Binomial Theorem and Logarithms: For the Use of the Midshipmen at the Naval School, PhiladelphiaPerkins & Purves, 1843 - 92 páginas |
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Página 20
... whence A — A ' — = 0 , B — B ' — 0 , & c . And if we put A ― A ' — M , B — B ' — = N , & c . , the equation will be , M + Nx + Px2 + Qx3 + & c . = 0 . Whenever , then , we have an equation of the form , M + Nx + P + Qx3 + & c . = 0 ...
... whence A — A ' — = 0 , B — B ' — 0 , & c . And if we put A ― A ' — M , B — B ' — = N , & c . , the equation will be , M + Nx + Px2 + Qx3 + & c . = 0 . Whenever , then , we have an equation of the form , M + Nx + P + Qx3 + & c . = 0 ...
Página 26
... whence B 2C + B = mB " C = = 2 2 2 3D + 2C = mC " D = mC - 2C C ( m - 2 ) m ( m - 1 ) ( m - 2 ) = 3 3 2.3 4E + 3DmD " E = mD - 3D D ( m - 3 ) m ( m - 1 ) ( m - 2 ) ( m - 3 ) = 4 4 & c . & c . 2.3.4 & c . These values of B , C , D ...
... whence B 2C + B = mB " C = = 2 2 2 3D + 2C = mC " D = mC - 2C C ( m - 2 ) m ( m - 1 ) ( m - 2 ) = 3 3 2.3 4E + 3DmD " E = mD - 3D D ( m - 3 ) m ( m - 1 ) ( m - 2 ) ( m - 3 ) = 4 4 & c . & c . 2.3.4 & c . These values of B , C , D ...
Página 41
... whence we derive b 2 = mam - 1 This value of z , substituted in the second term of ( A ) , gives m ( m - 1 ) b b = mam - iz ? am - 2zX . 2 mam - 1 = ( 1 mam - 1 ( m - 1 ) b 2a whence we derive a second value of z , z = 2ab 2ma " ( m - 1 ) ...
... whence we derive b 2 = mam - 1 This value of z , substituted in the second term of ( A ) , gives m ( m - 1 ) b b = mam - iz ? am - 2zX . 2 mam - 1 = ( 1 mam - 1 ( m - 1 ) b 2a whence we derive a second value of z , z = 2ab 2ma " ( m - 1 ) ...
Página 42
... be found to five places of figures with the aid of logarithms , ( for which see Chapter VI . , ) and by the above methods we may then extend it to any required number of decimals . whence we have N - a3 x = 2α That 42 BINOMIAL THEOREM .
... be found to five places of figures with the aid of logarithms , ( for which see Chapter VI . , ) and by the above methods we may then extend it to any required number of decimals . whence we have N - a3 x = 2α That 42 BINOMIAL THEOREM .
Página 43
For the Use of the Midshipmen at the Naval School, Philadelphia William Chauvenet. whence we have N - a3 x = 2α That is , having found an approximate root a , subtract its square from the given number , and divide the remainder by 2a for ...
For the Use of the Midshipmen at the Naval School, Philadelphia William Chauvenet. whence we have N - a3 x = 2α That is , having found an approximate root a , subtract its square from the given number , and divide the remainder by 2a for ...
Outras edições - Ver tudo
Binomial Theorem and Logarithms: For the Use of the Midshipmen At the Naval ... William Chauvenet Pré-visualização limitada - 2024 |
Binomial Theorem and Logarithms: For the Use of the Midshipmen At the Naval ... William Chauvenet Pré-visualização limitada - 2024 |
Binomial Theorem and Logarithms: For the Use of the Midshipmen at the Naval ... William Chauvenet Pré-visualização indisponível - 2017 |
Palavras e frases frequentes
2n³ 3n³ 3d power 3d root 4th power 4th root Algebra anti-logarithms approximate ax=b binomial theorem Briggs calculation CHAPTER common logarithms compute convenient convergent cube root decimal fraction decimal point denominator example exponential equation express the value find log find the square find the value finite number formula becomes fractional exponents given logarithm given number Hence Hutton indefinitely small infinite series integral exponents involution and evolution log.b loga m+1)th term manner method modulus multiply naperian logarithm Newton number is equal number of terms obtain places of decimals positive integer power of a+b power or root powers and roots prime numbers quantity reciprocal rithms root of a³ significant figure square root succeeding terms system of logarithms system whose base uneven unit's place unity values substituted whence
Passagens conhecidas
Página 50 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.
Página 49 - The logarithm of the product of two or more numbers is equal to the sum of the logarithms of those numbers.
Página 61 - The fourth term is found by multiplying the second and third terms together and dividing by the first § 14O.
Página 50 - The logarithm of the quotient of two numbers is equal to the logarithm of the dividend minus the logarithm of the divisor.
Página 19 - Cxz+, etc.=A'+B'x+C'z2 + , etc., must be satisfied for each and every value given to x, then the coefficients of the like powers of x in the two members are equal each to each.
Página 74 - The logarithm of a number in any system is equal to the Naperian logarithm of that number multiplied by the modulus of the system.
Página 49 - Corollary. When the base is less than unity, it follows, from art. 3, that the logarithms of all numbers greater than unity are negative, while those of all numbers less than unity are positive. But when, as is almost always...
Página 55 - ... place, the characteristic being positive when this figure is to the left of the units' place, negative when it is to the right of the units' place, and zero when it is in the units
Página 27 - I have no doubt that he made the difcovery himfelf, without any light from Briggs, and that he thought it was new for all powers in general, as it was indeed for roots and quantities with fractional and irrational exponents.
Página 50 - Bee that to divide one number by another, we subtract the log. of the divisor from the log. of the dividend, and the remainder is the log.