Algebra cheatsheet I

This algebra reference is important for cryptography and error control coding.

1. NUMBER SETS

  • set natural Natural numbers {1,2,3,…}- Does not include infinity as it is not a valid number.
  • Prime numbers– A subset of natural numbers that has no lower order factors except 1. See link
  • set integerIntegers {… , -3, -2, -1, 0, 1, 2, 3, …}- Negative and positive whole numbers including zero.
  • set rational Rational numbers. The result of a division operation of any two integers but without zero as the denominator. Could also be thought of as numbers that can be represented as a fraction(ratio of two integers).
  • Irrational numbers. A number that can’t be represented as a ratio of two integers exactly, without zero as the denominator.(NOT rational)
  • set real Real numbers. Includes all rational and irrational numbers thus  Natural numbers, integers, rational numbers, irrational numbers.
  • set imaginary Imaginary numbers. See link
  • set complex complex numbers. A combination of real and imaginary numbers. Example: 3+4i, -16.5+4.3i

 

number sets

Mathematical number set venn diagram. Source: http://www.mathsisfun.com/sets/number-types.html

2. MODULAR ARITHMETIC

STEP 1

If we have A, R, M which belong to the set of integers, and m>0:

A ≡ R mod M ≡ QM + R

Where R is the reminder, Q is the quotient, M is the divider, A is the unreduced number.

This means that an integer “A” can be reduced to R in a modulo “M” operation.

STEP2

Due to different quotients giving different reminders, another constraint is added to the reminder.

0 ≤ R ≤ M-1

Example(Modulo 9 addition):

(16+7) mod 9=23 mod 9

Without the step 2 constraint on the reminder, the following reminders are valid:

For the following set of Q{…, -1, 0, 1, 2, 3, …} gives this set of R {…, 32, 23, 14, 5, -4, …}

If we apply the step 2 constraint, the valid value of the modulo addition is 5 as 0 ≤ 5 ≤ 8

Python code

Calling the python interpreter on the command line

>>> #Compute the modulo of an integer with python's modulo operator
... -1%5
4
>>> #Trial of the example above
... (16+7)%9
5

3. ALGEBRAIC GROUP(or simply group)

A group is a mathematical entity that describes  a set of elements, say G that is generally defined with certain properties under a hypothetical binary operation •  . the general operation •  combines any two elements of G to give a result. (a•b=c)

PROPERTIES OF GROUPS

  1. The result of the binary operation “•” with the elements of G is closed, This  means that the result will still be in the set of elements G.
  2. The operation on elements of the group is associative.(a•b)•c=a•(b•c)
  3. There is an element “e” in G called the neutral/identity element of the set G under the binary operation “•”.
  4. Every element in the group G has an inverse under the operation. The following relation thus automatically holds. For ∀ a∈G, a•a-1=e,where e is the identity element of the group.
  5. If the group is commutative under the operation, the group is known as an abelian group.

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