Introduction to real numbers integers rational:
A real numbers integers rational number is a number that can be represented as a/b, where an a and b are integers and b ≠ 0. From the definition of a rational number, we see that fraction such as -1/2, 3/4 and 2/5 are rational numbers. If we replace b in a/b by an integer 1, we have a/b = a/1 =a. Hence the integers b = b/1, -8 = -8/1, 0 = 0/1 and so on are real number integers rational.
Properties of the Real Numbers Integers Rational:
In real numbers integers rational the division of two integers does not necessarily result in an integer. Hence, there is a need to include fractions to form a bigger set of numbers known as set of rational numbers. The rational numbers and the irrational numbers completely fill the number line and form the set of real numbers. We can summarise the properties of the rational number as given below. Let a, b and c be any rational numbers.
1) Commutative:
a + b = b + a.
a * b = b * a.
2) Associative:
(a + b)+c = a + (b + c)
(a * b)*c = a * (b * c)
3) Identity:
a + 0 = 0 + a = a
a * 1= 1 * a = a
4) Distributive:
Multiplication can be distributed over addition.
a * (b + c) = (a * b) + (a * c)
5) Closure:
The system of rational numbers is closed under addition, subtraction, multiplication and division (except by 0).
a + b, a – b, a * b, a / b, (b ≠ 0) are all rational numbers.
Between, if you have problem on these topics factoring complex polynomials, please browse expert math related websites for more help on math help free online.
Examples for Real Numbers Integers Rational:
1) Find three real numbers integers rational between 1/5 and 1/3.
Solution:
Let q1, q2, q3, be the three required the rational numbers. Then
q1 = 1/2 (1/5 + 1/3) = 1/2 (3 + 5/ 15) = 1/2 * 8/15 = 4/15
q2 = 1/2 (1/2 + 1/3) = 1/2 (4 + 5/15) = 1/2 * 9/15 = 3/10
q3 = 1/2 (3/10 + 1/3) = 1/2 (9 + 10/30) = 1/2 * 19/30 = 19/60
1/5 < 4/15 < 3/10 < 19/60 < 1/3.
Hence, three rational numbers between 1/5 and 1/3 are 4/15, 3/10 and 19/60.
2) Find nine real numbers integers rational numbers between 2/5 and 1/2.
Solution:
2/5 = 0.4, 1/2 = 0.5, clearly, we have
0.4 < 0.41 < 0.42 < 0.43 < 0.44 < 0.45 < 0.46 < 0.47 < 0.48 < 0.49 < 0.5
Therefore, nine rational numbers between 2/5 and 1/2 are
41/100, 42/100, 43/100, 44 /100, 45/100, 46/100, 47/100, 48/100, 49/100.
A real numbers integers rational number is a number that can be represented as a/b, where an a and b are integers and b ≠ 0. From the definition of a rational number, we see that fraction such as -1/2, 3/4 and 2/5 are rational numbers. If we replace b in a/b by an integer 1, we have a/b = a/1 =a. Hence the integers b = b/1, -8 = -8/1, 0 = 0/1 and so on are real number integers rational.
Properties of the Real Numbers Integers Rational:
In real numbers integers rational the division of two integers does not necessarily result in an integer. Hence, there is a need to include fractions to form a bigger set of numbers known as set of rational numbers. The rational numbers and the irrational numbers completely fill the number line and form the set of real numbers. We can summarise the properties of the rational number as given below. Let a, b and c be any rational numbers.
1) Commutative:
a + b = b + a.
a * b = b * a.
2) Associative:
(a + b)+c = a + (b + c)
(a * b)*c = a * (b * c)
3) Identity:
a + 0 = 0 + a = a
a * 1= 1 * a = a
4) Distributive:
Multiplication can be distributed over addition.
a * (b + c) = (a * b) + (a * c)
5) Closure:
The system of rational numbers is closed under addition, subtraction, multiplication and division (except by 0).
a + b, a – b, a * b, a / b, (b ≠ 0) are all rational numbers.
Between, if you have problem on these topics factoring complex polynomials, please browse expert math related websites for more help on math help free online.
Examples for Real Numbers Integers Rational:
1) Find three real numbers integers rational between 1/5 and 1/3.
Solution:
Let q1, q2, q3, be the three required the rational numbers. Then
q1 = 1/2 (1/5 + 1/3) = 1/2 (3 + 5/ 15) = 1/2 * 8/15 = 4/15
q2 = 1/2 (1/2 + 1/3) = 1/2 (4 + 5/15) = 1/2 * 9/15 = 3/10
q3 = 1/2 (3/10 + 1/3) = 1/2 (9 + 10/30) = 1/2 * 19/30 = 19/60
1/5 < 4/15 < 3/10 < 19/60 < 1/3.
Hence, three rational numbers between 1/5 and 1/3 are 4/15, 3/10 and 19/60.
2) Find nine real numbers integers rational numbers between 2/5 and 1/2.
Solution:
2/5 = 0.4, 1/2 = 0.5, clearly, we have
0.4 < 0.41 < 0.42 < 0.43 < 0.44 < 0.45 < 0.46 < 0.47 < 0.48 < 0.49 < 0.5
Therefore, nine rational numbers between 2/5 and 1/2 are
41/100, 42/100, 43/100, 44 /100, 45/100, 46/100, 47/100, 48/100, 49/100.
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