For the Love of Math’s Part 3, continued from: My Curiosity|Mathematics|2|For the Love of Math’s – Logic Searcher
Also, can read: My Curiosity Part 4: Mathematics |1| For the Love of Math’s – Logic Searcher
Class VIIIth NCERT Book:
Maths discussed in this series is Indian Mathematics taken from School books (NCERT): NCERT
Ch1: Rational Numbers
1.1 Introduction
Equation like x + 2 = 13 (eq1) is solved when x = 11, because this value of x satisfies the given equation. The solution is a natural number.
Equation x + 5 = 5(eq2) is solved for x = 0 which is a whole number. If we consider only natural numbers, equation 2 cannot be solved for x. To solve eq like 2, we added the number zero to the collection of natural numbers and obtained the whole numbers. Now consider this equation x + 18 = 5 (eq3), this cannot be solved by even whole numbers. It is solved for value of x = -13 which is an integer (positive and negative). Now we may think when we consider integers, we may find solution to every equation. But you will see that is not true.
2x = 3 (eq4)
5x + 7 = 0 (eq5) For both these eq 4&5 we cannot find a solution from the integers. We need the numbers 3/2 to solve equation 4 and -7/5 to solve eq5. This leads us to the collection of rational numbers.
1.2 Properties of Rational Numbers
1.2.1 Closure (Remember from My Curiosity|Mathematics|2|For the Love of Math’s – Logic Searcher) (The closure property in Mathematics states that when a set of numbers is closed under an arithmetic operation, operating on any two numbers in the set always results in a number belonging to the same group of numbers.)
i) Whole numbers:
Whole numbers are closed under addition and multiplication. Whole numbers are not closed under subtraction and division. I believe you can understand why it is so. You can check the same for Natural numbers.
ii) Integers:
Apart from division, Integers are closed under addition, subtraction, multiplication.
Let me explain here:
Addition: 1 + 4 = 5 (Integers add among themselves to give Integer again). Same is the case for subtraction, multiplication.
Division: 5 / 8 = 5/8 (Integer division give rise to a number which is not integer, but a rational number. Hence integers are not closed under division.
iii) Rational numbers
A number that can be written in the form p/q, where p and q are integers and q /=(is not equal) 0 is called a rational number.
Rational numbers are closed under addition, subtraction, multiplication but not under division.
1.2.2 Commutativity
For whole numbers addition, multiplication is commutative, but subtraction and division are not. Same is true for Integers and Rational numbers.
1.2.3 Associativity
Same as Commutativity, Whole numbers, Integers and Rational numbers addition, multiplication is associative, but subtraction and division are not.
1.2.4 The Role of zero
Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and whole numbers as well.
1.2.5 The role of 1
1 is the multiplicative identity for rational numbers, whole, and integers.
1.2.6 Distributivity of multiplication over addition for rational numbers
For all rational numbers a, b and c: a (b +c) = ab + ac | a(b-c) = ab – ac
Summary
- Rational numbers are closed under the operations of addition, subtraction and multiplication.
- The operations addition and multiplication are
(i) commutative for rational numbers.
(ii) associative for rational numbers. - The rational number 0 is the additive identity for rational numbers.
- The rational number 1 is the multiplicative identity for rational numbers.
- Distributivity of rational numbers: For all rational numbers a, b and c,
a(b + c) = ab + ac and a(b – c) = ab – ac - Between any two given rational numbers there are countless rational numbers. The idea of mean helps us to find rational numbers between two rational numbers.
Ch2: Linear Equations in One Variable
2.1 Introduction
In previous blogs we have discussed about algebraic expressions and equations. Equations use the equality sign and it is missing in expressions.
Linear expressions:
2x, 2x + 1, 3y – 7, 12 – 5z, 5/4(x – 4) + 10
Not linear expressions:
x^2 + 1, y + y^2, 1 + z + z^2 + z^3 (since highest power of variable > 1)
We will deal with equations with linear expressions in one variable only. Such equations are known as linear equations in one variable.
a) An algebraic equation is an equality involving variables. It has an equality sign. The expression on the left of the equality sign is the Left hand Side (LHS). The expression on the right of the equality sign is the Right Hand Side (RHS).
b) In an equation the values of the expressions on the LHS and RHS are equal. This happens to be true only for certain values of the variables. These values are the solutions of the equation.
c) How to find the solution of an equation? We assume that the two sides of the equation are balanced. We perform the same mathematical operations on both sides of the equation, so that the balance is not disturbed. A few such steps give the solution.
2.2 Solving Equations having the Variable on both Sides
2x – 3 = x + 2 => (add 3 to both sides) => 2x – 3 + 3 = x + 2 + 3 => 2x = x + 5 => (subtract x from both sides) => 2x – x = x + 5 – x => x = 5 (Hence 5 is the solution) Note: We did same operation on both sides.
2.3 Reducing Equations to Simpler Form
SUMMARY:
- An algebraic equation is an equality involving variables. It says that the value of the expression on one side of the equality sign is equal to the value of the expression on the other side.
- The equations we study in Classes VI, VII and VIII are linear equations in one variable. In such equations, the expressions which form the equation contain only one variable. Further, the equations are linear, i.e., the highest power of the variable appearing in the equation is 1.
- An equation may have linear expressions on both sides. Equations that we studied in Classes VI and VII had just a number on one side of the equation.
- Just as numbers, variables can, also, be transposed from one side of the equation to the other.
- Occasionally, the expressions forming equations have to be simplified before we can solve them by usual methods. Some equations may not even be linear to begin with, but they can be brought to a linear form by multiplying both sides of the equation by a suitable expression.
- The utility of linear equations is in their diverse applications; different problems on numbers, ages, perimeters, combination of currency notes, and so on can be solved using linear equations.
Ch3: Understanding Quadrilaterals
3.1 Introduction
Paper is a model for a plane surface. When you join a number of points without lifting a pencil from the paper (and without retracing any portion of the drawing other than single points), you got a plane curve.
3.1.1 Convex and concave polygons
A simple closed curve made up of only line segments is called a polygon.
Convex Polygons: They have no portions of their diagonals in their exteriors or any line segment joining any two different points, in the interior of the polygon, lies wholly in the interior of it. Concave polygons are opposite of that.
3.1.2 Regular and irregular polygons
A regular polygon is both ‘equiangular’ and ‘equilateral’. For example a polygon like Square has sides of equal length and angles of equal measure. Hence it is a regular polygon. A rectangle is not a regular polygon because of same reason.
3.2 Sum of the Measure of the exterior angles of a polygon==360 degrees.
On many occasions a knowledge of exterior angles may throw light on the nature of interior angles and sides.
This is very important:
3.3 Kinds of Quadrilaterals
3.3.1 Trapezium
Trapezium is a quadrilateral with a pair of parallel sides.
3.3.2 Kite
Kite is a special type of a quadrilateral. The sides with the same markings in each figure are equal.
i) A kite has 4 sides and hence it a quadrilateral.
ii) There are exactly two distinct consecutive pairs of sides of equal length.
3.3.3 Parallelogram
A parallelogram is a quadrilateral. The opposite sides of a parallelogram are of equal length and they are parallel.
Diagonals of a parallelogram bisect each other (at the point of their intersection, of course!)
3.4 Some Special Parallelograms
3.4.1 Rhombus
Sides of rhombus are all of same length, this is not the case with the kite. A rhombus is a quadrilateral with sides of equal length. Since the opposite sides of a rhombus have the same length, it is also a parallelogram. So, a rhombus has all the properties of a parallelogram. So, a rhombus has all the properties of a parallelogram and also that of a kite. The diagonals of a rhombus are perpendicular bisectors of one another.
3.4.2 A rectangle
A rectangle is a parallelogram with equal angles. The diagonals of a rectangle are of equal length. In a rectangle the diagonals, besides being equal in length bisect each other.
3.4.3 A square
A square is a rectangle with equal sides. In square diagonals are equal in length, bisect each other and perpendicular to each other. The diagonals are perpendicular bisectors of each other.
Ch4: Practical Geometry
Ch5: Data Handling
5.1 Looking for Information
In our day-to-day life we come across information, such as:
a) Runs made by a batsman in the last 10 test matches.
b) Number of wickets taken by a bowler in the last 10 ODIs.
c) Marks stored by the student of your class in Mathematics unit test.
d) Number of story books read by each of your friends etc.
The information collected in all such cases is called data. Data is usually collected in the context of a situation that we want to study. Apart from noting down information, we need to organise data in a systematic manner and then interpret it accordingly.
Sometimes, data is represented graphically to give clear idea of what represents.
- A Pictograph: Pictorial representation of data using symbols.
- A bar graph: A display of information using bars of uniform width, their heights being proportional to the respective values.
- Double Bar Graph: A bar graph showing two sets of data simultaneously. It is useful for the comparison of the data.
5.2 Circle Graph or Pie Chart
5.3 Chance and Probability
Sometimes it so happens that a student prepares 4 chapters out of 5, very well for a test. But a major question is asked from the chapter that she left unprepared. Everyone knows that a particular train runs in time but the day you reach well in time it is late.
The chances of the train being in time or being late are not the same. Here some experiments although been discussed whose results have an equal chance of occurring.
5.3.1 Getting a result
In a cricket match when you toss a coin, you cannot control whether it is going to come head/tail. As both results are likely. Such an experiment is called a random experiment. Head or Tail are the two outcomes of this experiment.
5.3.2 Equally likely outcome
Tossing of coins, and throwing of dice, the outcome is equally likely.
5.3.3 Linking chances to probability
Consider the experiment of tossing a coin once. What are the outcomes? There are only two outcomes- Head or Tail. Both the outcomes are equally likely. Likelihood of getting a head is one out of two outcomes, i.e., 1/2. In other words, we say that the probability of getting a head = 1/2.
5.3.4 Outcomes as events
Each outcome of an experiment or a collection of outcomes make an event. In the experiment of tossing a coin, getting a Head is an event and getting a Tail is also an event.
When you throw a dice, getting an even number is also an event, but here what will be the probability of the event happening? 3/6 => 3 <–Number of outcomes that make the event. 6<- Total number of outcomes of the experiment.
Example: A bag has 4 red balls and 2 yellow balls. A ball is drawn from the bag without looking into the bag. What is probability of getting a red ball? Is it more or less than getting a yellow ball?
Solution: Total outcomes = 6 (6 balls) | Probability of getting red ball is 4/6 = 2/3. | Probability of getting yellow ball = 2/6 = 1/3. | Hence probability of getting a red ball is more than that of getting a yellow ball.
Ch6: Squares and Square Roots
6.1 Introduction
Area of square is equal to side multiply by side. When you see numbers like 4,9,16,25,64, it immediately strike to you that these numbers are product of some number with its own number, (2×2, 3×3, 4×4, 5×5 etc.)
These numbers are known as square numbers.
By definition, if a natural number m can be expressed as a nxn, where n is also a natural number, then m is a square number.
6.2 Properties of Square Numbers
Square numbers end with 0,1,4,5,6,9.
6.3 Some More Interesting Patterns
- Numbers whose dot pattern can be arranged as triangles, is called triangular numbers. If we combine two consecutive triangular numbers, we get a square number, like:
2. There are 2n non perfect square numbers between the squares of the numbers n and (n+1). e.g.: Non perfect square number between 1 and 2 (1, 2, 3, 4) i.e. 2 (2×1) n = 1
Non perfect square numbers between 2,3 (4,5,6,7,8,9) i.e. 4(2×2) n=2
Non perfect square numbers between 3,4 (9,10,11,12,13,14,15,16) i.e. 6(2×3) n = 3
How beautiful these patterns are!!!
3. Adding odd numbers
a) sum of first n odd natural numbers is nxn.
e.g. 1×1 (1) | 2×2 = 4 (sum of first 2 odd numbers i.e. 1+3) | 3×3 = 9 (sum of first 3 odd numbers 1+3+5)
b) If the number is a square number, it has to be the sum of successive odd numbers starting from 1.
c) If a natural number cannot be expressed as a sum of successive odd natural numbers starting with 1, then it is not a perfect square.
4. A sum of consecutive natural numbers.
a) 9(3×3) = 4+5
b) 25(5×5) = 12+13
5. Product of two consecutive even or odd natural numbers.
a) 11×13 = 143 = 12×12 -1×1
b) 12×14 = 168 = 13×13 – 1×1
So, in general we can say that (a+1) x (a-1) = a2 – 1
6.4 Finding the Square of a Number
23 = 20 + 3
23×23 = (20 + 3)2 = 20(20+3) + 3(20+3) = 202 + 20×3 + 3×20 + 32 = 400 + 60+60+9 = 529
6.4.1 Other patterns in Squares
(25)2 = 625 = (2×3) hundreds + 25
(35)2 = 1225 = (3X4) Hundreds + 25
6.4.2 Pythagorean Triplets
This concept is a bit sketchy, hence not discussing it here, as not all triplets can be found via the way suggested through this.
6.5 Square Roots
If there is a square area, square root of that area helps us find the way to calculate the sides. Square root also can help us find the sides of a right-angle triangle via Pythagoras law. Now we will learn what are the various ways to find the square roots of a number.
6.5.1 Finding square roots
Like subtraction is opposite operation of addition, and division is inverse operation of multiplication, square root finding is the inverse operation of squaring.
There is a small discussion mentioned here, -2, 2 both have 4 as their squares, so we can say that square root of 4 are both 2, -2. But to not make discussion too complex for the kids right now, we will consider only positive roots values only.
e.g: √4 = 2 (not -2)
6.5.2 Finding square root through repeated subtraction
Sum of first n odd natural numbers is n2, so you can find the square root by subtracting first n odd natural numbers. But finding the square root via this manner would take very long time in case of large nu mbers.
6.5.3 Finding square root through prime factorization
By finding prime factors, find out the pairs, and then multiplying the unique pairs of prime numbers.
√324 = 2x2x3x3x3x3 = (2×2)x(3×3)x(3×3) = 22 x 32 x 32 = taking square root = 2x3x3 = 18 (Square root of 324)
6.5.4 Finding square root by division method
A perfect square of n-digits, then its square root will have n/2 digits if n is even, or (n+1)/2 if n is odd. A 5 digits perfect square 10000 has 3 digits 100 as its square root.
You can find square root of big numbers by division method, using bar starting from left most digit in the number.
==============================================================================================================================================
Ch7: Cube and Cube Roots
7.1 Introduction
1729 a remarkable number as pointed out by our India’s great mathematical geniuses, S. Ramanujan. 1729 it is the smallest number that can be expressed as a sum of two cubes in two different ways:
1729 = 1728 + 1 = 123 + 13
1729 = 1000 + 729 = 103 + 93
Since then, 1729 is known as Hardy – Ramanujan Number.
So, let’s learn more interesting patters of cubes and cube roots.
7.2 Cubes
The word ‘cube’ is used in geometry. A cube is a solid figure which has all its sides equal. Dice that we use in playing Ludo is an example of cube.
Numbers like 1, 8, 27 are called perfect cubes, as each of these are obtained by multiplying the number to itself 3 times. e.g.: 1x1x1=1 | 2x2x2=8| 3x3x3=27
There are only ten perfect cubes from 1 to 1000. Cubes of even number are even. Cubes of odd number are odd.
7.2.1 Some interesting patterns
- Adding consecutive odd numbers
1 = 1 = 13
3 + 5 = 8 = 23
7 + 9 + 11 = 27 = 33
13 + 15 + 17 + 19 = 64 = 43
21 + 23 + 25 + 27 + 29 = 125 = 53
31 + 33 + 35 + 37 + 39 + 41 = 216 = 63 - Cubes and their prime factors
Each prime factors appears three times in its cubes.
4 = 2×2 43 = 64 = 2 x 2 x 2 x 2 x 2 x 2 = 23 x 23
In prime factorization of any number, if each factor appears three times, then, the number is a perfect cube.
7.2.2 Smallest multiple that is a perfect cube
Raj made a cuboid of platicine. Lenght, breadth and height of the cuboid are 15cm, 30 cm, 15 cm respectively. How many such cuboids are needed to make a perfect square:
step: Calculate and find out factors: 15 X 30 X 15 => 3x5x2x3x5x3x5 => 2x3x3x3x5x5x5 i.e if we can multiply this with 4 (2×2) it will become a perfect cube, hence number of cuboid needed are 4.