Mathematical Logic And Set Theory
The two subjects of mathematical logic and set theory have both belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy, and was not specifically studied by mathematicians.
Before Cantor‘s study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor’s work offended many mathematicians not only by considering actually infinite sets, but by showing that this implies different sizes of infinity and the existence of mathematical objects that cannot be computed, or even explicitly described . This led to the controversy over Cantor’s set theory.
In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. Examples of such intuitive definitions are “a set is a collection of objects”, “natural number is what is used for counting”, “a point is a shape with a zero length in every direction”, “a curve is a trace left by a moving point”, etc.
This approach of the foundations of the mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic, which explicitly lacks the law of excluded middle.
Modular Forms And Theta Functions
The constant is connected in a deep way with the theory of modular forms and theta functions. For example, the Chudnovsky algorithm involves in an essential way the j-invariant of an elliptic curve.
- . }\int _^}.}
The constant is the unique normalizing factor such that H defines a linear complex structure on the Hilbert space of square-integrable real-valued functions on the real line. The Hilbert transform, like the Fourier transform, can be characterized purely in terms of its transformation properties on the Hilbert space L2: up to a normalization factor, it is the unique bounded linear operator that commutes with positive dilations and anti-commutes with all reflections of the real line. The constant is the unique normalizing factor that makes this transformation unitary.
Fourier Transform And Heisenberg Uncertainty Principle
The constant also appears as a critical spectral parameter in the Fourier transform. This is the integral transform, that takes a complex-valued integrable function f on the real line to the function defined as:
- f . }=\int _^fe^\,dx.}
Although there are several different conventions for the Fourier transform and its inverse, any such convention must involve somewhere. The above is the most canonical definition, however, giving the unique unitary operator on L2 that is also an algebra homomorphism of L1 to L.
The Heisenberg uncertainty principle also contains the number . The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: with our conventions for the Fourier transform,
- |^\,dx\right)\left|^\,d\xi \right)\geq \left|^\,dx\right)^.}
The physical consequence, about the uncertainty in simultaneous position and momentum observations of a quantum mechanical system, is discussed below. The appearance of in the formulae of Fourier analysis is ultimately a consequence of the Stonevon Neumann theorem, asserting the uniqueness of the Schrödinger representation of the Heisenberg group.
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< Less Than And > Greater Than
This symbol < means less than, for example 2 < 4 means that 2 is less than 4.
This symbol > means greater than, for example 4 > 2.
These symbols mean less than or equal to and greater than or equal to and are commonly used in algebra. In computer applications < = and > = are used.
These symbols are less common and mean much less than, or much greater than.
Ten Frames Help Develop Subitizing Skills
What on earth is subitizing?, I hear you ask.
Subitizing means that you can tell how many objects there are without having to count each individual one.
So for example when you roll a dice, you can tell what number you have rolled without having to count each dot on the dice.
Young children start out by having to count each individual object. As they develop, they begin to be able to see a group of 2 or of 3 without having to count each object.
So there you go! Who knew the little old ten frame could have so much going for it?!
And now you know why theyre so important, we will turn our attention to how to practise them.
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Number Theory And Riemann Zeta Function
Prüfer groupL-functionspWeil conjecturehyperbolicmodular group
The Riemann zeta function is used in many areas of mathematics. When evaluated at s = 2 it can be written as
- + }}+}}+}}+\cdots }
Finding a simple solution for this infinite series was a famous problem in mathematics called the Basel problem. Leonhard Euler solved it in 1735 when he showed it was equal to 2/6. Euler’s result leads to the number theory result that the probability of two random numbers being relatively prime is equal to 6/2. This probability is based on the observation that the probability that any number is divisible by a prime p is 1/p Hence the probability that two numbers are both divisible by this prime is 1/p2, and the probability that at least one of them is not is 1 1/p2. For distinct primes, these divisibility events are mutually independent so the probability that two numbers are relatively prime is given by a product over all primes:
- % . \prod _^\left& =\left^\\& =}}+}}+\cdots }}\\& =}=}}\approx 61\%.\end}}
This probability can be used in conjunction with a random number generator to approximate using a Monte Carlo approach.
The zeta function also satisfies Riemann’s functional equation, which involves as well as the gamma function:
- . }.}
Numbers Bigger Than A Trillion
The digit zero plays an important role as you count very large numbers. It helps track these multiples of 10 because the larger the number is, the more zeroes are needed. In the table below, the first column lists the name of the number, the second provides the number of zeros that follow the initial digit, and the third tells you how many groups of three zeros you would need to write out each number.
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What Is Special About Polynomials
Because of the strict definition, polynomials are easy to work with.
For example we know that:
- If you add polynomials you get a polynomial
- If you multiply polynomials you get a polynomial
So you can do lots of additions and multiplications, and still have a polynomial as the result.
Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines.
Math Problems Sorted By Engineering Topics
If you are looking for math problems related to engineering topics, here you will find problems organized by topic areas. Click on the topic in the grid below to go to the associated problem listing.
Design Issues | Rocketry, Launches, and Launch Vehicles | Telescopes and Remote Sensing | Properties of Orbits | Data and Telemetry | Mission Planning | Spacecraft Design
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How Many Times Vs How Many Times Greater
I keep on coming into these problems while my gre prepration
How many times vs how many times greater or
a is what % of b or a is what % greater than b
What is the right convention to solve these problems? Some websites treat both of these scenarios similar? However I believe that in case of larger you have to do
if how many times b greater than ado b-a/a.
Let’s say a=9, b=3
So I get these answers. Are these answers correct?
1: a is 3 times b.2: a is 2 times greater than b.3: a is 300% times b.4: a is 200% times greater than b.
My confusion is that in this question, author does the opposite of what he should do.
- $\begingroup$I am with you, but this may be rather a linguistic question and interpretation 2 is probably not as widespread $\endgroup$
The phrase “A is what % of B” should be written as $A=x\cdot B$. And now solve for x, and then multiply by 100.
Example 1a: If A is 100, and B is 50, then $100=x\cdot 50$, means that $x = 2$, and A is 200% of B.
The phrase “A is what % greater than B,” should be written as $A=x\cdot B$, just as before. But now, when you solve for x, and multiply by 100, you want to take the additional step of subtracting 100. Notice that this will only work if A is actually greater than B.
Example 1b: In the above example, A would be 100% greater than B.
Example 2: if A is 150, and B is 100, then solving for x in $A=x\cdot B$, would give us $x = 1.5$, and so A is 150% of B. But A is 50% greater than B.
What Is The Definition Of Mathematics
Mathematics simply means to learn or to study or gain knowledge. The theories and concepts given in mathematics help us understand and solve various types of problems in academic as well as in real life situations.
Mathematics is a subject of logic. Learning mathematics will help students to grow their problem-solving and logical reasoning skills. Solving mathematical problems is one of the best brain exercises.
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What Does How Many More Mean
By Team MeaningKosh – 1 August 2022
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Table Of Content:
how many more means how many persons having more than one person. how much more means the measurement/quantity more than entity/person.
Math Problems Sorted By Space Science Topic
Here you will find hundreds of math problems related to all of the major astronomical objects from asteroids and planets to galaxies and black holes! Click on the topic below to see which problems are available.
Earth | Moon | Sun | Planets | Stars | Universe | Space Travel | Astrobiology | Black Holes
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What Are Common Keywords For Word Problems
The following is a listing of most of the more-common keywords for word problems:
sold for, cost
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Note that “per”, in “Division”, means “divided by”, as in “I drove 90 miles on three gallons of gas, so I got 30 miles per gallon”. Also, “a” sometimes means “divided by”, as in “When I tanked up, I paid $12.36 for three gallons, so the gas was $4.12 a gallon”.
Warning: The “less than” construction, in “Subtraction”, is backwards in the English from what it is in the math. If you need, for instance, to translate “1.5 less than x“, the temptation is to write “1.5 x“. Do not do this!
You can see how this is wrong by using this construction in a “real world” situation: Consider the statement, “He makes $1.50 an hour less than me.” You do not figure his wage by subtracting your wage from $1.50. Instead, you subtract $1.50 from your wage. So remember: the “less than” construction is backwards.
Also note that order is important in the “quotient/ratio of” and “difference between/of” constructions. If a problems says “the ratio of x and y“, it means “x divided by y“, not “y divided by x“. If the problem says “the difference of x and y“, it means “x y“, not “y x“.
You’ll be expected to know that a “dozen” is twelve you may be expected to know that a “score” is twenty. You’ll be expected to know the number of days in a year, the number of hours in a day, and other basic units of measure.
ten divided by one-half:
Common Mathematical Symbols And Terminology: Maths Glossary
Mathematical symbols and terminology can be confusing and can be a barrier to learning and understanding basic numeracy.
This page complements our numeracy skills pages and provides a quick glossary of common mathematical symbols and terminology with concise definitions.
Are we missing something? Get it touch to let us know.
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Make Your Own Ten Frame
If your child is new to ten frames, a fun way to start is by exploring ten them in a hands-on way.
Draw out a ten frame on a big piece of paper or whiteboard.
Next grab some items you can place inside the ten frame. You could use toy cars, blocks, stuffies or something along those lines. In our case, weve used our menagerie of small plastic animals.
Have your child choose a number and then show it in the frame by placing one counter/object per box.
For example, the number 4:
Challenge your child put those 4 objects in a different pattern, like so:
How many different ways can you show the number 4 in the ten frame?
Ask your child how many more objects you would need to add to make 10. Help them count the empty squares. Then switch numbers and try a different one.
Another thing you could do is take it in turns to make a number on the ten frame and then have the other person say what number they see. This gives your child practice at both constructing numbers in the ten frame and also recognising them.
Although these activities are quick and simple things to try, hands-on math like this is really important. Its a way for young children to make sense of new math ideas by doing.
Adoption Of The Symbol
In the earliest usages, the Greek letter was used to denote the semiperimeter of a circle. and was combined in ratios with or to form circle constants. The first recorded use is Oughtred’s ” “, to express the ratio of periphery and diameter in the 1647 and later editions of Clavis Mathematicae.Barrow likewise used ” ” to represent 6.28… .
The earliest known use of the Greek letter alone to represent the ratio of a circle’s circumference to its diameter was by Welsh mathematician William Jones in his 1706 work Synopsis Palmariorum Matheseos or, a New Introduction to the Mathematics. The Greek letter first appears there in the phrase “1/2 Periphery ” in the discussion of a circle with radius one. However, he writes that his equations for are from the “ready pen of the truly ingenious Mr. John Machin“, leading to speculation that Machin may have employed the Greek letter before Jones. Jones’ notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767.
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In The Mandelbrot Set
An occurrence of in the fractal called the Mandelbrot set was discovered by David Boll in 1991. He examined the behaviour of the Mandelbrot set near the “neck” at . When the number of iterations until divergence for the point is multiplied by , the result approaches as approaches zero. The point at the cusp of the large “valley” on the right side of the Mandelbrot set behaves similarly: the number of iterations until divergence multiplied by the square root of tends to .
What Does How Many Mean In Math
For this case we use the question “how many” to refer to things that we can count numerically.
For example, we can say:
How many oranges are there?
How many numbers are between 1 and 10?
How many people are in the car?
We must differentiate this question from “how much”. This last question we must use it for amounts that we can not count exactly.
For example, we can say:
How much orange juice is in the jar?
How much flour is placed in the food?
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Awards And Prize Problems
The most prestigious award in mathematics is the Fields Medal, established in 1936 and awarded every four years to up to four individuals. It is considered the mathematical equivalent of the Nobel Prize.
Other prestigious mathematics awards include:
- The Abel Prize, instituted in 2002 and first awarded in 2003
- The Chern Medal for lifetime achievement, introduced in 2009 and first awarded in 2010
- The Wolf Prize in Mathematics, also for lifetime achievement, instituted in 1978
A famous list of 23 open problems, called “Hilbert’s problems“, was compiled in 1900 by German mathematician David Hilbert. This list has achieved great celebrity among mathematicians, and, as of 2022, at least thirteen of the problems have been solved.
A new list of seven important problems, titled the “Millennium Prize Problems“, was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert’s problems. A solution to any of these problems carries a 1 million dollar reward. To date, only one of these problems, the Poincaré conjecture, has been solved.