Monday, November 27, 2023

What Does Both Mean In Math

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How Converting Between Addition And Multiplication Makes Math Easier

Finding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy

The power of logarithms, zero and the word or

Multivariable calculus was never my best subjectI dont have great intuition about graphs and equations, so I often had trouble visualizing the questions. I tend to approach shapes more like a topologist;or a toddler with some Play-Doh: the exact equation isnt terribly important to me, just the large-scale features. I think about mathematical shapes as if they are made of clay, not graphed using precise formulas. All that to say I still feel a little intimidated when someone shows me an equation and Im supposed to know what it will look like if you graph it, or when Im supposed to make the picture I see using equations.

Earlier this summer at an enjoyable conference about illustrating mathematics, one of the speakers introduced us to Surfer, and I started playing. Surfer is a free program you can download from the open mathematics website Imaginary, and it has an easy learning curve. The program has a lot of equations and their graphs built in, and you can start tinkering with them as soon as youve installed it.

While we were playing with Surfer, another conference attendee made a shape that astonished me: two linked hollow rings, or tori, pictured at the top of this post. How did he do that? I wondered, It must be so difficult to find an equation that will create both of those tori at once!

To see what I mean, lets look at how to graph an equation in Surfer.;

What Is The Difference Between Only If And Iff

I have read this question. I am now stuck with the difference between “if and only if” and “only if“. Please help me out.


  • 1$\begingroup$Also try to understand in terms of plain translation. AiffB means A is true ‘if’ B is true & A is true ‘only if’ B is true.The ‘only if’ means that A is true in no other cases.’A if B’ can be written as B => A.And ‘A only if B’ can be written as notB => notA. It is the property of => sign that c=>d is same as notd=>notc. Thus , you can replace notB=>notA by A=>B. Thus A iff B can be written as A=>B and B=>A . Of course what I am saying is same as what others have already said . I just wanted to emphasise how we can intuitively try to understand the logic from the meaning of ‘if’ and ‘only if’.$\endgroup$Feb 11 ’14 at 11:18
  • 1Jun 7 ’15 at 12:52
  • 4$\begingroup$The mathematician R.L. Moore used “only if” to mean “if and only if”. This sounds weird to us now, because it goes against the accepted convention, but I can see what Moore was thinking. The statement “A only if B” sounds like the statement “A if B”, except that you are also given an extra piece of information: not just A if B, but A only if B.$\endgroup$

Let’s assume A and B are two statements. Then to say “A only if B” means that A can only ever be true when B is true. That is, B is necessary for A to be true. To say “A if and only if B” means that A is true if B is true, and B is true if A is true. That is, A is necessary and sufficient for B. Succinctly,

is the same as saying

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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Life’s Extremes: Math Vs Language

06 November 2011

In this weekly series, LiveScience examines the psychology and sociology of opposite human behavior and personality types.

Do you know what “abecedarian” means? What about the solution to 250 x 11?

Most people would agree they are better at verbal or math subjects in school, as grades usually do attest. Highly intelligent individuals often do well in both subjects, and may know the answers to both questions above, lickety-split, while less intelligent people can struggle. But a minority of us excels in the language department and bombs at mathematics, or vice versa. ;

These extremes in ability speak to the very makeup of our brains. “The brain systems for maths and language are quite different,” said Brian Butterworth, emeritus professor of cognitive neuropsychology at University College London, using British English’s dialect for “math.” “So perhaps it is not surprising that these two capacities are rather independent.”

By learning more about the regions of our brains responsible for language and math processing, researchers hope to someday better help those with severe deficits, such as in reading ability, called dyslexia, and general numeracy, called dyscalculia.;

Wordly wise

Verbal ability reading, writing and speaking arises from across much of our brain, requiring key elements to harmonize. ;

In the case of well-studied dyslexia, several candidate genes have emerged that code for how neurons in the brain form interconnections.

A head for numbers

E In Scientific Notation And The Meaning Of 1e6

What does both mean in math.

You don’t need a calculator to use E to express a number in scientific notation. You can simply let E stand for the base root of an exponent, but only when the base is 10. You wouldn’t use E to stand for base 8, 4 or any other base, especially if the base is Euler’s number, e.

When you use E in this way, you write the number xEy, where x is the first set of integers in the number and y is the exponent. For example, you would write the number 1 million as 1E6. In regular scientific notation, this is 1 × 106, or 1 followed by 6 zeros. Similarly 5 million would be 5E6, and 42,732 would be 4.27E4. When writing a number in scientific notation, whether you use E or not, you usually round to two decimal places.

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Iv Distributive Property Of Multiplication Over Addition

Multiplication distributes over Addition

Multiplying a factor to;a group of real numbers that are being added;together is equal to the;sum of the products of the factor;and each;addend;in the parenthesis.

In other words, adding two or more real numbers and multiplying;it to an outside number is the same as multiplying the outside number to every number inside the parenthesis, then adding their products.


The following is the summary of the properties of real numbers discussed;above:

+ Addition Plus Positive

The addition symbol + is usually used to indicate that two or more numbers should be added together, for example, 2 + 2.

The + symbol can also be used to indicate a positive number although this is less common, for example, +2. Our page on Positive and Negative Numbers explains that a number without a sign is considered to be positive, so the plus is not usually necessary.

See our page on Addition for more.

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Multiplication Property Of Equality

Algebra plays an important role in mathematics. One of the fundamental algebraic concepts states that an equation is a mathematical sentence with an equal sign. We can translate day to day activities and transactions to algebraic equations.;;

Example: 5 + 3 = 8

;;;;;;;;12 7 = 5

Balanced operations of addition, subtraction, multiplication, and division on both sides do not change the truth value of any equation.

The multiplication property of equality states that when we multiply both sides of an equation by the same number, the two sides remain equal.

That is, if a, b, and c are real numbers such that a = b, then

;;;;;;;;;;;a × c = b × c

Example 1 : Lisa and Linda have got the same amount of money. If both of them double their money, that is both of them multiply their money by 2; they still have the same amount of money.;

Note that the property holds true even when the multiplicand is zero as zero times any number is zero.;

We use this property to solve equations.

Example 2: x4;= 5

Multiply both sides by 4.

x4;× 4 = 5 × 4;

x = 20;

To check we can substitute the value of x in the original equation.

204;= 5;

5 = 5;

Example 3: One-fourth of the kids who visited the amusement park Jump & Slide on holiday tried their new ride loop-O-loop. If 75 kids tried the ride, how many kids visited the park that day?

Let a be the number of kids who visited the park that day. One-fourth of this number is given to be 75. That is,

a4;= 75

You need to solve the equation for a.

a4;× 4 = 75 × 4;

a = 300;

What Does If And Only If Mean In Mathematics

Algebra Basics: Exponents In Algebra – Math Antics

To understand if and only if, we must first know what is meant by a conditional statement. A conditional statement is one that is formed from two other statements, which we will denote by P and Q. To form a conditional statement, we could say if P then Q.

The following are examples of this kind of statement:

  • If it is raining outside, then I take my umbrella with me on my walk.
  • If you study hard, then you will earn an A.
  • If n is divisible by 4, then n is divisible by 2.

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The Associative Property For Products And Sums

The associative property means that if you are performing an arithmetic operation on more than two numbers, you can associate or put brackets around two of the numbers without affecting the answer. Products and sums have the associative property while differences and quotients do not.

For example, if an arithmetical operation is performed on the numbers 12, 4 and 2, the sum can be calculated as

A product example is

But for quotients

and for differences

Multiplication and addition have the associative property while division and subtraction do not.

Advanced And Rarely Used Logical Symbols

These symbols are sorted by their Unicode value:

  • U+0305;;COMBINING OVERLINE, used as abbreviation for standard numerals . For example, using HTML style “4” is a shorthand for the standard numeral “SSSS0”.
  • Overline is also a rarely used format for denoting Gödel numbers: for example, “A B” says the Gödel number of “”.
  • Overline is also a way for denoting negation used primarily in electronics: for example, “A B” is the same as “¬”.
  • U+2191UPWARDS ARROW or U+007C|VERTICAL LINE: Sheffer stroke, the sign for the NAND operator .
  • U+2193DOWNWARDS ARROWPeirce Arrow, the sign for the NOR operator .
  • U+2299CIRCLED DOT OPERATOR the sign for the XNOR operator .
  • U+2235BECAUSE: because
  • U+22A7MODELS: is a model of ” rel=”nofollow”>valuation satisfying”)
  • U+22A8TRUE: is true of
  • U+22ACDOES NOT PROVE: negated , the sign for “does not prove”, for example T P says “P is not a theorem of T
  • U+22ADNOT TRUE: is not true of
  • U+2020DAGGER: Affirmation operator
  • U+22BCNAND: NAND operator.
  • U+22BDNOR: NOR operator.
  • U+25C7WHITE DIAMOND: modal operator for “it is possible that”, “it is not necessarily not” or rarely “it is not probably not”
  • U+22C6STAR OPERATOR: usually used for ad-hoc operators
  • U+22A5UP TACK or U+2193DOWNWARDS ARROW: Webb-operator or Peirce arrow, the sign for NOR. Confusingly, “” is also the sign for contradiction or absurdity.
  • U+27DBLEFT AND RIGHT TACK: semantic equivalent
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    A Note On Equal Precedence

    Since multiplication and division are of equal precedence, it may be helpful to think of dividing by a number as multiplying by the reciprocal of that number. Thus 3 \div 4 = 3 \cdot \frac. In other words, the quotient of 3 and 4 equals the product of 3 and \frac.

    Similarly, as addition and subtraction are of equal precedence, we can think of subtracting a number as the same as adding the negative of that number. Thus 34=3+. In other words, the difference of 3 and 4 equals the sum of positive three and negative four.

    With this understanding, think of 13+7 as the sum of 1, negative 3, and 7, and then add these terms together. Now that youve reframed the operations, any order will work:

    • +7=2+7=5
    • ;+1=4+1=5

    The important thing is to keep the negative sign with any negative number .

    Examples Of Both In A Sentence

    What does both mean in math.

    bothThe Mercury NewsbothThe KnowOrange County RegisterbothThe CannabistBothTwin CitiesbothThe Denver Postboth clevelandboth oregonliveboth New York Timesboth WSJboth House Beautifulboth Essenceboth Quartz Indiaboth Dallas News

    These example sentences are selected automatically from various online news sources to reflect current usage of the word ‘both.’ Views expressed in the examples do not represent the opinion of Merriam-Webster or its editors. Send us feedback.

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    Where Does Euler’s Number E Come From

    The number represented by e was discovered by mathematician Leonard Euler as a solution to a problem posed by another mathematician, Jacob Bernoulli, 50 years earlier. Bernoulli’s problem was a financial one.

    Suppose you put $1,000 in a bank that pays 100% annual compound interest and leave it there for a year. You’ll have $2,000. Now suppose the interest rate is half that, but the bank pays it twice a year. At the end of a year, you’d have $2,250. Now suppose the bank paid only 8.33%, which is 1/12 of 100%, but paid it 12 times a year. At the end of the year, you’d have $2,613. The general equation for this progression is:

    where r is 1 and n is the payment period.

    It turns out that, as n approaches infinity, the result gets closer and closer to e, which is 2.7182818284 to 10 decimal places. This is how Euler discovered it. The maximum return you could get on an investment of $1,000 in one year would be $2,718.

    List Of Logic Symbols

    This article contains logic symbols. Without proper rendering support, you may see question marks, boxes, or other symbolsinstead of logic symbols.

    In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subsequent columns contains an informal explanation, a short example, the Unicode location, the name for use in HTML documents, and the LaTeX symbol.

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    Symbols That Do Not Belong To Formulas

    In this section, the symbols that are listed are used as some sorts of punctuation marks in mathematical reasoning, or as abbreviations of English phrases. They are generally not used inside a formula. Some were used in classical logic for indicating the logical dependence between sentences written in plain English. Except for the first two, they are normally not used in printed mathematical texts since, for readability, it is generally recommended to have at least one word between two formulas. However, they are still used on a black board for indicating relationships between formulas.

    Used for marking the end of a proof and separating it from the current text. The initialismQ.E.D. or QED is often used for the same purpose, either in its upper-case form or in lower case.
    Bourbaki dangerous bend symbol: Sometimes used in the margin to forewarn readers against serious errors, where they risk falling, or to mark a passage that is tricky on a first reading because of an especially subtle argument.
    Abbreviation of “therefore”. Placed between two assertions, it means that the first one implies the second one. For example: “All humans are mortal, and Socrates is a human. Socrates is mortal.”
    Abbreviation of “because” or “since”. Placed between two assertions, it means that the first one is implied by the second one. For example: “11 is prime it has no positive integer factors other than itself and one.”

    Why Subtraction And Division Are Not Associative

    Algebra Shortcut Trick – how to solve equations instantly (2)

    If we want Associative Property to work with subtraction and division, changing the way on how we group the numbers should not affect the result.

    Associative Property for Subtraction

    Does the problem \left – c = a – \left hold?

    These examples clearly show;that changing the grouping of numbers in subtraction;yield different answers. Thus, associativity is not a property of subtraction.

    Associative Property for Division

    Does the property \left \div c = a \div \left hold?

    I hope this single example seals the deal that changing how you group numbers when dividing indeed affect the outcome. Therefore, associativity is not a property of division.


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    The Meaning Of The Product Of A Number

    The product of a number and one or more other numbers is the value obtained when the numbers are multiplied together. For example, the product of 2, 5 and 7 is

    While the product obtained by multiplying specific numbers together is always the same, products are not unique. The product of 6 and 4 is always 24, but so is the product of 2 and 12, or 8 and 3. No matter which numbers you multiply to obtain a product, the multiplication operation has four properties that distinguish it from other basic arithmetic operations, Addition, subtraction and division share some of these properties, but each has a unique combination.

    Look Up The Meaning Of Math Words

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    • Ph.D., Biomedical Sciences, University of Tennessee at Knoxville
    • B.A., Physics and Mathematics, Hastings College

    This is a glossary of common mathematical terms used in arithmetic, geometry, algebra, and statistics.

    Abacus:;An early counting tool used for basic arithmetic.

    Absolute Value:;Always a positive number, absolute value refers to the distance of a number from 0.

    Acute Angle:;An angle whose measure is between 0° and 90° or with less than 90° radians.

    Addend:;A number involved in an addition problem; numbers being added are called addends.

    Algebra: The branch of mathematics that substitutes letters for numbers to solve for unknown values.

    Algorithm: A procedure or set of steps used to solve a mathematical computation.

    Angle: Two rays sharing the same endpoint .

    Angle Bisector: The line dividing an angle into two equal angles.

    Area: The two-dimensional space taken up by an object or shape, given in square units.

    Array: A set of numbers or objects that follow a specific pattern.

    Attribute: A characteristic or feature of an objectsuch as size, shape, color, etc.that allows it to be grouped.

    Average: The average is the same as the mean. Add up a series of numbers and divide the sum by the total number of values to find the average.

    Base: The bottom of a shape or three-dimensional object, what an object rests on.

    Base 10: Number system that assigns place value to numbers.

    Bar Graph: A graph that represents data visually using bars of different heights or lengths.

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