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[...] Dividing Decimals Quick method: use Long Division without the decimal point, then re-insert the decimal point in the answer. Dividing a Decimal Number by a Whole Number [...]

[...] Decimals A Decimal Number (based on the number 10) contains a Decimal Point. First, let's have an example: Here is the number "forty-five and six-tenths" [...]

[...] Convert Decimals to Percents Quick Method: move the decimal point 2 places to the right and add a "%" sign! Example: Convert 0.35 to percent Move the [...]

[...] Multiplying Decimals Multiply without the decimal point, then re-insert it in the correct spot! How to Multiply Decimals Just follow these steps: [...]

[...] m is square-free, and the result follows. Here is a more convenient way to characterize the algebraic integers of a quadratic field. 2.2.6 Proposition The set B of algebraic integers of Q( √ m), m [...]

[...] of K/Q and L/Q one at a time, thus allowing an induction argument. Let R, S, T be the algebraic integers of K,L,KL respectively. Define RS as the set of all finite sums aibi with ai ∈ R, bi ∈ S. [...]

[...] ; click here for PDF file of this chapter. As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the [...]

[...] In the AKLB setup, if A is a Dedekind domain, then so is B. In particular, the ring of algebraic integers in a number field is a Dedekind domain.Proof. In view of (3.1.2), it suffices to show that [...]

[...] ) = v(x) + v(y); (c) v(x + y) ≥ min(v(x), v(y)). A discrete valuation induces a nonarchimedean absolute value via |x| = cv(x), where c is a constant with 0 < c < 1. 9.1.2 Example Let A be a [...]

[...] Values B1. Theorem Let L/K be a finite extension of fields, with n = [L : K]. If | | is an absolute value on K and K is locally compact (hence complete) in the topology induced by | |, then there is [...]

[...] . Section 9.1 1. This follows from (6.1.5) andthe observation that a root of unity must have absolute value 1. 2. If the characteristic is p = 0, then there are only p integers, andthe result follows [...]

[...] is a parallelotope whose sides are the σ(xi), and the volume of the fundamental domain is the absolute value of the determinant whose rows (or columns) are the σ(xi). Consequently [see (5.1.1) for [...]

[...] difference between the number factors of 32 & 64 is? Option: (a) 3 (b) 2 (c) 1 (d) 0 (e) None Explore: Let us write down all the factors of 32 and 64 32 : 1, 2, 4, 8, 16, 32 64 : 1, 2, 4, 8, 16, [...]

[...] : 12 If (3 - X)/ (1 + X) = 1, then X = ? Option: (a) 10 (b) 1 (c) 2 (d) -2 (e) None Explore:       (3 - X)/(1 + X) = 1 Or, 3 - X = 1 + X  Or, 3 - 1 = 2X Or, [...]

[...] times a number less 3 is 32. What is twice the number? Option: (a) 3 (b) 5 (c) 16 (d) 7 (e) None Explore: Here the equation is not given as a mathematical equation. But it is given as an English [...]

[...] which of the following must be true? Option: (a) X <7 (b) X <13 (c) X > 9 (d) None Explore:      4X - 7 < 2X + 13 => 4X -2X < 13 + 7 => 2X < 20 =& [...]

[...] the remaining 3 & hang it in the dining room. So, there are 3x4 =12 ways of doing this. Venn Diagram: A Venn diagram is a drawing in which circular areas represent groups of items sharing common [...]

[...] Counting & Venn diagram: Counting: In some questions you are asked to count something: How many apples did Jorge buy, [...]

[...] French? Explore: Let, no. of students study both Latin & French = x     Using Venn diagram  7-x +x+8-x+3 = 15 => 18-x = 15 => x = 18-15 => x = 3 Answer: 3 students study [...]

[...] ?  (A)30%  (B)0%  (C)20%  (D)15%  (E)5% Explore: The following Venn diagram will help to visualize the situation: Let, x be the required percentage. So, we can write 45 + [...]

[...] Place Value Our Number System is based on ten. Using only ten symbols (called Digits) we can write any number. The Ten [...]

Even and Odd Numbers Even Numbers Any integer that can be divided exactly by 2. The last digit will be 0, 2, 4, 6 or 8 Example: -24, 0, 6 and 38 are [...]

Long Division Below is the process written out in full. You will often see other versions, which are generally just a shortened version of the proces [...]

Subtracting Decimals Subtracting decimals is easy if you keep your work neat To subtract decimals, follow these steps: Write down the two numbers, o [...]

[...] . By (2.1.6), m = bc where c is a product of conjugates of b. But a conjugate of an algebraic integer is an algebraic integer. (If a monomorphism is applied to an equation of integral [...]

[...] over A. If x is a real or complex number that is integral over Z, then x is called an algebraic integer. Thus for every integer d, √ d is an algebraic integer, as is any nth root of unity. ( [...]

[...] not all ci’s are zero.) Let a(h) = (a1h, . . . , arh), and let xh be the corresponding algebraic integer xa(h). Then by the displayed equation above and the choice of t to exceed the right side, we [...]

[...] , so are δ and all the γi. Now xi = γi δ = γiδ δ2 = γiδ e so exi = γiδ is an algebraic integer. By definition of xi, exi = n j=1 ecij r bj , an algebraic integer in RS. But [...]

[...] :. Y=8/2=4 :. X = 2+Y=2+4= 6 :. The solution of this system of equations is (X, Y) = (6, 4) Quadratic Equations: When the highest power of the variable in an equation is 2 (i.e square) it is called a [...]

[...] :. Y=8/2=4 :. X = 2+Y=2+4= 6 :. The solution of this system of equations is (X, Y) = (6, 4) Quadratic Equations: When the highest power of the variable in an equation is 2 (i.e square) it is called a [...]

Question: 1 If X+Y = 9 and X/Y =  2, then X equals to ....? Option: (a) 3 (b) 4 (c) 5 (d) 6 (e) 7 Explore: x+y = 9 => y = 9-x x/ = 2 => x/( [...]

Question: 31 The sum of square of two numbers is 80 and the square of their difference is 16. Determine the product of the two numbers. Explore: Let, [...]

If you face any problem to loading this page,  click here for PDF file of this chapter.  8.1 Decomposition and Inertia Groups We return to [...]

We continue building our algebraic background to prepare for algebraic number theory. 2.1 Norms and Traces 2.1.1 Definitions and Comments If E/F is a [...]

If you face any problem to loading this page,  click here for PDF file of this chapter. Chapter 1 Section 1.1 1. Multiply the equation by an−1 [...]

If you face any problem to loading this page,  click here for PDF file of this chapter. We collect some results that might be covered in a first [...]

[...]   75 = 15 = 3 100 20 4   ÷5 ÷ 5 Answer = 3/4 Note: 75/100 is called a decimal fraction and 3/4 is called a common fraction ! Example: Express 0.625 as a fraction Step 1: write [...]

[...]   75 = 15 = 3 100 20 4   ÷5 ÷ 5 Answer = 3/4 Note: 75/100 is called a decimal fraction and 3/4 is called a common fraction ! Example: Express 62.5% as a fraction Step 1: Write [...]

[...] ;hundredths" position So, 13.76 is "13 and 7 tenths and 6 hundredths" ... as a Decimal Fraction Or, you could think of a decimal number as a Decimal Fraction. A Decimal Fraction is a [...]