Significant Figures

Linear Algebra:

Significant Figures:

When we speak about the word significant,the first thing that comes to our mind is Fitted or designed to signify or make known, or that which has significance; a sign; a token; a symbol.The basic introduction is we use specific number of digits to denote an exact value of a number for required accuracy.The digits used for such a purpose are called significant figures.Next in line is how do we measure these significant figure,to understand the measurement of these figures is of utmost importance. The common question we face is what does significant number depend on???The number of significant figures depends on the instrument used and accuracy required.It is wrong to record diameter as 2 cm. It means the reading is approximated to the nearest 1 cm. If the exact reading is 2 cm then it should be recorded as 2.0 cm.The significant figures express number of units, rounded to the nearest such unit.

Let us now understand the importance of the expression of digits.The outcome of any situation or thing solely depends on how well it is expressed.If something is expressed to the best of its ability the better will be the outcome.The reliability of a measurement is indicated by the number of digits used to represent it. To express it more accurately we express it with digits that are known with certainty. These are called as Significant figures. They contain all the certain digits plus one doubtful digit in a number.

If the volume of the liquid is reported to be 23।5mL, it means that digits 2 and 3 are certain while 5 is uncertain. Therefore, the number 23.5 has three significant figures. The number of significant figures conveys that except the last digits all others are correct. The number of digits depends on the precision of the scale used.

Approximation:

We know that the digits of a number, one by one in order from left to right decrease in value rapidly.Numbers can be rounded off to any given place.When we round off a number to any given place we get an approximate value of the number.Absolute Error and Percentage Relative Error.The difference between these two values is called the absolute error.

Absolute Error = Original number - Approximated number
The ratio of the absolute error and the original number expressed in percentage form is called percentage relative error.

Summary:

1. We use specific number of digits to denote the value of a number to a certain degree of accuracy.

2. For a number between 0 and 1, the successive zeros after the decimal are not significant.

3. For a number with a decimal point, the final zeros are significant.

Hope you like the above example of Significant Figures.Please leave your comments, if you have any doubts.

Similarity

Similarity:

Introduction to similarity:

Let us learn one of the most interesting topics,that of Similarity.We can see many things around us of the same shape and size,now let us learn about the concept of similarity in detail.
Two geometrical objects are called similar if they both have the same shape. More precisely, one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. One can be obtained from the other by uniformly "stretching" the same amount on all directions, possibly with additional rotation and reflection, i.e., both have the same shape, and one has the same shape as the mirror image of the other.

Study Axioms of Similarity:


We can understand the concept of similarity by studying about the different axioms as well.
AAA axiom of similarity: If in any two triangles, corresponding angles are equal then the two triangles are said to be similar to each other.

SSS axiom of similarity: If in any two triangles, corresponding sides are in same proportion then the two triangles are similar.

SAS axiom of similarity: If in any two triangles, two sides are in same proportion and if the included angle is equal then the triangles are similar.

Special Note 1: In a triangle ABC if a parallel line is drawn to any of the sides( Suppose BC) touching other sides at D and E(AB and AC respectively), then triangle ABC will be similar to the triangle formed, i.e ADE.
Special Note 2: In a right triangle ABC right angled at B, if a perpendicular is drawn from B to the side AC to meet at D, then the triangles thus formed and the original triangle are similar to each other.
Study Properties of Similarity in Triangles:
When two triangles(trABC,tr DEF) are similar to each other
1) The corresponding sides are in same proportion to each other, i.e
AB/DE = BC/EF = CA/FD
2) The corresponding angles in these triangles are equal
angle A = Angle D; angle B = angle E; angle C = angle F
3) The perimeters are in same proportion as the corresponding sides ratio.
perimeter(tr ABC) / perimeter(tr DEF) = AB/DE = BC/EF = CA/FD
4) The areas of the triangles are in same proportion as the square of ratios of sides
area(ABC) / area(DEF) = (AB/DE)^2 = (BC/EF)^2 = (CA/FD)^2.

Hope you like the above example of Similarity.Please leave your comments, if you have any doubts.

Ratio

Ratios:

In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient.Example: For every Spoon of sugar, you need 2 spoons of flour.Now let us understand the more specific meaning of a Ratio-Ratios means pairs of numbers and they are used to formulate comparisons of numbers.Now let us understand the basic use of a Ratio,a ratio can be used to compares two numbers using a fraction.Ratios can be written in three different ways:
(1) 5 to 3
(2) 5:3
(3) 5/3
They all mean the same thing. I have a class of 35 students, 20 students are boys and 15 are girls. I can compare number of boys to girls using the ratio (or 20:15; or 20 to 15). You can reduce this fraction to by dividing both numerator and denominator by 5.
Notation and Terminology of Ratios
The ratio of quantity A and B can be expressed as
• The ratio of A to B
• A is to B
• A: B
The quantity A and B are sometimes called terms with A being the antecedent and B being the consequent.
Finding Ratios
A ratio is a fraction that is numerator and denominator can have different units. For example: if you travel 100 miles in 2 hours, we can write that as a ratio:
2hours
100miles
And we can simplify that ratio, using the same techniques for simplifying fractions:
2hours100miles = 1hour50miles
If there are 4 oranges and 5 apples, the ratios of oranges to apples is exposed as 2:3; where as the fraction of oranges to total fruit is
Ex 1: In a bag of red and green balls, the ratios of red ball to green ball are 3:4. If the bag contains 120 green balls, how many red balls are there?
Solution:
Step 1: Assign variables:
Let x = red sweets
Write down the items in the ratio as a fraction.
= =
Step 2: Solve the equation
Cross multiply the equation
4 × 120 = 5 × x
480 = 5x
Isolate variable x
X= =96
Ans: There are 96 red balls.
Ex 2: A special mixture contains rice, water and corn in the ratio of 2:3:6. If a bag contains 3 pounds of rice, how much corn it contains?
Solution:
Step 1: Assign variables:
Let x = amount of corn
Write down the items in the ratio as a fraction.
= =
Step 2: Solve the equation
Cross multiply the equitation
2 × x = 3 × 6
2x = 16
Isolate variable x
X= = 9
Ans: The bag contains mixture 9 pounds of corn.

Hope you like the above example of Ratios.Please leave your comments, if you have any doubts.

Algebraic Expressions:

Algebraic Expressions:

Introduction to algebraic expressions:

In mathematics,an expression is a finite combination of symbols that are well-formed,according to the rules applicable in the context at hand. Algebraic expressions are formed from variables and constants.We simply use the combination of operations of addition, subtraction, multiplication and division on the variables and constants to form expressions. We are faced with this question most of the time-What are Algebraic Expressions?? In general terms it is easy to explain algebraic Expressions,"An algebraic expression consists of the signs and symbols."Algebraic expressions does not contain equal sign.In algebra, an expression may be used to designate a value, which value might depend on values assigned to variables occurring in the expression; the determination of this value depends on the semantics attached to the symbols of the expression. These semantic rules may declare that certain expressions do not designate any value; such expressions are said to have an undefined वलुए.Another way of understanding the concept of Algebraic Expression is by looking at a few example problems of Algebraic Expression,similarly given below are few practice problems on algebraic expressions.

Some examples of algebraic expressions:

2x + 2y , 2xy , x2 - y2 , 3m + n - 5.
There are one or more terms in the algebraic expressions. For example, 3x2 + 4x + 2 is algebraic expression. It has three terms.

Let us discuss about some example problems on Algebraic expressions.

Example Problems - Algebraic Expressions
Problem 1:
Find the number of terms in the algebraic expression. 6x - 7
Solution:
The given expression 6x - 7 has two terms.
Problem 2:
Find the number of terms in the algebraic expression. a2 - 2ab + b2
Solution:
The given expression a2 - 2ab + b2 has three terms.
Problem 3:
What is the coefficient of x2 in the given algebraic expression? 5x2 + 2x + 6
Solution:
The coefficient of x2 is 5 in the expression.
Problem 4:
Add the two algebraic expressions: 3x + 21 and 4x + 7
Solution:
Sum of the two algebra expressions: 3x + 21 + 4x + 7
Rearrange the expression by grouping like terms.
3x + 4x + 21 + 7
Adding like terms in the expression.
7x + 28
Hence, 3x + 4x + 21 + 7 = 7x + २८
Problem 5:
Add the two algebraic expressions: 8x + 11 + 6z and 5x – 7
Solution:
Sum of the two expressions: 8x + 11 + 6z + 5x – 7
Rearrange the expression by grouping like terms.
8x + 5x + 6z + 11 – 7
Adding like terms in the expression.
13x + 6z + 4
Hence, 8x + 11 + 6z + 5x - 7 = 13x + 6z + 4
Problem 6:
Simplify the algebraic expression: (3a2 + 5a – 9) – (8a – a2 – 7)
Solution:
The given expression: (3a2 + 5a – 9) – (8a – a2 – 7)
Expand the expression.
3a2 + 5a – 9 – 8a + a2 + 7
Rearrange the expression by grouping like terms.
3a2 + a2 + 5a – 8a – 9 + 7
Adding like terms in the expression.
4a2 – 3a - 2
Hence, (3a2 + 5a – 9) – (8a – a2 – 7) = 4a2 – 3a - 2.
Problem 7:
Evaluate the algebraic expression b3 + 6b2 + 6b – 4 when b = -3
Solution:
We get a value for the algebraic expression when we substitute b is – 3.
b3 + 6b2 + 6b – 4
=(-3)3 + 6(-3)2 + 6(-3) – 4
= + 6 – 18 - 4
= -27 + 6(9) – 18 – 4
= -27 + 54 – 18 – 4
= - (27 + 18 + 4) + 54
= - 49 + 54
= 5.

Practice Problems - Algebraic Expressions:

Problem 1:
Find the number of terms in the expression. 64mn - 8
Answer:
Two terms.
Problem 2:
Find the number of terms in the expression. 6x2 - 3xy + 8y2 + 6
Answer:
Four terms.
Problem 3:
What is the coefficient of x in the given algebraic expression? 10x2 + 20x + 8
Answer:
20 .
Problem 4:
Add the two expressions: 6ab + 5a + 2ab and 7a - 5
Answer:
8ab + 12a - 5
Problem 5:
Simplify the algebraic expression: a2 + 2ab + b2 + 5ab + 2b2
Answer:
a2 + 7ab + 3b2
Problem 6:
Evaluate the algebraic expression: 10x2 + 20x when x = 2.
Answer:
80.

Hope you like the above example of Algebraic Expressions.Please leave your comments, if you have any doubts.

Percent and Example Problems:

Percent and Example Problems:

The term percent in math is used to represents the fraction number and also we represent the term with denominator 100.For example:3/6 =50 /100 .Then the percentage of the given fraction number as 50%.and the fraction number 3/6 can be represented as a decimal as 0.50, 3/6,50% and 0.50 are represents the same..

Let us now learn about the uses of Percentages,its is important to understand the uses of a Percentage.
Percentages are used to express how large/small one quantity is, relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity, which should be greater than zero. For example, an increase of $ 0.15 on a price of $ 2.50 is an increase by a fraction of 0.15 / 2.50 = 0.06. Expressed as a percentage, this is therefore a 6% increase.

Example Problems to Find Percentage in Math:

As we know percentages have various uses altogether,but let us now look at the most common uses of a percentage.Percentage is mostly used in banking sector to find the interest.The formula to find interest is,
Interest =Principle amount x time x rate of interest.The easiest way to learn about percentages is by solving a few problems related to Percentages

Example 1:
Suppose we need to find the interest will $2000 earn in one year at an annual interest rate of 5%
Solution:
5% = 5 /100
(5/100 ) x 2000 =100
So the answer is $

Example 2:
Find the interest if amount if $1000 and month is 6 and rate of interest is 6%
Solution:
Interest =Amount x time x rate of interest.
Note : 6 month = ½ of years.
Therefore,
6% =>6/100 or 3/50
Substitute these values into interest formula. We get,
Interest = $1000 x ½ x 6/100
=$1000 x1/2 x 3/50
=$500 x 3/50
=$10 x 3
=$३०

Example 3:
There are 45 boys out of 90 students in the class room. Find the percentage of boys in the class room.
Solution:
Total students =90
Out of these 90 students 45 students are boys ,it is represented in terms of fraction.
= 45/90 [fraction form]
Convert the fraction into percentage,
So multiply the fraction number by 100,we get,
(45/90) * 100
=50%
Example 4:
40 percent of X = 200 then find x?
Solution:
40 % * X =200
40 x */100 =200
40x =200 *100
40x = 20000
X =20000/40
X= 2000/4
X=500
Therefore, X value = 500
Verification:
40 percent of 500 =200
(40 x 500)/100 =200
20000/100 =200
200 =200
Hence Verified.

Hope you like the above example of Percents.Please leave your comments, if you have any doubts.

Integers

Integers:
Positive numbers, zero and negative numbers together form an Integer. A fraction is a part or parts of a whole. Decimal fraction is the special fraction whose denominators are 10, 100, 1000 etc. These fractions are called decimal fractions. Let us see about Integers, Fractions and decimals in this article.The numbers 0, 1, −1, 2, −2, … are called integers of which 1, 2, 3, … are called positive integers and −1, −2, −3,… are called negative integers. The collections of all integers are denoted by the letter Z. Thus Z = {…, −3, −2, −1, 0, 1, 2, 3…}.Let us now learn about the Types of Interger Factors:If two numbers are in a / b form, where the two numbers are integer then it said to be a integers fraction,Here, a (numerator)/ b (denominator) à both ‘a’ and ‘b’ not equal to zero.Types of Integer fraction:1. Proper Fraction.2. Improper Fraction.3. Mixed Fraction.Proper Fraction:In the proper fraction is the fraction in which the numerator (the top number) is less than the denominator (the bottom number).Example:1/2, 2/3, 5/7.

Improper Fraction:An improper fraction is the numerator (the top number) is greater than or equal to the denominator (the bottom number).Example:4/3, 5/2, 7/5.Mixed Fraction:Mixed Fraction is the combination of whole number and proper fraction.लेट usIn mixed fraction addition we do the following steps,1. First convert mixed fraction into the proper fraction.[Multiply the denominator (Bottom of the number) of the fraction by the whole number and then add the numerator (top of the number) keep the answer over the original denominator.]2. Add the numerators if the denominators are same.3. If the denominators are Unequal make the common denominator using LCM method.4. Then simplify the answer that is divided the numerator and denominator by the same number.Solved Example on Integer Fraction There are 225 students are in science group and 350 students are in arts group. Total 600 students in a college. What fraction of the college are the arts students?Choices: A. 7/12B. 5/12C. 1/12D. 25/35Correct Answer: ASolution: Step 1: Total number of students in a college = 600.Step 2: Number of science group students= 225.Step 3: Number of arts group students= 350.Step 4:Fraction of arts group students = Number of arts students / Total number of students in college = [Substitute the values]= (50*7)/ (50*12) [Write 350 as 50*7 and 600 as 50*12] = [Reduced term of the fraction] Step 5: The fraction of arts students in a college Practice problem on Integer fraction:Answer : Answer : 2 - 4 Answer :-6 Hope you like the above example of Integers.Please leave your comments, if you have any doubts.