Factor tree

Mind tricks

Math quotes

Inspirational math quotes

1. “Do not worry about your difficulties in mathematics. I can assure you mine are still greater.” – Albert Einstein

2. “The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.” – Charles Caleb Colton

3. “Without mathematics, there’s nothing you can do. Everything around you is mathematics. Everything around you is numbers.” – Shakuntala Devi

4. “In mathematics the art of proposing a question must be held of higher value than solving it.” – Georg Ferdinand Ludwig Philipp Cantor

5. “The only way to learn mathematics is to do mathematics.” – Paul Halmos

6. “Obvious is the most dangerous word in mathematics.” – Eric Temple Bell

7. “Mathematics is the music of reason.” –James Joseph Sylvester  

8. “You don’t have to be a mathematician to have a feel for numbers.” – John Forbes Nash, Jr.

9. “Life is a math equation. In order to gain the most, you have to know how to convert negatives into positives.” –Anonymous

10. “Arithmetic is being able to count up to twenty without taking off your shoes.” –Mickey Mouse

Maths in Environment studies

Mathematics plays a key role in environmental studies, modeling, etc. Basic mathematics - calculus, percents, ratios, graphs and charts, sequences, sampling, averages, a population growth model, variability and probability - all relate to current, critical issues such as pollution, the availability of resources, environmental clean-up, recycling, CFC's, and population growth. In January of this year the annual winter meeting of the national mathematics societies held theme sessions on Mathematics and the Environment. Several presentations were made. Papers are available on request as described below. Fred Roberts - Department of Mathematics, Rutgers University Moving Traffic So As To Use Less Fuel and Reduce Pollution Two of the ways in which mathematics is used in traffic management are in the phasing of traffic lights and in the design of patterns of one-way streets. Mathematical methods first developed in the early stages of sequencing the DNA molecule have turned out to be useful in deciding when to give different streams of traffic a green light. Related mathematical methods are useful in deciding how to make streets one-way so as to move traffic more efficiently. Robert McKelvey - Department of Mathematics, Univ. of Montana Global Climate Change: How We Set Policy How we deal with uncertainty in making environmental decisions, focusing on some of the interlocking environmental problems of today: 1) global warming; 2) biodiversity and genetic diversity (loss of species); and 3)impending losses of resources (land, energy, clean air, water). Mary Wheeler - Department of Mathematics, Rice University, and Kyle Roberson, Pacific Northwest Laboratories Bio-remediation Modeling: Using Indigenous Organisms to Eliminate Soil Contaminants An explanation of laboratory, field, and simulation work to validate remediation strategies at U.S. Department of Energy sites, such as Hanford, WA. A project goal is to formulate and implement accurate and efficient algorithms for modeling biodegradation processes. Numerical simulation results that utilize realistic data and parallel computational complexity issues are discussed. Simon Levin - Section of Ecology and Systematics, Cornel The Problem of Scale in Ecology: Why this is Important in Resolving Global Problems Global environmental problems have local and regional causes and consequences, such as, linkages between photosynthetic dynamics at the leaf level, regional shifts in forest composition, and global changes in climate and the distribution of greenhouse gases. The fundamental problem is relating processes that are operating on very different scales of space and time. Mathematical methods provide the only way such problems can be approached, and techniques of scaling,

Application of algebra

 Real Life Applications of Algebra Objectives

      

 
Too often students think of algebra as an abstract topic completely disconnected from the real world. This may in part be attributed to the way in which many algebra curricula are written or presented, causing students to see the subject as valueless. Fortunately, real-life applications of algebra objectives abound, and learners can investigate them throughout the course.

How Much Can You Buy?

Writing and solving various types of equations is one of the key objectives of algebra. Many of the most widely useful applications pertaining to equations involve the transfer of money. Such problems are often of the type “You have x amount of money, how much of y product can you buy with it?” For instance, envision a scenario in which you’re helping prepare for a party, and are sent to the store with $30 to buy as many liter bottles of soda as possible, as well as a package of plastic cups. Each liter of soda costs $1.80, and a package of cups costs $4.50. To quickly calculate how many sodas you can buy, you can write and solve an algebraic equation: 1.8x+4.5=30. Other real-life applications in this category could include figuring out how many bottles you could buy if they are priced at two for $3.50 or are being offered as a part of a buy-one-get-one-free special.

Markup and Markdown

Interpreting and solving problems involving ratios, proportions and percents comprises another objective of algebra. Real-world scenarios in this genre can easily be created around the idea of store sales. Types of problems could include determining the percent off, percent saved, new cost or original cost. For example, suppose a shirt is priced at $22 and a sign says the price is 30 percent off. You want to know how much the shirt cost originally to see whether your savings would be significant. Thirty percent off equates to 70 percent of the original price, so, using the algebraic proportion formula to write a proportion: 22/x=70/100, and solve it by using cross-products.

Article

Advantages of Mathematics

Many of us considered about the key benefits of Arithmetic during our youth. Many of us were not able to view the key benefits of mathematics beyond the everyday use of determining simple figures. Let us see in details what are some of the key benefits of learning mathematics and marveling at this difficult topic at beginning age.

The significance of mathematics is two-fold, it is essential in the progression of technological innovation and two, it is essential in our knowing of the technicalities of the galaxy. And in here and now you should individuals for self improvement, both psychologically and in the office.

Mathematics provides students with a exclusively highly effective set of resources to understand and change the globe. These resources include sensible thinking, problem-solving capabilities, and the capability to think in subjective ways. Arithmetic is essential in lifestyle, many types of career, technological innovation, medication, the economic system, the surroundings and growth, and in public decision-making.

One should also be aware of the extensive significance of Arithmetic, and the way in which it is improving at a amazing rate. Arithmetic is about routine and structure; it is about sensible research, reduction, computation within these styles and components. When styles are found, often in commonly different areas of technological innovation, the mathematics of these styles can be used to describe and control natural events and circumstances. Arithmetic has a persistent impact on our life, and plays a role in the prosperity of the individual.

The research of mathematics can fulfill a number of passions and capabilities. It produces the creativity. It teaches in clear and sensible thought. It is a task, with types of challenging concepts and unresolved problems, because it offers with the questions coming up from complex components. Yet it also has a ongoing drive to generality, to finding the right principles and methods to make challenging things easy, to describing why a situation must be as it is. In so doing, it produces a variety of terminology and concepts, which may then be used to make a essential participation to our knowing and admiration around the globe, and our capability to find and make our way in it.

Application of statistics

List of fields of application of statistics

Statistics is the mathematical science involving the collection, analysis and interpretation of data. A number of specialties have evolved to apply statistical and methods to various disciplines. Certain topics have "statistical" in their name but relate to manipulations of probability distributions rather than to statistical analysis.

Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in theinsurance and finance industries.Astrostatistics is the discipline that applies statistical analysis to the understanding of astronomical data.      
Biostatistics is a branch of biologythat studies biological phenomena and observations by means of statistical analysis, and includesmedical statistics.   Business analytics is a rapidly developing business process that applies statistical methods to data sets (often very large) to develop new insights and understanding of business performance & opportunities.            Chemometrics is the science of relating measurements made on achemical system or process to the state of the system via application of mathematical or statistical methods.     Demography is the statistical study of all populations. It can be a very general science that can be applied to any kind of dynamic population, that is, one that changes over time or space.                      Econometrics is a branch ofeconomics that applies statistical methods to the empirical study of economic theories and relationships.                 Environmental statistics is the application of statistical methods toenvironmental science. Weather, climate, air and water quality are included, as are studies of plant and animal populations.                 Epidemiology is the study of factors affecting the health and illness of populations, and serves as the foundation and logic of interventions made in the interest of public health and preventive medicine.                     Geostatistics is a branch ofgeography that deals with the analysis of data from disciplines such as petroleum geology, hydrogeology,hydrology, meteorology,oceanography, geochemistry,geography.                      Machine learning is the subfield ofcomputer science that formulates algorithms in order to make predictions from data.    Operations research (or operational research) is an interdisciplinary branch of applied mathematics and formal science that uses methods such as mathematical modeling, statistics, and algorithms to arrive at optimal or near optimal solutions to complex problems.                             Population ecology is a sub-field ofecology that deals with the dynamics of species populations and how these populations interact with theenvironment.  Psychometrics is the theory and technique of educational and psychological measurement of knowledge, abilities, attitudes, and personality traits.                                          Quality control reviews the factors involved in manufacturing and production; it can make use ofstatistical sampling of product items to aid decisions in process control or in accepting deliveries.

Least common multiple

The least common multiple of two numbers is the smallest number that can be divided evenly by your two original numbers. 

Least Common Multiple

The least common multiple is a math topic you usually use when you're trying to find a common denominator between two fractions, and it's one of those things that you learn pretty early on in your education, but it can easily be forgotten or mistaken for a different math idea, usually the greatest common factor. So, let's start by reminding you exactly what it is.

The least common multiple of two numbers, often written as the LCM, is the smallest number that can be divided evenly by those original two numbers. For example, the LCM of 5 and 6 is 30, because it is the smallest number that both 5 and 6 go into. And, that's it - least common multiple.

What Is a Multiple?

But, in order to not forget what a least common multiple is in the future, it's probably best to understand some of the vocabulary in the name - mainly, what is a multiple? Well, the multiples of a number are just what you get when you multiply that number by 1, then 2, then 3 and so on. For example, the multiples of 2 are 2, 4, 6, 8, 10, 12 and on and on. Or, the multiples of 7 are 7, 14, 21, 28, 35 and on and on and on. Notice that 2 is a multiple of 2, and 7 is a multiple of 7. That means that any number is a multiple of itself; we'll need to remember that a little bit later on. Anyways, now that we know this vocabulary, we can say that the least common multiple just means the smallest multiple that two numbers have in common.

How to Find the Least Common Multiple

The most fool proof way to find a least common multiple is to list out all the multiples of each number and then find the first one they have in common, like I've been showing you so far. LCM of 8 and 6? Well, multiples of 8 are 8, 16, 24, 32... and the multiples of 6: 6, 12, 18, 24 - Ans  - 24.

Pascal Traingle comparing with christmas tree

Have you ever heard or Pascal’s Triangle? It is a cool number pattern named after Blaise Pascal, a famous French Mathematician and Philosopher. I took this triangle and turned it into a Christmas Tree math activity.

top 10 success

Numeracy vocabulary

types of slope

properties of numbers

math seven

maths in nature

SHAPES

Shapes - Perfect

Earth is the perfect shape for minimising the pull of gravity on its outer edges - a sphere (although centrifugal force from its spin actually makes it an oblate spheroid, flattened at top and bottom). Geometry is the branch of maths that describes such shapes.

SYMMETRY

Symmetry

Five axes of symmetry are traced on the petals of this flower, from each dark purple line on the petal to an imaginary line bisecting the angle between the opposing purple lines. The lines also trace the shape of a star.

puzzles

Ans: 7.

Sol. 91 ÷ 13 = 7.

Ans: 0.

Sol. Looking at lines of numbers from the top : 9×8 = 72; 72×8 = 576; 576×8 = 4608;

mutiplication trick

How to divide a large number by 5 in seconds?:
EXAMPLE: 195 / 5,

 Step 1: 195 x 2 = 390.0

 Step 2: Move the decimal one place to the left: 39.0 

birthday trick

maths tricks

Mental calculation

Mental calculation

Mental calculation comprises arithmetical calculations using only the human brain, with no help from calculators, computers, or pen and paper. People use mental calculation when computing tools are not available, when it is faster than other means of calculation (for example, conventional methods as taught in educational institutions), or in a competitive context. Mental calculation often involves the use of specific techniques devised for specific types of problems.

Many of these techniques take advantage of or rely on the decimalnumeral system. Usually, the choice of radix determines what methods to use and also which calculations are easier to perform mentally. For example, multiplying or dividing by ten is an easy task when working in decimal (just move the decimal point), whereas multiplying or dividing by sixteen is not; however, the opposite is true when working in hexadecimal.

Methods and techniques

Casting out nines

Main article: Casting out nines

After applying an arithmetic operation to two operands and getting a result, you can use this procedure to improve your confidence that the result is correct.

Sum the digits of the first operand; any 9s (or sets of digits that add to 9) can be counted as 0.If the resulting sum has two or more digits, sum those digits as in step one; repeat this step until the resulting sum has only one digit.Repeat steps one and two with the second operand. You now have two one-digit numbers, one condensed from the first operand and the other condensed from the second operand. (These one-digit numbers are also the remainders you would end up with if you divided the original operands by 9; mathematically speaking, they're the original operands modulo 9.)Apply the originally specified operation to the two condensed operands, and then apply the summing-of-digits procedure to the result of the operation.Sum the digits of the result you originally obtained for the original calculation..If the result of step 4 does not equal the result of step 5, then the original answer is wrong. If the two results match, then the original answer may be right, though it isn't guaranteed to be.

Example

Say we've calculated that 6338 × 79 equals 500702Sum the digits of 6338: (6 + 3 = 9, so count that as 0) + 3 + 8 = 11Iterate as needed: 1 + 1 = 2Sum the digits of 79: 7 + (9 counted as 0) = 7Perform the original operation on the condensed operands, and sum digits: 2 × 7 = 14; 1 + 4 = 5Sum the digits of 500702: 5 + 0 + 0 + (7 + 0 + 2 = 9, which counts as 0) = 55 = 5, so there's a good chance that we were right that 6338 × 79 equals 500702.

You can use the same procedure with multiple operations just repeat steps 1 and 2 for each operation.

Importance of math in day today life

Mathematics is a methodical application of matter. It is so said because the subject makes a man methodical or systematic. Mathematics makes our life orderly and prevents chaos. Certain qualities that are nurtured by mathematics are power of reasoning, creativity, abstract or spatial thinking, critical thinking, problem-solving ability and even effective communication skills. 

Mathematics is the cradle of all creations, without which the world cannot move an inch. Be it a cook or a farmer, a carpenter or a mechanic, a shopkeeper or a doctor, an engineer or a scientist, a musician or a magician, everyone needs mathematics in their day-to-day life. Even insects use mathematics in their everyday life for existence. 

Snails make their shells, spiders design their webs, and bees build hexagonal combs. There are countless examples of mathematical patterns in nature's fabric. Anyone can be a mathematician if one is given proper guidance and training in the formative period of one's life. A good curriculum of mathematics is helpful in effective teaching and learning of the subject. 

Experience says learning mathematics can be made easier and enjoyable if our curriculum includes mathematical activities and games. Maths puzzles and riddles encourage and attract an alert and open-minded attitude among youngsters and help them develop clarity in their thinking. Emphasis should be laid on development of clear concept in mathematics in a child, right from the primary classes. 

If a teacher fails here, then the child will develop a phobia for the subject as he moves on to the higher classes. For explaining a topic in mathematics, a teacher should take help of pictures, sketches, diagrams and models as far as possible. As it is believed that the process of learning is complete if our sense of hearing is accompanied by our sense of sight. Open-ended questions should be given to the child to answer and he/she should be encouraged to think about the solutions in all possible manners. The child should be appreciated for every correct attempt. And the mistakes must be immediately corrected without any criticism. 

The greatest hurdle in the process of learning mathematics is lack of practice. Students should daily work out at least 10 problems from different areas in order to master the concept and develop speed and accuracy in solving a problem. Learning of multiplication-tables should be encouraged in the lower classes. 

Another very effective means of spreading the knowledge of mathematics among children is through peer-teaching. Once a child has learned a concept from his teacher, the latter should ask him to explain the same to fellow students. Moreover, in the process all the children will be able to express their doubts on the topic and clear them through discussions in a group. 

The present age is one of skill-development and innovations. The more mathematical we are in our approach, the more successful we will be. Mathematics offers rationality to our thoughts. It is a tool in our hands to make our life simpler and easier. Let us realize and appreciate the beauty of the subject and embrace it with all our heart. It is a talent which should be compulsorily honed by all in every walk of life. 

Angles

Types of angles.                                    Acute angle:

An angle whose measure is less than 90 degrees. The following is an acute angle.

Right angle:

An angle whose measure is 90 degrees. The following is a right angle.

Obtuse angle:

An angle whose measure is bigger than 90 degrees but less than 180 degrees. Thus, it is between 90 degrees and 180 degrees. The following is an obtuse angle.

Straight angle

An angle whose measure is 180 degrees.Thus, a straight angle look like a straight line. The following is a straight angle.

Reflex angle:

An angle whose measure is bigger than 180 degrees but less than 360 degrees.The following is a reflex angle.

Adjacent angles:

Angle with a common vertex and one common side. <1 and <2, are adjacent angles.

Complementary angles:

Two angles whose measures add to 90 degrees. Angle 1 and angle 2 are complementary angles because together they form a right angle.

Note that angle 1 and angle 2 do not have to be adjacent to be complementary as long as they add up to 90 degrees

Supplementary angles:

Two angles whose measures add to 180 degrees. The following are supplementary angles.

Vertical angles:

Angles that have a common vertex and whose sides are formed by the same lines. The following(angle 1 and angle 2) are vertical angles.

When two parallel lines are crossed by a third line(Transversal), 8 angles are formed. Take a look at the following figure

Angles 3,4,5,8 are interior angles

Angles 1,2,6,7 are exterior angles

Alternate interior angles:

Pairs of interior angles on opposite sides of the transversal.

For instance, angle 3 and angle 5 are alternate interior angles. Angle 4 and angle 8 are also alternate interior angles.

Alternate exterior angles:

Pairs of exterior angles on opposite sides of the transversal.

Angle 2 and angle 7 are alternate exterior angles.

Corresponding angles:

Pairs of angles that are in similar positions.

Angle 3 and angle 2 are corresponding angles.

Angle 5 and angle 7 are corresponding angles

Formula

What are Illustrations of  formulas?

Formulas are statements of algebra, which apply to numbers of a definition set.

They can be proved except the axioms. You can deduce new formulas by well-known formulas by logical reasoning. This procedure is called a proof.

The proof  ideas and also the proof ways can be described by pictures. In addition the formulas themselves become more alive.

You can find illustrations of well known formulas on this page.

Simple Formulas  top

Commutative law of multiplication (Axiom) 
 
ab=baDistributive law (Axiom)

 
(a+b)c=ac+bc 

Product of a difference and a number

 
(a-b)c=ac-bc 

Product of two sums

 
(a+b)(c+d)=ac+ad+bc+bd 

Product of two differences

 
(a-b)(c-d)=  ac+bd  -ad-bc 

Product of a sum and a difference

 
(a-b)(c+d)=  ac+ad   -bc-bd 

Looking for a parallelogram with the same area

 
a²=bx

Binomial Formulas   top

First binomial formula

 
(a+b)²=a² + 2ab + b² 

 Second binomial formula

 
(a-b)²=  b²+a²  -2ab 

Third binomial formula

 
a²-b²=(a+b)(a-b) 

Tri-nomial formula

(a+b+c)²=a²+b²+c²+2ab+2ac+2bc 

Difference of the squares of a sum and a difference

 
(a+b)²-(a-b)²=4ab

Pythagoras's Theorem  top

The Pythagorean theorem (Pythagoras or one of his students, Pythagoras of Samos, 580-500 BC)

 
a²+b²=c²

The Pythagorean theorem (Euklid, ~300 BC)  
Classical proof with triangles

 
a²+b²=c² 

 The Pythagorean theorem (Euklid, ~300 BC) 
Proof with four-sided figures

 
a²+b²=c² 

Euklid's theorem (Euklid, ~300 BC)

 
a²=cp (You can show b²=cq in analogy.) 

Height formula

 
a²=p²+h² (Pythagorean theorem), a²=pc=p²+pq (Euklid's theorem), 
hence h²=pq 

The Pythagorean theorem (Liu Hui, ~300, China)

 
a²+b²=c²

The Pythagorean theorem ("The bride's chair", ~900, India)

 
a²+b²=c² 

The Pythagorean theorem (Atscharja Bhaskara, Indien,  ~1150)

 
c²=(a-b)²+2ab oder c²=a²+b² 

The Pythagorean theorem (Leonardo da Vinci, 1452-1519)

 
a²+b²=c² 

The Pythagorean theorem (Arthur Schopenhauer's case was a=b, 1788-1860)

 
a²+b²=c² 

The Pythagorean theorem (James Garfield 1876, later on the 20th US President)

 
You use the formula of the area of a trapezium [A=mh, here h=a+b and m=(a+b)/2] 
(a+b)²/2=c²/2+2*(1/2*ab) or a²+b²=c² 

The Pythagorean theorem (Hermann Baravalle 1945)

.........

c²=a²+b² 

The Pythagorean theorem

(a+b)²=c²+4*(1/2ab) oder a²+b²=c²

Cubes     top

Cube of a sum 
 
(a+b)³=a³+3a²b+3ab²+b³

You can see both cubes and the six rectangular parallelepipeds in 3D-view:

Cube of a difference

The formula is (a-b)³=a³-3a²b+3ab²-b³. You convert it to (a-b)³=a³-3ab(a-b)-b³ for an illustration.......You take the drawing of the formula (a+b)³=a³+3a²b+3ab²+b³ from above and replace a by the difference a-b. 

Then the edges are (a-b)+b with different combinations (on the left).

The term (a-b)³ is illustrated by the blue cube (on the right). 
 

......You  recieve the blue cube, too, if you take away the three green bodies and the yellow cube from the red cube:

 
(a-b)³ = a³-3ab(a-b)-b³ = a³-3a²b+3ab²-b³