0 } cannot be a divisor).
Example 3
  1. Convert the following fractions to decimals.
\latex{\frac{3}{4};\frac{117}{6};-\frac{5}{3};\frac{38}{14}}
  1. Convert the following decimals to fractions.
\latex{-0.5;12.8;0.\dot{3};-1.125}
Solution
  1. Express the fractions as divisions and perform the operations:
\latex{\frac{3}{4}=3\div 4=0.75}
\latex{\frac{117}{6}=117\div 6=19.5}
The results are terminating decimals.
\latex{117\div 6=19.5}
   \latex{57}
     \latex{30}
        \latex{0}
\latex{-\frac{5}{3}=(-5)\div 3=-1.\dot{6}}
\latex{5\div 3=1.66...}
\latex{20}
  \latex{20}
     \latex{2}
The results are recurring decimals.
\latex{\frac{38}{14}=\frac{19}{7} =19\div 7=2.\dot{7}1428\dot{5}}
\latex{19\div 7=2.7142857...}
   \latex{50}
      \latex{10}
         \latex{30}
            \latex{20}
               \latex{60}
                  \latex{40}
                    \latex{50}
                     \latex{1}
The points above the digits mark the sequence of decimals that are repeated.
Since the remainders are repeated, the digits of the quotient are also repeated. Thus, the result is a recurring decimal.
\latex{0.\dot{3}=0.3333...}
  1. \latex{-0.5=-\frac{5}{10}=-\frac{1}{2}}
\latex{0.\dot{3}=\frac{1}{3}}
\latex{12.8=\frac{128}{10}=\frac{64}{5}}
\latex{-1.125=-\frac{1125}{1000}=-\frac{9}{8}}
\latex{0.\dot{3}=\frac{1}{3}}
\latex{1\div 3=0.333...}
\latex{10}
  \latex{10}
     \latex{10}
        \latex{1}
\latex{1\div 3=\frac{1}{3}}
Rational numbers can be expressed as either recurring or terminating decimals.
Non-terminating and non-recurring decimals are not rational numbers. For example:
\latex{0.101001000100001...}
\latex{2007.200820092010201120122013...}
Exercises
{{exercise_number}}. Convert the fractions to decimals or vice versa to make calculation easier.
  1. \latex{0.8+\frac{7}{2}-\frac{3}{4}+1.2 }
  1. \latex{0.8+\frac{7}{2}-1.125+\frac{3}{4} }
  1. \latex{0.8+\frac{2}{5}-\frac{3}{4}+3.6 }
  1. \latex{0.6\times \frac{2}{3} }
  1. \latex{1.125\div \frac{1}{8} }
  1. \latex{3\frac{2}{3}\times 0.\dot{3}}
{{exercise_number}}. Choose some of the cards  \latex{2}    \latex{3}    \latex{4}    \latex{8}    \latex{-}    \latex{\div}    \latex{\times}    \latex{+}  (you do not have to choose all the operations) and arrange them so that the result of each operation is
  1. a natural number;
  2. an integer, but not a natural number;
  3. a rational number, but not an integer.
Find several solutions.
{{exercise_number}}. Decide whether the following statements are true or false.
  1. Every decimal is a rational number.
  2. There is a number that is neither negative nor positive.
  3. Every natural number is a rational number.
  4. There is an integer that is not a rational number.
  5. If a number is a natural number, it is not negative.
{{exercise_number}}. Arrange the following numbers in ascending order and show them on a number line.
  1. \latex{-2}; \latex{\frac{20}{25}}; \latex{1.21}; \latex{-2.2}; \latex{\frac{6}{5}}; \latex{-\frac{5}{4}}
  1. \latex{\frac{2}{3}}; \latex{-\frac{4}{9}}; \latex{-1}; \latex{-\frac{1}{2}}; \latex{\frac{4}{12}}; \latex{\frac{9}{4}}
{{exercise_number}}. Choose one element from sets A and B so that
  1. their sum;
  1. their difference;
  1. their product;
  1. their quotient is an integer.
\latex{A=\left\{-\frac{1}{3};0;\frac{4}{5};1\frac{3}{5} \right\}}
\latex{B=\left\{-3.8;-\frac{2}{5};\frac{2}{3};1\frac{1}{4} \right\}}
{{exercise_number}}.Write the following numbers into the correct sets.
\latex{-1.5};     \latex{3\frac{1}{7}};    \latex{-7};    \latex{-\frac{10}{5}};   \latex{1.8};    \latex{\frac{9}{21}}

\latex{0.717117111};   \latex{0};   \latex{-1\frac{4}{5}};   \latex{\frac{9}{3}};   \latex{0.8\dot{3}1\dot{2}}
\latex{\Q}
\latex{\Z}
\latex{\N}
Quiz
Calculate the quotient.
\latex{\frac{10-9+8-7+6-5+4-3+2-1}{1-2+3-4+5-6+7-8+9}=}
nfki_banner

Bizning veb-saytimiz sizga eng yaxshi onlayn tajribani taqdim etishini ta’minlash uchun cookie-fayllardan foydalanadi. Batafsil ma’lumot