Category: Maths Notes PPT

Decimal Addition and Subtraction

Decimals in Mathematics are defined as the numbers whose whole part and the fractional part. These two parts are separated by a decimal point. The values on the left of the decimal point represent whole numbers which can be greater than equal to zero. The values on the right of the decimal point are always smaller than one. Decimals are used every day in our daily lives. Weights, money, lengths etc. can be expressed in decimals. Decimals are used in situations which require a higher level of precision level in terms of measurement metrics. Just like whole numbers, we can also carry out simple mathematical operations of addition, subtraction, multiplication, and division with decimals. Further ahead in this article, we will look at the addition and subtraction of decimals in greater detail. 

Examples of Decimals: 

An example of the decimal is 17.48, where, 17 is the whole number part, and 0.48 is the fractional part.

Place Value in Decimals 

As we move from left to right in decimals, the place value of the digits is divided by 10 subsequently.so, tenth in decimal place value on the right side of the decimal point is 1/10 which is 0.1. Similarly, hundredth is 1/100 or 0.001.

Now, we will look at the rules and solved examples for decimal addition and subtraction one after the other.

Addition of Decimal Numbers

Let us look at how to perform the addition of decimal numbers. There are certain rules to be followed while adding decimals. The rules for adding decimals are as follows: 

• Firstly, the numbers are padded with zeros. This depends on the maximum number of digits past the decimal point.

• So, if there are two decimal numbers to be added and one number has three digits after the decimal point, and the other number has one digit after the decimal point, we add two zeros to the latter number to make the number of digits after the decimal place equal for both numbers. This is known as padding with zeros. 

• This process is known as the conversion of decimals into like decimals. 

• After this process, the numbers have to be lined up in a vertical manner so that the place values of both numbers are in sync. 

• This means the tenth place digit of one number should be just above or below the tenth-place digit of another number. 

• Then, for decimal number addition, it should be done digit wise starting from the rightmost digit and moving towards the left-most digit. 

Steps to Add Decimals

Step 1: Convert decimals into like decimals 

Step 2: Line up the numbers as per their place values 

Step 3: Perform addition of digits from rightmost digit to left-most digit

Addition of Decimals Examples 

Example 1) Add 45.01 and 34.986

Solution) first step is to convert the decimals to like decimals. 

We add one zero to 45.01 at the end to make the decimals like. 

Next, we will line up the numbers according to the place values and then perform addition.

45.010

+ 34.986

   79.996

Example 2) Add 39.999 and 2.9

Solution) first step is to convert the decimals to like decimals. 

We add two zeros to 2.9 at the end to make the decimals like. 

Next, we will line up the numbers according to the place values and then perform addition.

39.999

+ 02.900

42.899

Subtraction of decimals  

Now, we will learn how to subtract decimals. There are certain rules to be followed while subtracting decimals. The rules for the subtraction of decimals are as follows: 

• Firstly, the numbers are padded with zeros. This depends on the maximum number of digits past the decimal point.

• So, if there are two decimal numbers to be added and one number has three digits after the decimal point, and the other number has one digit after the decimal point, we add two zeros to the latter number to make the number of digits after the decimal place equal for both numbers. This is known as padding with zeros. 

• This process is known as the conversion of decimals into like decimals. 

• After this process, the numbers have to be lined up in a vertical manner so that the place values of both numbers are in sync. The numbers have to line up such that the larger number is on the top of the smaller number.

Steps to Subtract Decimals

Step 1: Convert decimals into like decimals 

Step 2: Line up the numbers as per their place values with a larger number above the smaller number

Step 3: Perform subtraction of digits from rightmost digit to the left-most digit

Subtraction of Decimals Examples

1. Subtract 3.78 from 10.5 

Solution) First step is to make the decimals like, and in the second step then line them up and lastly perform the subtraction. 

10.50

– 03.78

6.72 

Algebraic Equations Examples

An equation says that two things are equal. This means that the equation will always have an equal sign ‘=’ like this:

x+5=10

The above equation says that both sides are balanced: what is on the left (x+5) is equal to what is on the right (10). The equation is balanced as both sides have the same value. If you solve the equation, the value of x will be 5, the equation will be mathematically balanced. To avoid committing an error that tips the equation out of balance, ensure that any change on either side of the equation is reciprocated on the opposite side. For example, if you want to add a number 2 to the left side of the equation you will have to add the same 2 to the other side of the equation i.e.

x+5=10

x+5+2=10+2

The process goes the same for subtraction, multiplication, and division. The equation will remain balanced as long as you do the same thing to both sides.

Types of Algebraic Equations

Algebraic equations questions are solved based on their position of variables, the functions and the types of operators used. A few of the equations in algebra are discussed below.

Polynomial Equations: All polynomial equations are different algebraic equations, but all algebraic equations are not polynomial equations. To recall the basics, a polynomial equation is an equation that consists of variables, exponents and coefficients.

Linear equations: ax+b=c (a not equal to 0)

Quadratic Equations: A quadratic equation is a polynomial equation of degree 2 in one variable of type.

The quadratic equations are polynomials with degree 2 in one variable of type  f(x) = ax² + bx + c.

Quadratic Equations: ax²+bx+c=0 (a not equal to 0)

Cubic Equations: A cubic equation is a polynomial equation of degree 2. All the cubic polynomials are also algebraic equations.

Cubic Polynomials: ax³+bx²+cx+d=0

Rational Polynomial Equations

P(x)/Q(x)=0

Trigonometric Equations: All the trigonometric equations are considered as algebraic functions. For a trigonometry equation, the expression includes the trigonometric functions of a variable.

Trigonometric Equations: cos2x = 1+4sinx

How to Solve Algebraic Equations?

Consider the following situation. I am going to the market. In one carry bag, I carry some potatoes, onions and papayas. The bag can carry a total of 8 items in it. So I purchased 4 potatoes and 2 onions. How many papayas can I now carry?

Consider the number of towels to be ‘x’. Let’s form the equation now.

4 potatoes + 2 onions + ‘x’ papaya = 8 items

The left-hand side (LHS) of our equation is being compared to the right-hand side (RHS) of the equation.

Now, let’s solve this equation:

4+2+x=8

6+x=8

6+x-6=8−6

x=2

I can carry 2 papayas for my trip.

In the same way, what would certainly represent an inequality? Definitely, when the left-hand side is not equal to the right-hand side. How would this happen?

Taking the same example, 6 + x = 8, and change that equal to a greater than or a lesser than sign. They aren’t equations anymore. Look for some more examples to clarify this concept.

x + 5 = 10 is an equation

xy + 3 = z is an equation

But, 6p < 77 is not.

Let’s see some of the algebraic equations examples with answers

Question: Simplify the given equation : 2(x+4)+3(x–5)–2y=0

Solution

Given equation: 2(p+4)+3(p−5)–2q=0

2p+2×4+3p–3×5–2q=0 (Using Distributive property to get rid of parenthesis)

2p+8+3p–15–2q=0 (Simplifying).

5p–2q–7=0 (on further simplifying terms).

Geometry is one of the topics that many students love, no matter if they like calculations. Making angles using scale and compass gives us a different kind of joy and relaxation in the world of numbers and multiplication tables. But then comes trigonometry to add a bit of complexity and along with trigonometry, you get your first taste of vectors. 

Yes, vectors are also a part of Mathematics and geometry. An angle between two vectors is the smallest angle that can be used for one vector to rotate on its axis so that it aligns with the other vector. Two vectors are needed to produce a scalar quantity, which is said to be a real number. 

Today, we will be trying to find the angle between the two vectors using trigonometric formulas. We will be doing it in such a way that it will become easier for students to understand.  

(Two vectors connected via dot making angle theta.)

If you are looking to find an angle between two vectors using a calculator, you might be in for a surprise. Still, there are many websites online that can show you the direct answer, but that’s not how you will get marks in your exams. So, we will be helping to solve it. 

Angle between Two Vectors Formula

To find the angle between the vectors, we first need to take two vectors in the equation. Let’s assume two vectors and name them vector (X) and vector (Y). Now separate these two vectors with angle. 

Here, we have now set up the situation to help us find out the angle between the two vectors. To find out the angle, we first need to find out the given vectors’ dot product. As a result, vector (X) and vector (Y) = |X| |Y| Cos.

Thus, making the angle between the two vectors given in the formula will be as follows:

[ theta  = Cos^{-1}frac{overrightarrow{x}.overrightarrow{y}}{|overrightarrow{x}||overrightarrow{y}|}]

In the above equation, we can find the angle between the two vectors. 

This was the easy way to find the angle between two vectors. Let us now go through the two common ways to determine this angle, and then we will decide which one to use for our case.

Two Methods to Calculate the Angle between Two Vectors

There are two major formulas that are generally used to determine the angle between two vectors: one is in terms of dot product and the other is in terms of the cross product. However, the most widely used formula to determine the angle between two vectors involves the dot product method. Now, we will see what problem arises when we use the cross-product method. Consider x and y to be two vectors and θ to be the angle between them. The following are the two formulas that can be used to find the angle between them. These formulas use both the dot product and the cross product.

• The angle between two vectors can be determined using the dot product as [ theta  = cos^{-1} [ frac{x . y}{ left | x right | left | y right |}]

• The angle between two vectors can be determined using the cross product as [ theta  =sin^{-1} [ frac{x times y}{ left | x right | left | y right |}].

Here, x · y is the dot product and x × y is the cross product of x and y. It is to be noted that the cross product formula requires the magnitude of the numerator, while the dot product formula does not.

Note: When it comes to finding out the angle between two equal vectors, you don’t need to solve any equation as the angle will be zero. The main reason behind it is that two equal vectors will have the same direction and magnitude as one another.  

Solved Example

Let’s try to use the following equation to determine the angle between the two vectors 3i + 4j – k and 2i – j + k. 

The first vector is 3i + 4j – k. 

The second vector is 2i – j + k. 

Now, let’s find the dot product of these two. 

= (3i + 4j – k ).(2i – j + k).

= (3)(2) + (4)(-1) + (-1)(1)

= (6-4-1)

= -1

Thus, the dot product of the two vectors  = 1.

Now, we have to find out the magnitude of the vectors. 

For the first one, [sqrt{3^{2}+4^{2}+(-1)^{2}}] = [sqrt{26}] = 5.09

For the second one,  [sqrt{2^{2}+(-1)^{2}+1^{2}}] = [sqrt{6}] = 2.45

Now, putting the values in the formula.,

[theta=Cos^{-1}frac{overrightarrow{x}.overrightarrow{y}}{|overrightarrow{x}||overrightarrow{y}|}] 

= [Cos^{-1}frac{1}{(5.09)(2.45)}]

= [Cos^{-1}frac{1}{(12.47)}]

= [Cos^{-1} (0.0802)]

= 85.39^o

Conclusion

To summarise, let us go through the major points that we have learned about this topic. The angle between the tails of two vectors is known as the angle between these vectors. There are two ways in which we can find this angle, that is, either by using the dot product (scalar product) or the cross product (vector product). It must be noted that the angle between two vectors will always lie somewhere between 0° and 180°.

Assume that you are multiplying a number twice. You do this to find the square root of a number. Again, if the same number is multiplied three times by itself, you would be finding the cube of the number. Now, let’s say you reverse this entire process. What are you finding here? The root of a number. It can be a square root or it can be the cube root of the number. Here, you will be learning a new topic called antilogs. Understanding this concept will help you ease your problems. In addition to that, antilogs have a wide range of applications in the field of maths.

Definition of Antilog

Antilog, also known as “Anti-Logarithms” of a number, is the opposite way of finding a logarithm of the same number. Consider, if x is the logarithm of the number y with the base b, then we can say that y is the antilog of x to b. It is described as

If log y = x Then, y = antilog x

Both the logarithm and the antilog have their base as 2.7183. If the logarithm and antilogarithm have their base 10, that should be converted to the natural logarithm and antilog by multiplying by 2.303.

How to calculate Antilog

There are two methods using which one can calculate the log to Antilog of a number:

• Using an antilog table

• Antilog calculation 

Before we go ahead, here are a few things you need to know about the characteristics and the mantissa parts. 

Let’s consider a number 7.345

Here, 

• 7 is the characteristic

• 345 is the mantissa

Characteristics is the whole number while the mantissa is all the numbers after the decimal point.

Method 1

Calculating the antilog using Antilog table

Follow the steps given below to calculate the antilog of a number using the antilog table.

Let us consider a number: 2.5463

• Step 1: The first thing to do is to separate the characteristic and the mantissa part. In the above example, the characteristic part is 2 while the mantissa part is 5463.

• Step 2: Using the antilog table, find the corresponding value of mantissa. Find the row number that is equivalent to .54 and then choose column number 6. The corresponding value is 3516.

• Step 3: Now move to the mean difference column. Again use the .54 row and see what’s the corresponding value under the column 3. In this case, the value is 2.

• Step 4: Add the values you found out in step 2 and step 3. Here, it is – 3516 + 2 = 3518

• Step 5: In this step, we add the decimal. According to Step 1, we find the characteristic part. Add 1 to the characteristic part. In this case, we found out that the characteristic part is 2. So here, there have to be 3 numbers before the decimal point.

Therefore, the antilog of 2.5463 = 351.8

Method 2

Calculating the antilog 

How to take antilog in a calculator? This might be a question running in your head. Well, it is simple to do that. Follow the steps given below to calculate the antilog of a number using a simple calculator. 

Let us consider a number: 2.5463

• Step 1: The first thing to do is to separate the characteristic and the mantissa part. In the above example, the characteristic part is 2 while the mantissa part is 5463.

• Step 2: In this method, you’ll have to know the base. Generally for numeric computations, the base is always assumed to be 10. So, to calculate the antilog you need to use the base 10.

• Step 3: In this step, you calculate the 10x. Since the base of the number is always assumed to be 10, the calculation of antilog becomes easier. And if the mantissa is 0 and we just have a whole number, the calculation becomes even simpler. So, 10 times to the power of the given number, gives us the antilog. 

Therefore, 102.5463 = 351.8102.5463 = 351.8

You can use any method. Both of them will give you the same outcome. 

Antilog Table

The table given below helps you find the antilog of a number. Here’s antilog table pdf 1 to 100.

Examples of Antilog

Question 1: Find the antilog of 2.7531

Solution: Given, number = 2.7531

• Step 1: The first thing to do is to separate the characteristic and the mantissa part. Here, the characteristic part is 2 while the mantissa part is 7531

• Step 2: Using the antilog table, find the corresponding value of mantissa. Find the row number that is equivalent to .75 and then choose the column number. The corresponding value is 5662.

• Step 3: Now move to the mean difference column. Again use the .75 row and see what’s the corresponding value under column 1. In this case, the value is 3.

• Step 4: Add the values you found out in steps 2 and step 3. Here, it is 5662 + 3 = 5664.

• Step 5: In this step, we add the decimal. According to Step 1, we find the characteristic part. Add 1 to the characteristic part. In this case, we found out that the characteristic part is 2. So here, there have to be 3 numbers before the decimal point.

Therefore, the antilog of 2.7351 = 566.4

Question 2: Find the antilog of 1.4265.

Solution: Given, number = 1.4265

• Step 1: The first thing to do is to separate the characteristic and the mantissa part. Here, the characteristic part is 1 while the mantissa part is 4265

• Step 2: Using the antilog table, find the corresponding value of
  mantissa. Find the row number that is equivalent to .42 and then choose the column number. The corresponding value is 2667.

• Step 3: Now move to the mean difference column. Again use the .42 row and see what’s the corresponding value under column 5. In this case, the value is 4.

• Step 4: Add the values you found out in steps 2 and step 3. Here, it is 2667 + 4 = 2671.

• Step 5: In this step, we add the decimal. According to Step 1, we find the characteristic part. Add 1 to the characteristic part. In this case, we found out that the characteristic part is 1. So here, there have to be 2 numbers before the decimal point.

Therefore, the antilog of 1.4265 = 26.71

The easiest way to think about the area between two curves: the area between the curves is the area below the upper curve minus the area underneath the lower curve. You can figure out the area between two curves by calculating the difference between the definite integrals of two functions. In 2-D geometry, the area is a volume that describes the region occupied by the two-dimensional figure. Two functions are needed to determine the area, say f(x) and g(x), and the integral limits from ‘a’ to ‘b’ (b should be >a) of the function, that acts as the bespoke of the curve.

Formula to Find the Area between Two Curves

The basic mathematical expression written to compute the area between two curves is as follows:

If P: y = f(x) and Q : y = g(x) and x1 and x2 are the two limits,

Now the standard formula of- Area Between Two Curves, A=∫x2x1[f(x)−g(x)]

Through this topic, you should be able to:

ü  find the area between two curves

ü  find the area between two curves by integration

Calculating Areas Between Two Curves by Integration

1. Area under a curve – Region encircled by the given function, vertical lines and the x –axis.

2. Area Under a Curve – region encircled by the given function, horizontal lines and the y –axis.

3. Area between curves expressed by given two functions.

In case f(x) is a nonnegative and continuous function of x on the closed interval [a, b], then the area of the region enclosed by the graph of ‘f’, the x-axis and the vertical lines x=a and x=b is given by:

b

a

Area f (x)dx

()

When computing the area under a curve f(x), follow the below set of instructions:

1. Shade the area.

2. Identify the boundaries a and b,

3. Establish the definite integral,

4. Integrate.

Calculating Areas Between Curves Using Double Integrals

The common application of the single variable integral is to compute the area under a curve f(x) over some interval [a,b] by integrating f(x) over that interval. That being said, you can sometimes also apply double integrals to compute areas between curves. However, the proposition is not the same. It’s fairly simple to understand the tactic to achieve this once you can envision how to use a single integral to find the length of the interval.

Now you must be thinking as to What happens if you integrate the function f(x)=1 over the interval [a,b]? You can compute that

∫baf(x)dx=∫ba1dx=x∣∣ba=b−a.

The integral of the function f(x) =1 is merely the length of the interval [a,b]. Fact is that it also comes about as the area of the rectangle of height 1 and length (b−a), but we can explain it as the length of the interval [a,b].

You can apply the similar trick for finding areas with double integrals. The integral of a function f(x,y) over a region D can be simplified as the quantity beneath the surface z=f(x,y) over the region D. As executed above, we can attempt the tactic of integrating the function f(x,y)=1 over the region D. This would give the volume under the function f(x,y)=1 over D. But the integral of f(x,y)=1 is also the area of the region D. This can be a nifty way of calculating the area of the region D. Hence, if we If we entitle ‘A’ be the area of the region D, we can write it in the form of :-

A=∬DdA.

()

Solved Example

As we said above, practice is the key to master over calculating area curves. So let’s begin with some fun exercises.

Problem

Find the area encircled by the following curves: 4,= 0, x = y – x =y

()

Solution

Determining the boundaries: y = x² – 4, y=0 which implies x²- 4=0, therefore, (x-2) (x+2) = 0 x = – 2 or x = 2.

With the preview of the graph we can observe that 2=x is the boundary at ‘a’. The assessment of 2- =x is long away from encircling the area of the region. This is why the graph here plays a crucial part in helping identify the appropriate outcome to the problem. The value of the other boundary is provided by the equation of the vertical line 4,=x .

Boundaries are:  2,=a  and  4,=b  

Now, Establish the integral:

A= ò ò A f (x)dx (x 4)dx

Solving,

ò (x² – 4) dx= [(frac{1}{3} x^{3} – 4x)]  ò = ([frac{1}{3}.(4)^{3}) – 4.4) – (frac{1}{3}.{2}^{3} – 4.2)]

= [(frac{64}{3} – 16) – (frac{8}{3} – 8) = frac{64}{3} – 16 – frac{8}{3} + 8 – frac{56}{3} – 8 – frac{32}{3}]

Thus, the area encircled by the curves y – x² -4, y=0, x-4 = 32ö ç3 square units.

Fun Facts

1. Multiple integrals are much easier to use than single integrals’ in finding area with integrals 

2. Drawing the sketch or graph beforehand makes it easy to find areas of the region that should be subtracted.

3.  It may be a requisite to find the areas of curves in several parts and add up the outcomes to achieve the final result. 

Area of Isosceles Triangle

There are three types of a triangle based on the length of the sides. They are as follows:

1.  Equilateral triangle

2.  Isosceles triangle, and 

3. Scalene triangle.

Equilateral Triangle: An equilateral triangle is a triangle whose all three sides are equal and each interior angle measures 60o.

Isosceles Triangle:  An isosceles triangle is a triangle whose two sides are equal.

Scalene Triangle:  A scalene triangle is a triangle whose all three sides are unequal.

Area of Isosceles Triangle

The area of an isosceles triangle is the amount of space that it occupies in a 2-dimensional surface.

So, the area of an isosceles triangle can be calculated if the length of its side is known.

Let us consider an isosceles triangle whose two equal sides length is ‘a’ unit and length of its base is ’b’ unit. Then, 

Derivation of area of an Isosceles Triangle

The formula for the area of an isosceles triangle can be derived using any of the following two methods.

1. Using basic area of triangle formula

2. Using Heron’s formula

METHOD: 1 

Deriving area of an isosceles triangle using basic area of triangle formula

Since, the altitude of an isosceles triangle drawn from its vertical angle bisects its base at point D.

So,

We can determine the length of altitude AD by using Pythagoras theorem.

In ∆ADC, Right angled at angle D. Then, hypotenuse = AC, altitude = AD and base = DC.

According to Pythagoras theorem,

METHOD: 2 

Deriving area of an Isosceles Triangle Using Heron’s Formula

Solved Examples:

Q.1. Find the Perimeter and Area of an Isosceles Triangle Whose two Equal Sides and Base Length is 5 cm and 6 cm Respectively.

Ans.   Given, length of two equal sides of an isosceles triangle = a = 5 cm 

          And length of its base = b = 6 cm

Perimeter of an isosceles triangle = 2a + b 

                                                          = 2(5) + 6

                                                           = 10 + 6 = 16 cm

Q.2. If the Base and Area of an Isosceles Triangle are 8 cm and 12 cm2 respectively. Then find its perimeter.

Ans.  Given, length of base = b = 8 cm

                      And, area = 12 cm2 

Q.3. Find the Altitude of an Isosceles Triangle Whose Two Equal Sides and Base Length is 7 cm and 4 cm Respectively.

Ans. Given, length of two equal sides of an isosceles triangle = a = 7 cm 

          And length of its base = b = 4 cm

Q.4. Find the Area of Right Isosceles Triangle Whose Hypotenuse is 5[sqrt{2}] cm.

Ans.  Let the two equal sides AB and BC of the right isosceles triangle ABC be ‘a’ cm each and AC be the hypotenuse of length 5[sqrt{2}] cm.

                                                                                                                    = 12.5 cm2