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Significant Figures
The SIGNIFICANT FIGURES of a number are digits that carry meaning contributing to its measurement resolution . This includes all digits except: * All leading zeros ; * Trailing zeros when they are merely placeholders to indicate the scale of the number (exact rules are explained at identifying significant figures ); and * Spurious digits introduced, for example, by calculations carried out to greater precision than that of the original data, or measurements reported to a greater precision than the equipment supports. Significance arithmetic are approximate rules for roughly maintaining significance throughout a computation. The more sophisticated scientific rules are known as propagation of uncertainty
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Kinetic Energy
Ek = ½m v 2 Ek = Et+Er CLASSICAL MECHANICS F = m a {displaystyle {vec {F}}=m{vec {a}}} Second law of motion * History * Timeline Branches * Applied * Celestial * Continuum * Dynamics * Kinematics
Kinematics
* Kinetics * Statics * Statistical Fundamentals *
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Positional Notation
POSITIONAL NOTATION or PLACE-VALUE NOTATION is a method of representing or encoding numbers . Positional notation
Positional notation
is distinguished from other notations (such as Roman numerals ) for its use of the same symbol for the different orders of magnitude (for example, the "ones place", "tens place", "hundreds place"). This greatly simplified arithmetic , leading to the rapid spread of the notation across the world. With the use of a radix point (decimal point in base-10), the notation can be extended to include fractions and the numeric expansions of real numbers . The Babylonian numeral system , base-60, was the first positional system developed, and its influence is present today in the way time and angles are counted in tallies related to 60, like 60 minutes in an hour, 360 degrees in a circle
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Base (exponentiation)
In exponentiation , the BASE is the number b in an expression of the form bn. CONTENTS * 1 Related terms * 2 Roots * 3 Logarithms * 4 See also RELATED TERMSThe number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b. It is more commonly expressed as "the nth power of b", "b to the nth power" or "b to the power n". For example, the fourth power of 10 is 10,000 because 104 = 10 × 10 × 10 × 10 = 10,000. The term power strictly refers to the entire expression, but is sometimes used to refer to the exponent. ROOTSWhen the nth power of b equals a number a, or a = bn, then b is called an "nth root " of a. For example, 10 is a fourth root of 10,000. LOGARITHMSThe inverse function to exponentiation with base b (when it is well-defined ) is called the logarithm to base b, denoted logb. Thus: logb a = n. For example, log10 10,000 = 4
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Logarithm
In mathematics , the LOGARITHM is the inverse operation to exponentiation , just as division is the inverse of multiplication 10 is used as a factor three times. More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm can be calculated for any two positive real numbers b and x where b is not equal to 1. The logarithm of x to base b, denoted logb(x), is the unique real number y such that by = x. For example, log2(64) = 6, as 64 = 26. The logarithm to base 10 (that is b = 10) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the number e (≈ 2.718) as its base; its use is widespread in mathematics and physics , because of its simpler derivative . The binary logarithm uses base 2 (that is b = 2) and is commonly used in computer science
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Area Of A Disk
In geometry , the area enclosed by a circle of radius r is π r2. Here the Greek letter π represents a constant , approximately equal to 3.14159, which is equal to the ratio of the circumference of any circle to its diameter . One method of deriving this formula, which originated with Archimedes , involves viewing the circle as the limit of a sequence of regular polygons . The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides , and the corresponding formula (that the area is half the perimeter times the radius, i.e.  1⁄2 × 2πr × r) holds in the limit for a circle. Although often referred to as the AREA OF A CIRCLE in informal contexts, strictly speaking the term disk refers to the interior of the circle, while circle is reserved for the boundary only, which is a curve and covers no area itself. Therefore, the AREA OF A DISK is the more precise phrase for the area enclosed by a circle
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Absolute Value
In mathematics , the ABSOLUTE VALUE or MODULUS x of a real number x is the non-negative value of x without regard to its sign . Namely, x = x for a positive x, x = −x for a negative x (in which case −x is positive), and 0 = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers , the quaternions , ordered rings , fields and vector spaces . The absolute value is closely related to the notions of magnitude , distance , and norm in various mathematical and physical contexts
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Plus-minus Sign
The PLUS-MINUS SIGN (±) is a mathematical symbol with multiple meanings. * In mathematics , it generally indicates a choice of exactly two possible values, one of which is the negation of the other. * In experimental sciences , the sign commonly indicates the confidence interval or error in a measurement, often the standard deviation or standard error . The sign may also represent an inclusive range of values that a reading might have. * In engineering the sign indicates the tolerance , which is the range of values that are considered to be acceptable, safe, or which comply with some standard, or with a contract. * In botany it is used in morphological descriptions to notate "more or less". * In chemistry the sign is used to indicate a racemic mixture . * In chess , the sign indicates a clear advantage for the white player; the complementary sign ∓ indicates the same advantage for the black player. The sign is normally pronounced "plus or minus"
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Significand
The SIGNIFICAND (also MANTISSA or COEFFICIENT) is part of a number in scientific notation or a floating-point number , consisting of its significant digits . Depending on the interpretation of the exponent , the significand may represent an integer or a fraction . The word mantissa seems to have been introduced by Arthur Burks in 1946 writing for the Institute for Advanced Study
Institute for Advanced Study
at Princeton , although this use of the word is discouraged by the IEEE floating-point standard committee as well as some professionals such as the creator of the standard William Kahan . CONTENTS * 1 Example * 2 Significands and the hidden bit * 3 Use of "mantissa" * 4 See also * 5 References EXAMPLEThe number 123.45 can be represented as a decimal floating-point number with the integer 12345 as the significand and a 10−2 power term, also called characteristics , where −2 is the exponent (and 10 is the base)
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Floor And Ceiling Functions
In mathematics and computer science , the FLOOR FUNCTION is the function that takes as input a real number x {displaystyle x} and gives as output the greatest integer floor ( x ) = x {displaystyle {text{floor}}(x)=lfloor xrfloor } that is less than or equal to x {displaystyle x} . Similarly, the CEILING FUNCTION maps x {displaystyle x} to the least integer ceiling ( x ) = x {displaystyle {text{ceiling}}(x)=lceil xrceil } that is greater than or equal to x {displaystyle x}
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Normalized Number
In applied mathematics , a number is NORMALIZED when it is written in scientific notation with one nonzero decimal digit before the decimal point. Thus, a real number when written out in normalized scientific notation is as follows: d 0 . d 1 d 2 d 3 10 n {displaystyle pm d_{0}.d_{1}d_{2}d_{3}dots times 10^{n}} where n is an integer , d 0 , {displaystyle d_{0},} d 1 , {displaystyle d_{1},} d 2 {displaystyle d_{2}} , d 3 {displaystyle d_{3}} ... are the digits of the number in base 10, and d 0 {displaystyle d_{0}} is not zero. That is, its leading digit (i.e. leftmost) is not zero and is followed by the decimal point. This is the standard form of scientific notation . An alternative style is to have the first non-zero digit after the decimal point
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Common Logarithm
In mathematics , the COMMON LOGARITHM is the logarithm with base 10. It is also known as the DECADIC LOGARITHM and also as the DECIMAL LOGARITHM, named after its base, or BRIGGSIAN LOGARITHM, after Henry Briggs , an English mathematician who pioneered its use, as well as "standard logarithm". Historically, it was known as logarithmus decimalis or logarithmus decadis. It is indicated by log10(x), or sometimes Log(x) with a capital L (however, this notation is ambiguous since it can also mean the complex natural logarithmic multi-valued function ). On calculators it is usually "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when they write "log". To mitigate this ambiguity the ISO 80000 specification recommends that log10(x) should be written lg (x) and loge(x) should be ln (x)
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Benford's Law
BENFORD\'S LAW, also called the FIRST-DIGIT LAW, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data . The law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small. For example, in sets that obey the law, the number 1 appears as the most significant digit about 30% of the time, while 9 appears as the most significant digit less than 5% of the time. By contrast, if the digits were distributed uniformly, they would each occur about 11.1% of the time. Benford's law
Benford's law
also makes predictions about the distribution of second digits, third digits, digit combinations, and so on
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