In the previous lesson, we learned about pattern of numbers. In this lesson we discuss about Sequences.
A sequence is an ordered list of numbers. The sum of the terms of a sequence is called a series.
Each number of a sequence is called a term (or element) of the sequence.
A finite sequence contains a finite number of terms (you can count them). 1, 4, 7, 10, 13
An infinite sequence contains an infinite number of terms (you cannot count them). 1, 4, 7, 10, 13, . . .
The terms of a sequence are referred to in the subscripted form shown below, where the natural number subscript refers to the location (position) of the term in the sequence.
(If you study computer programming languages such as C, C++, and Java, you will find that the first position in their arrays (sequences) start with a subscript of zero.)
The general form of a sequence is represented:
The domain of a sequence consists of the counting numbers 1, 2, 3, 4, … and the range consists of the terms of the sequence.
The terms in a sequence may, or may not, have a pattern, or a related formula.
For some sequences, the terms are simply random.
Let’s examine some sequences that have patterns:
Sequences often possess a definite pattern that is used to arrive at the sequence’s terms.
It is often possible to express such patterns as a formula. In the sequence shown at the left, an explicit formula may be:
When two integers are multiplied together, the answer is called a product. The integers that were multiplied together are called the factors of the product. 3 • 6 = 18 (3 and 6 are factors of 18)
The greatest common factor of two (or more) integers is the largest integer that is a factor of both (or all) numbers.
Consider the numbers 18, 24, and 36. The greatest common factor is 6. (6 is the largest integer that will divide evenly into all three numbers)
The greatest common factor, (GCF), of two (or more) monomials is the product of the greatest common factor of the numerical coefficients (the numbers out in front) and the highest power of every variable that is a factor of each monomial.
Example: Consider 10x²y3 and 15xy² The greatest common factor is 5xy² . The largest factor of 10 and 15 is 5. The highest power of x that is contained in both terms is x. The highest power of y that is contained in both terms is y² .
When factoring polynomials, first look for the largest monomial which is a factor of each term of the polynomial. Factor out (divide each term by) this largest monomial.
Example 1: Factor: 4x + 8y The largest integer that will divide evenly into 4 and 8 is 4. Since the terms do not contain a variable (x or y) in common, we cannot factor any variables. The greatest common factor is 4. Divide each term by 4. Answer: 4(x + 2y)
Example 2: Factor: 15x2y3 + 10xy² The largest integer that will divide evenly into 15 and 10 is 5. The largest power of x present in both terms is x. The largest power of y present in both term is y². The GCF is 5xy². Divide each term by the GCF. Answer: 5xy²(3xy + 2)
The absolute value of a number can be considered as the distance between 0 and that number on the real number line.
Remember that distance is always a positive quantity (or zero). The distance in the diagram below from -3 to 0 is 3 units. These units are never negative values.
The absolute value of an integer is the numerical value (magnitude) of an integer regardless of its sign (direction). It is denoted by the symbol | |. The absolute value of an integer is either zero or positive. Also, the corresponding positive and negative integers have the same absolute value. Examples: The absolute value of -2 is | -2 | = 2. The absolute value of 5 is | 5 | = 5. The absolute value of 0 is | 0 | = 0
A rational number is a number which can be put in the form \(\frac { p }{ q }\), where p and q are both integers and q ≠ 0. p is called numerator (Nr) and q is called denominator (Dr).
A rational number is either a terminating or non-terminating but recurring (repeating) decimal.
A rational number may be positive, negative or zero. Examples:
The sum, difference and the product of two rational numbers is always a rational number.
The quotient of a division of one rational number by a non-zero rational number is a rational number. Rational numbers satisfy the closure property under addition, subtraction, multiplication and division.
Results Since every number is divisible by 1, we can say that :
Every natural number is a rational number, but every rational number need not be a natural number. For example, 3 = \(\frac { 3 }{ 1 }\) , 5 = \(\frac { 5 }{ 1 }\), 9 = \(\frac { 9 }{ 1 }\) and so on. but, \(\frac { 7 }{ 9 } ,\frac { 11 }{ 13 } ,\frac { 5 }{ 7 }\) are rational numbers but not natural numbers.
Zero is a rational number because \(\left( 0=\frac { 0 }{ 1 } =\frac { 0 }{ 2 } =…. \right)\).
Every integer is a rational number, but every rational number may not be an integer. For example \(\frac { -2 }{ 1 } ,\frac { -5 }{ 1 } ,\frac { 0 }{ 1 } ,\frac { 3 }{ 1 } ,\frac { 5 }{ 1 }\), etc. are all rationals, but rationals like \(\frac { 3 }{ 2 } ,\frac { -5 }{ 2 }\) etc. are not integers.
Rational numbers can be positive and negative.
Every positive rational number is greater than zero.
Every negative rational number is less than zero.
Every positive rational number is greater than every negative rational number.
Every negative rational number is smaller than every positive rational number.
Equivalent Rational Numbers
∵ Rational no. can be written with different Nr and Dr. Such rational number that are equal to each other are said to be equivalent to each other. Example: Write \(\frac { 2 }{ 5 }\) in an equivalent form so that the numerator is equal to –56. Solution: Multiplying both the numerator and denominator of \(\frac { 2 }{ 5 }\) by –28, we have
Lowest Form of a Rational Number
A rational number is said to be in lowest form if the numerator and the denominator have no common factor other than 1.
Example: Write the following rational numbers in the lowest form : Solution:
Standard Form of a Rational Number
A rational number \(\frac { p }{ q }\) is said to be in its standard form if (i) its denominator ‘q’ is positive (ii) the numerator and denominator have no common factor other than 1. For example : \(\frac { 3 }{ 2 } ,\frac { -5 }{ 2 } ,\frac { 1 }{ 7 }\), etc.
Example: Express the rational number \(\frac { 14 }{ -21 }\) in standard form. Solution: The given rational number is \(\frac { 14 }{ -21 }\). 1. Its denominator is negative. Multiply both the numerator and denominator by –1 to change it to positive, i.e., 2. The greatest common divisor of 14 and 21 is 7. Dividing both numerator and denominator by 7, we have which is the required answer.
Equality of Rational Numbers
Method-1: If two or more rational numbers have the same standard form, we say that the given rational numbers are equal. Example: Are the rational numbers \(\frac { 8 }{ -12 }\) and \(\frac { -50 }{ 75 }\) equal? Solution: We first express these given rational numbers in the standard form. The first rational number is \(\frac { 8 }{ -12 }\). (i) Multiplying both the numerator and denominator by –1. (ii) Dividing both the numerator and denominator by the greatest common divisor of 8 and 12, which is 4. Again, the second rational number is \(\frac { -50 }{ 75 }\). (i) The denominator is positive. (ii) Dividing both numerator and denominator by the greatest common divisor of 50 and 75, which is 25. Clearly, both the rational numbers have the same standard form. Therefore, \(\frac { 8 }{ -12 }\) = \(\frac { -50 }{ 75 }\)
Method-2: In this method, to test the equality of two rational numbers, say \(\frac { a }{ b }\) and \(\frac { c }{ d }\), we use cross multiplication in the following way : \(\frac { a }{ b } =\frac { c }{ d }\) Then a × d = b × c If a × d = b × c, we say that the two rational numbers \(\frac { a }{ b }\) and \(\frac { c }{ d }\) are equal. Example: Check the equality of the rational numbers \(\frac { -7 }{ 21 }\) and \(\frac { 3 }{ -9 }\). Solution:
Comparison of Rational Numbers
Comparing fraction. We compare two unequal fractions, each is written as another equal fraction so that both have the same denominators. Then the fraction with greater numerator is greater. Example : To compare \(\frac { 7 }{ 6 }\) and \(\frac { 5 }{ 8 }\), find the L.C.M. of 6 and 8 (it is 24) and To compare two negative rational numbers, we compare them ignoring their negative signs and then reverse the order. Note : Every positive rational number is greater than negative rational number.
Representation of Rational Numbers on Number Line
We know that the natural numbers, whole numbers and integers can be represented on a number line. For representing an integer on a number line, we draw a line and choose a point O on it to represent ‘0’. We can represent this point ‘O’ by any other alphabet also. Then we mark points on the number line at equal distances on both sides of O. Let A, B, C, D be the points on the right hand side and A’, B’, C’, D’ be the points on the left of O as shown in the figure. The points on the left side of O, i.e., A’, B’, C’, D’, etc. represent negative integers –1, –2, –3, –4 whereas, points on the right side of O, i.e., A, B, C, D represent positive integers 1, 2, 3, 4 etc. Clearly, the points A and A’ representing the integers 1 and –1 respectively are on opposite sides of O, but at equal distance from O. Same is true for B and B’ ; C and C’ and other points on the number line. (1) Natural Numbers (2) Whole Numbers (3) Integers Negative numbers are in left side of zero (0) & positive numbers are in right side. ∵ negative numbers are less than positive numbers ∴ If we move on number line from right to left we are getting smaller numbers. Also OA = distance of 1 from 0 OD’ = distance of –4 from 0 D’A = distance between –4 and 1. etc. (4) Rational Numbers (a) If Nr < Dr: We divide line segment OA (i.e. distance between 0 & 1) in equal parts as denominator (Dr). (b) If Nr > Dr: Example: Represent \(\frac { 13 }{ 3 }\) and \(-\frac { 13 }{ 3 }\) on number line. Solution: Therefore, from O mark OA, AB, BC, CD and DE to the right of O such that OA = AB = BC = CD = DE = 1 unit. Clearly, Point A,B,C,D,E represents the Rational numbers 1, 2, 3, 4, 5 respectively. Since we have to consider 4 complete units and a part of the fifth unit, therefore divide the fifth unit DE into 3 equal parts. Take 1 part out of these 3 parts. Then point P is the representation of number \(\frac { 13 }{ 3 }\) on the number line. Similarly, take 4 full unit lengths to the left of 0 and divide the fifth unit D’E’ into 3 equal parts. Take 1 part out of these three equal parts. Thus, P’ represents the rational number \(-\frac { 13 }{ 3 }\).
Rational Number Example Problems With Solutions
Example 1: Is zero a rational number? can you write it in the form , where p and q are integers and q ≠ 0? Solution:
Example 2: Find five rational numbers between 3/5 and 4/5. Solution:
Example 3: Find six rational numbers between 3 and 4. Solution:
Example 4: Find two rational & two irrational numbers between 4 and 5. Solution:
Therefore the 3 rational numbers between 4 & 5 are \(\frac { 9 }{ 2 } \), \(\frac { 19 }{ 2 } \)
The 2 irrational numbers between 4 and 5 are as follows: a = 4.101001000…. b = 4.20002000….
