Math article for VI grade Algebra (Number system: Rational numbers, Factors/multiples, GCF)
Hey guys! Welcome to another descriptive session of grade VI math. In previous article we have explored some initial topics of Algebra but in this article we are going to explore another unit of grade VI math content and that is Number system. Most of the students get relaxed when this unit is included in their syllabus because of simple arithmetic operations execution is the only clause they need to handle in this unit. But to reach to that level student has to learn various kinds of number presentation on which actual arithmetic operations are implied. Today in this article we will discuss all about rational numbers and its related operations like multiple factors and GCF (greatest common factor).
Let us start with introduction of Rational numbers, all the fraction form presentation of the numbers is known as Rational numbers but with a condition that the number in denominator is not equal to zero. For example “k/l” and “m/n” are rational numbers for all numbers but with condition “l and n is not equal to zero”. Now let’s talk about general arithmetic operations on rational numbers:
1. Addition of rational numbers: One common thing to be remembered while doing addition, multiplication or division with Rational numbers is that these numbers are following Associative law, Distributive law and commutative law. On the basis of above two statements, the addition procedure of rational expressions includes following three assumptions:
1. If two rational numbers are x and y so their simple addition will results (x + y) and it should satisfy associative and commutative law.
2. (x + 0) = x; x is a rational number.
3. x + x* = 0, let us prove this;
Suppose a and b are two fractions then from statement (1)
a + b = (general addition)
Subtract 3rd statement equation from first one
a* + b* = (a + b)*; e.g. 3* + 8* = 11*
a + b = (a− b) if a > b, e.g., 7 + 4' = (7 − 4)' = 3.
a + b' = (b− a)' if a< b, e.g., 2 + 8' = (8 − 2)' = 6'.
2. Multiplying rational numbers: Almost same approach is taken for implying this arithmetic operation on rational numbers, similar kind of assumptions for this are as follows:
1. If two rational numbers are x and y then there is a way to multiply
them that results to another rational number as xy, that, if x and y are infraction form, xy is the usual product of them and also satisfies the associative, commutative and distributive law.
2. 1 · x = x; when x is a rational number.
3. 0 · x = 0
If x and y both are non-zero fractions then the following can be proved:
(−x)y = −(xy)
x(−y) = −(xy)
(−x)(−y) = xy
4. Dividing rational numbers: The process of division of rational numbers is as same as that of dividing fractions. But before exploring it in detail a required theorem is needed to be explained here:
theorem states that: “if x, y are two rational numbers with y != 0. Then there is one and only one rational number’ z’ such that product of z and y equals to x; x = zy”.
So if we write x = zy
Then it can be rewritten as x/y = z
Therefore ‘z’ is called the division of x by y.
Form of x/y is also known as Quotient of x and y, where y is not equal to zero.
5. Comparing rational numbers: this process is executed by the actual relation of numbers in respect of number line, for example if student requires to explain a<b then it will be as that ‘a’ is to the left of ‘b’ on the number line. There are three mutual exclusive conditions that arise when the comparison of rational expression is done, these conditions are: a = b or a <b or a> b.
This phenomenon is also called trichotomy law.
If a, b and c are three rational numbers then their possible inequalities are as follows:
(i) For any a, b, a < b is equivalent to:−a > −b.
(ii) For any a, b, c, a < b is equivalent to: a + c < b + c.
(iii) For any a, b, a < b is equivalent to: b − a > 0.
(iv) For any a, b, c, if c > 0, then a < b is equivalent to:ac < bc.
(v) For any a, b, c, if c < 0, then a < b is equivalent to: ac > bc.
(vi) For any a, a > 0 is equivalent to: 1/a > 0
This is all about Rational numbers and general arithmetic operations implementation on them. Now let us move towards the next topic of Number system and that is Factors. Factors are two numbers that, when they are multiplied together then it results a new number called product. Every integer number except 1 has at least two factors and Composite numbers may have more than two factors as their solution (also read on how to factor polynomials).
One more important term exhibits when factor is explained and that term is multiples. Multiples are the whole integer numbers of any particular value that can be easily divisible with all of them or we can say multiples are the result of two integer numbers from which one is always the same( whose multiples are to be find).
For example:
If we need to find multiples of 3 then;
3 X 1 , 3 X 2 ……. 3 X n results the set of its multiples as
3, 6, 9 , 12…….
Now when any number is decomposed into its multiples then that form of presentation is called its factors. For example 15 can be divided into 3 and 5, similarly 12 can be divided into 6 and 2, and 6 can be further divided in to 2 and 3; therefore the final factors of 12 are 2, 2, and 3.
A number can have different sets of factors, for example 12 can also be factorized as 3 and 4, and then 4 can be further decomposed in 2 and 2. In this case the final factors of 12 are: 3, 2, and 2. While finding factors of any number, student can use the following clues to make the task easier.
· Any even number must include 2 in its factors
· Any integer number ending with numeral 5 has a factor of 5.
· Any number that is more than 0 and ends with numeral 0 will have 2 and 5 in its factors list.
Now let us talk about the last term of today’s session and that is Greatest Common factor, it is pretty understood that two or more numbers may have similar factors those are termed as common factors but the largest factor of them, as far as numeral value concern, is called greatest common factor. In general two methods are preferred to find out the GCF of numbers:
First method: it is known as listing factors and include following steps;
a. First List all the factors of each number.
b. Then identify their common factors.
c. Greatest of the common factors is the GCF of the numbers.
Second method (Use prime factors) : this method based on the calculation of prime factors therefore includes following steps:
a. Fine prime factors of each number
b. Then identify their common prime factors.
c. Then product of these common prime factors is the GCF of the numbers.
Let us take an example to explain it practically:
Q. Determine the GCF of 15 and 18.
Solution: by using Listing Factors:
Factors of 15 are: 1, 3, 5, 15
Factors of 18 are: 1, 2, 3, 6, 9 , 18
Common factor of 15 and 18 are 1 and 3, greater of them is 3 so it is said to be GCF of 15 and 18.
That’s all for today, in this article we have explained the major terms of Number system unit and remaining of them will be included in its successive article. Content of grade VI math is simple but still it has some depth so proper attention to each topic is the key to pass out this class with extreme knowledge of important math topics.
At any stage if students like you feels any sort of difficulty to sort out the math query then they can prefer Online math tutoring websites. This math tutoring provider has all the relevant data of every math topic, which is categorized according to math grades for your ease. Internet is the friendliest environment in present time that’s why math tutoring services has chosen this platform for spreading their service and it is working for them, in past few years this moderate educational scheme has gained much positive feedback from its users because of it is 24 x 7 hours assistance given by expert online math tutors.
In next post we will talk on Fractions in Grade Vi. For more information on ICSE class 8 syllabus, you can visit our website
Hey guys! Welcome to another descriptive session of grade VI math. In previous article we have explored some initial topics of Algebra but in this article we are going to explore another unit of grade VI math content and that is Number system. Most of the students get relaxed when this unit is included in their syllabus because of simple arithmetic operations execution is the only clause they need to handle in this unit. But to reach to that level student has to learn various kinds of number presentation on which actual arithmetic operations are implied. Today in this article we will discuss all about rational numbers and its related operations like multiple factors and GCF (greatest common factor).
Let us start with introduction of Rational numbers, all the fraction form presentation of the numbers is known as Rational numbers but with a condition that the number in denominator is not equal to zero. For example “k/l” and “m/n” are rational numbers for all numbers but with condition “l and n is not equal to zero”. Now let’s talk about general arithmetic operations on rational numbers:
1. Addition of rational numbers: One common thing to be remembered while doing addition, multiplication or division with Rational numbers is that these numbers are following Associative law, Distributive law and commutative law. On the basis of above two statements, the addition procedure of rational expressions includes following three assumptions:
1. If two rational numbers are x and y so their simple addition will results (x + y) and it should satisfy associative and commutative law.
2. (x + 0) = x; x is a rational number.
3. x + x* = 0, let us prove this;
Suppose a and b are two fractions then from statement (1)
a + b = (general addition)
Subtract 3rd statement equation from first one
a* + b* = (a + b)*; e.g. 3* + 8* = 11*
a + b = (a− b) if a > b, e.g., 7 + 4' = (7 − 4)' = 3.
a + b' = (b− a)' if a< b, e.g., 2 + 8' = (8 − 2)' = 6'.
2. Multiplying rational numbers: Almost same approach is taken for implying this arithmetic operation on rational numbers, similar kind of assumptions for this are as follows:
1. If two rational numbers are x and y then there is a way to multiply
them that results to another rational number as xy, that, if x and y are infraction form, xy is the usual product of them and also satisfies the associative, commutative and distributive law.
2. 1 · x = x; when x is a rational number.
3. 0 · x = 0
If x and y both are non-zero fractions then the following can be proved:
(−x)y = −(xy)
x(−y) = −(xy)
(−x)(−y) = xy
4. Dividing rational numbers: The process of division of rational numbers is as same as that of dividing fractions. But before exploring it in detail a required theorem is needed to be explained here:
theorem states that: “if x, y are two rational numbers with y != 0. Then there is one and only one rational number’ z’ such that product of z and y equals to x; x = zy”.
So if we write x = zy
Then it can be rewritten as x/y = z
Therefore ‘z’ is called the division of x by y.
Form of x/y is also known as Quotient of x and y, where y is not equal to zero.
5. Comparing rational numbers: this process is executed by the actual relation of numbers in respect of number line, for example if student requires to explain a<b then it will be as that ‘a’ is to the left of ‘b’ on the number line. There are three mutual exclusive conditions that arise when the comparison of rational expression is done, these conditions are: a = b or a <b or a> b.
This phenomenon is also called trichotomy law.
If a, b and c are three rational numbers then their possible inequalities are as follows:
(i) For any a, b, a < b is equivalent to:−a > −b.
(ii) For any a, b, c, a < b is equivalent to: a + c < b + c.
(iii) For any a, b, a < b is equivalent to: b − a > 0.
(iv) For any a, b, c, if c > 0, then a < b is equivalent to:ac < bc.
(v) For any a, b, c, if c < 0, then a < b is equivalent to: ac > bc.
(vi) For any a, a > 0 is equivalent to: 1/a > 0
This is all about Rational numbers and general arithmetic operations implementation on them. Now let us move towards the next topic of Number system and that is Factors. Factors are two numbers that, when they are multiplied together then it results a new number called product. Every integer number except 1 has at least two factors and Composite numbers may have more than two factors as their solution (also read on how to factor polynomials).
One more important term exhibits when factor is explained and that term is multiples. Multiples are the whole integer numbers of any particular value that can be easily divisible with all of them or we can say multiples are the result of two integer numbers from which one is always the same( whose multiples are to be find).
For example:
If we need to find multiples of 3 then;
3 X 1 , 3 X 2 ……. 3 X n results the set of its multiples as
3, 6, 9 , 12…….
Now when any number is decomposed into its multiples then that form of presentation is called its factors. For example 15 can be divided into 3 and 5, similarly 12 can be divided into 6 and 2, and 6 can be further divided in to 2 and 3; therefore the final factors of 12 are 2, 2, and 3.
A number can have different sets of factors, for example 12 can also be factorized as 3 and 4, and then 4 can be further decomposed in 2 and 2. In this case the final factors of 12 are: 3, 2, and 2. While finding factors of any number, student can use the following clues to make the task easier.
· Any even number must include 2 in its factors
· Any integer number ending with numeral 5 has a factor of 5.
· Any number that is more than 0 and ends with numeral 0 will have 2 and 5 in its factors list.
Now let us talk about the last term of today’s session and that is Greatest Common factor, it is pretty understood that two or more numbers may have similar factors those are termed as common factors but the largest factor of them, as far as numeral value concern, is called greatest common factor. In general two methods are preferred to find out the GCF of numbers:
First method: it is known as listing factors and include following steps;
a. First List all the factors of each number.
b. Then identify their common factors.
c. Greatest of the common factors is the GCF of the numbers.
Second method (Use prime factors) : this method based on the calculation of prime factors therefore includes following steps:
a. Fine prime factors of each number
b. Then identify their common prime factors.
c. Then product of these common prime factors is the GCF of the numbers.
Let us take an example to explain it practically:
Q. Determine the GCF of 15 and 18.
Solution: by using Listing Factors:
Factors of 15 are: 1, 3, 5, 15
Factors of 18 are: 1, 2, 3, 6, 9 , 18
Common factor of 15 and 18 are 1 and 3, greater of them is 3 so it is said to be GCF of 15 and 18.
That’s all for today, in this article we have explained the major terms of Number system unit and remaining of them will be included in its successive article. Content of grade VI math is simple but still it has some depth so proper attention to each topic is the key to pass out this class with extreme knowledge of important math topics.
At any stage if students like you feels any sort of difficulty to sort out the math query then they can prefer Online math tutoring websites. This math tutoring provider has all the relevant data of every math topic, which is categorized according to math grades for your ease. Internet is the friendliest environment in present time that’s why math tutoring services has chosen this platform for spreading their service and it is working for them, in past few years this moderate educational scheme has gained much positive feedback from its users because of it is 24 x 7 hours assistance given by expert online math tutors.
In next post we will talk on Fractions in Grade Vi. For more information on ICSE class 8 syllabus, you can visit our website
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