• Real numbers: Numbers which can be quantified and represented by a unique point on the number line are called real numbers.
• Complex numbers: Complex numbers are the numbers which have both real and imaginary part.
• Rational numbers: Numbers of the form p/q, where p and q are integers and q≠0 are called rational numbers.
• Irrational numbers: Numbers which are not rational but can be represented on the number line are called irrational numbers.
• Integers: Rational numbers of the form p/q, where p and q are integers and q= ±1 are called integers.
• Fractions : Fractions are a type of rational numbers, which are of the form p/q, where p and q are integers and q≠0 and whose numerator is less than the denominator and both are in the lowest terms.
• Whole numbers: Whole numbers are the set of positive integers from 0. They do not have any decimal or fractional part.
• Negative integers: Negative integers are the set of negative numbers before 0. They do not have any fractional or decimal part.
• Natural numbers: Natural numbers are the set of positive integers, that is, integers from 1 to ∞, excluding fractional n decimal part. They are whole numbers excluding zero.
• Even numbers: Numbers divisible by 2 are called even numbers.
• Odd numbers: Numbers which are not divisible by 2 are called odd numbers. Odd numbers leave 1 as the remainder when divided by 2.
• Prime numbers: Any number other than 1 which does not have any factor apart from one and the number itself is called a prime number.
• Co-prime/relatively prime numbers: Two numbers are said to be co-prime or relatively prime if they do not have any common factor other than one.
• Composite numbers: a number that has more than two distinct factors is called a composite number.
• Perfect numbers: A number is said to be a perfect number if the sum of all its factors, excluding itself (but including 1) is equal to the number itself.
Wednesday 25 January 2017
Types of numbers
History of algebra
No single person discovered algebra, since various people in different parts of the world discovered it at different times. Some aspects of algebra were even discovered multiple times by different people who were unaware of each other. Virtually every major civilization worked out some portion of the algebraic puzzle, although certain people like Diophantus, Muhammad Ibn Musa al-Khwarizmi and Gottfried Leibniz made more significant contributions.
The Babylonians pioneered many of the basic usages of algebra. A tablet dated between 1900 and 1600 B.C. contains Pythagorean triples and other advanced mathematics. There is also evidence of rudimentary algebra in Ancient Egypt, including a document on linear equations that is one of the earliest mathematical proofs ever discovered. While the Ancient Greeks were better known for other forms of mathematics, they did devise a form of geometric algebra that used the sides of objects to represent algebraic terms. Mathematicians from present day India and China also developed early versions of algebra, with the modern algebraic term "Modus Indorum" referring specifically to an algebraic method devised in India.
Two of the most important people in the history of algebra are Diophantus and al-Khwarizmi. The former is frequently referred to as "the father of algebra," and his treatise "Arithmetica" was the first to use symbols to represent unknown numbers. Al-Khwarizmi was the first to distinguish algebra from geometry and arithmetic, and he pioneered the concept of balancing and reducing the sides of an equation. The word algebra refers to his work "Hidab al-Jabr wal-Muqubala," or "The Book of Restoration and Balance."
Since then, Europeans like Francois Viete and Gottfried Leibniz and scholars from other parts of the world, such as Seki Kowa, have refined mathematicians understanding of algebra.
Reference: reference.com
Tuesday 24 January 2017
Mathematician's
Math. It's one of those things that most people either love or hate. Those who fall on the hate side of things might still have nightmares of showing up for a high school math test unprepared, even years after graduation. Math is, by nature, an abstract subject, and it can be hard to wrap your head around it if you don't have a good teacher to guide you.
But even if you don't count yourself a fan of mathematics, it's hard to argue that it hasn't been a vital factor in our rapid evolution as a society. We reached the moon because of math. Math allowed us to tease out the secrets of DNA, create and transmit electricity over hundreds of miles to power our homes and offices, and gave rise to computers and all that they do for the world. Without math, we'd still be living in caves getting eaten by cave tigers.
Our history is rich with mathematicians who helped advance our collective understanding of math, but there are a few standouts whose brilliant work and intuitions pushed things in huge leaps and bounds. Their thoughts and discoveries continue to echo through the ages, reverberating today in our cellphones, satellites, hula hoops and automobiles. We picked five of the most brilliant mathematicians whose work continues to help shape our modern world, sometimes hundreds of years after their death. Enjoy!
Carl Gauss (1777-1855)
Isaac Newton is a hard act to follow, but if anyone can pull it off, it's Carl Gauss. If Newton is considered the greatest scientist of all time, Gauss could easily be called the greatest mathematician ever. Carl Friedrich Gauss was born to a poor family in Germany in 1777 and quickly showed himself to be a brilliant mathematician. He published "Arithmetical Investigations," a foundational textbook that laid out the tenets of number theory (the study of whole numbers). Without number theory, you could kiss computers goodbye. Computers operate, on a the most basic level, using just two digits — 1 and 0, and many of the advancements that we've made in using computers to solve problems are solved using number theory. Gauss was prolific, and his work on number theory was just a small part of his contribution to math; you can find his influence throughout algebra, statistics, geometry, optics, astronomy and many other subjects that underlie our modern world.
John von Neumann (1903-1957)
John von Neumann was born János Neumann in Budapest a few years after the start of the 20th century, a well-timed birth for all of us, for he went on to design the architecture underlying nearly every single computer built on the planet today. Right now, whatever device or computer that you are reading this on, be it phone or computer, is cycling through a series of basic steps billions of times over each second; steps that allow it to do things like render Internet articles and play videos and music, steps that were first thought up by John von Neumann.
Von Neumann received his Ph.D in mathematics at the age of 22 while also earning a degree in chemical engineering to appease his father, who was keen on his son having a good marketable skill. Thankfully for all of us, he stuck with math. In 1930, he went to work at Princeton University with Albert Einstein at the Institute of Advanced Study. Before his death in 1957, von Neumann made important discoveries in set theory, geometry, quantum mechanics, game theory, statistics, computer science and was a vital member of the Manhattan Project.
Reference:Wikipedia
Basic math formulas
Average formula:
Let a1,a2,a3,......,an be a set of numbers, average = (a1 + a2 + a3,+......+ an)/n
Percent:
Percent to fraction: x% = x/100
Percentage formula: Rate/100 = Percentage/base
Rate: The percent.
Base: The amount you are taking the percent of.
Percentage: The answer obtained by multiplying the base by the rate
Consumer math formulas:
Discount = list price × discount rate
Sale price = list price − discount
Discount rate = discount ÷ list price
Sales tax = price of item × tax rate
Interest = principal × rate of interest × time
Tips = cost of meals × tip rate
Commission = cost of service × commission rate
Geometry formulas:
Perimeter:
Perimeter of a square: s + s + s + s
s:length of one side
Perimeter of a rectangle: l + w + l + w
l: length
w: width
Perimeter of a triangle: a + b + c
a, b, and c: lengths of the 3 sides
Area:
Area of a square: s × s
s: length of one side
Area of a rectangle: l × w
l: length
w: width
Area of a triangle: (b × h)/2
b: length of base
h: length of height
Area of a trapezoid: (b1 + b2) × h/2
b1 and b2: parallel sides or the bases
h: length of height
volume:
Volume of a cube: s × s × s
s: length of one side
Volume of a box: l × w × h
l: length
w: width
h: height
Volume of a sphere: (4/3) × pi × r3
pi: 3.14
r: radius of sphere
Volume of a triangular prism: area of triangle × Height = (1/2 base × height) × Height
base: length of the base of the triangle
height: height of the triangle
Height: height of the triangular prism
Volume of a cylinder:pi × r2 × Height
pi: 3.14
r: radius of the circle of the base
Height: height of the cylinder
Maths equations
Some Important Maths Equations
1. a(b+c) = ab + ac
2. (a+b)2 = a2 + 2ab + b2
3. (a-b)2 = a2-2ab + b2
4. (a+b)2 = (a-b)2 + 4ab
5. (a-b)2 = (a+b)2 - 4ab
6. (a-b) (a+b) = a2 - b2
7. (a+b)3 = a3 + 3ab(a+b) + b3
8. (a-b)3 = a3 - 3ab(a-b) - b3
9. a3 + b 3 = (a+b) (a2 - ab + b2)
10. a3 - b 3 = (a-b) (a2 + ab + b2)
11. (a3 - b3) / (a2 + ab + b2) = a-b
12. (a3 + b3) / (a2 - ab + b2) = a+b
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How to use these equations in exams.
1. (4.75 x 4.75 x 4.75 + 1.25 x 1.25 x 1.25) / ( 4.75 x 4.75 + 1.25 x 1.25 - 4.75 x 1.25) = ?
If you saw a question like this, you can use these equations to find its answer. If try to solve these question without using equation you gonna take too much time , thats what the examiners want.
= (4.753 + 1.253) / (4.752 + 1.253 - 4.75 x 1.25)
now it is same like in the 12th equation (a3 + b3) / (a2 - ab + b2) = a+b
ie. = 4.75 + 1.25
= 6
Maths short cut tricks
Let me explain this rule by taking examples
If you had 343, you would double the last digit to get six, and subtract that from 34 to get 28.
If you get an answer divisible by 7 (including zero), then the original number is divisible by
seven. If you don't know the new number's divisibility, you can apply the rule again
EXAMPLE.
Example -:
623: 62 - (3 × 2) = 56 = 7 × 8
483: 48 − (3 × 2) = 42 = 7 × 6
MULTIPLICATION OF ANY TWO NUMBERS, LIES BETWEEN 11 AND 19
Let me explain this rule by taking examples
13*19 = (13+9)*10 + (3*9) = 220 + 27 = 247
Means add first number and last digit of the second number take zero in the third place of this number then add product of last digit of the two numbers in it.
EXAMPLE.
18*14 = (18+4)*10 + (8*4) = 220 + 32 = 252
If a number is divisible by two it will end in an even number or a 0.
EXAMPLE.
For example
28964
given number end with 2 then it will be divisible by 2.
and
89450
given number end with 0 it will also be divisible by 2.
If one is allowed as a prime, then any number could be written as a product of primes in many ways.
EXAMPLE.
24 = 1*2*2*2*3
or 24 = 1*1*2*2*2*3
or 24 = 1*1*1*1*1*2*2*2*3
The fact that factoring into primes can only be done in one way is important in mathematics.
If a number is divisible by three the sum of its digits will be divisible by 3.
EXAMPLE.
372 = 3 + 7 + 2 = 12
A corollary of this is that any number made by rearranging the digits of a number divisible by 3 will also be divisible by 3.
EXCEPTIONAL DIVISION RULE FOR 4
If a number is divisible by four the last two digit are divisible by 4 (or are zeros).
EXAMPLE.
45624
last two digit is 24
24 is divisible by 4 then 45624 is divisible by 4.
Reference:M4maths.com
Thursday 5 January 2017
Famous mathematicians
Greek philosopher and mathematician Pythagoras lived around the year 500 BC and is known for his Pythagorean theorem relating to the three sides of a right angle triangle: a² + b² = c²
Greek mathematician Euclid is often referred to as the ‘Father of Geometry’ for his revolutionary ideas and influential textbook called ‘Elements’ that he wrote around the year 300 BC.
Archimedes of Syracuse lived around the year 250 BC and among other things, developed a method for determining the volume of objects with irregular shapes.
Italian mathematician Leonardo of Pisa (better known as Fibonacci) lived between the years 1170 and 1250 and is best known today for Fibonacci numbers, the number sequence named after him. Fibonacci introduced the number sequence to Western Europe in his book ‘Liber Abaci’ after they had been described earlier by Indian mathematicians.
The Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ....
In the 17th century Galileo Galilei and Johannes Kepler made important discoveries relating to planetary motion and orbits.
German mathematician Gottfried Leibniz lived between 1646 and 1716, developing important calculus concepts and mathematical notation practices.
Isaac Newton discovered the laws of physics and brought together many important concepts of infinitesimal calculus.
Much of the work done by Leibniz and Newton is based on theories by French philosopher Rene Descartes. As well as his many contributions to philosophy, Descartes also had a huge impact on mathematics, creating analytical geometry, developing a system that describes geometry using algebra, contributing to optics and much more.
Born in France, Pierre de Fermat was an amateur mathematician who is best known for Fermat’s Last Theorem.
In 1642 French mathematician Blaise Pascal invented the mechanical calculator.
Swiss mathematician Leonhard Euler was probably the most influential mathematician of the 18th century, making discoveries in graph theory and introducing many modern mathematical words and notations among other things.
Born in 1777, German mathematician Carl Friedrich Gauss contributed brilliant work in geometry, statistics, number theories, algebra and much more.
Bernhard Riemann was an influential German mathematician who contributed to differential geometry and analysis, paving the way for the development of general relativity by Albert Einstein.
Born in 1882, Emmy Noether was a German mathematician who made important contributions to abstract algebra and theoretical physics, described by Einstein as the most important woman in the history of mathematics.
Alan Turing was a British mathematician and computer scientist who cracked German ciphers (codes) in the Second World War, contributed to mathematical logic and played an important role in the development of algorithms, artificial intelligence and the modern computer.
Born in 1953, British mathematician Andrew Wiles is most famous for proving Fermat’s Last Theorem.
Reference:math facts
Top ten facts about maths
1. In 2010 on World Maths Day, 1.13 million students from more than 235 countries set a record correctly answering 479,732,613 questions.
2. Americans called mathematics ‘math’, arguing that ‘mathematics’ functions as a singular noun so ‘math’ should be singular too.
3. They have been calling maths ‘math’ for much longer than we have called it ‘maths’.
4. ‘Mathematics’ is an anagram of ‘me asthmatic’.
5. The only number in English that is spelled with its letters in alphabetical order is ‘forty’.
6. The only Shakespeare play to include the word ‘mathematics’ is The Taming Of The Shrew.
7. Notches on animal bones show that people have been doing mathematics, or at least making computations, since around 30,000BC.
8. The word ‘hundrath’ in Old Norse, from which our ‘hundred’ derives, meant not 100 but 120.
9. “Pure mathematics is, in its way, the poetry of logical ideas.” (Albert Einstein).
10. “Mathematics [is] the subject in which we never know what we are talking about nor whether what we are saying is true.” (Bertrand Russell).