Algebra

Natural numbers

All positive integers starting from $1$ $1, 2, 3, 4, 5, 6, 7, 8, ...$

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Sum

Sum of first $n$ natural numbers is $\frac{n*(n+1)}{2}$

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Factorial

Denoted by the exclamation mark symbol.

n! is called n factorial and its value is defined as:

$n! = 1 \times 2 \times 3 \times ... \times n$

i.e., n! is the product of the first n natural numbers.

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Factors and multiples

Factors: A factor of a number is a smaller or equal number such that it divides the number exactly and gives the remainder zero. For example:

Factors of $12 \to 1, 2,3,4,6,12$ Multiples: A multiple of a number, $n$, is a larger number such that the $n$ is a factor of that number. For example:

Multiples of $4 \to 4, 8, 12, 16, ...$

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Prime numbers

Prime numbers are natural numbers greater than 1 that cannot be expressed as a product of two smaller natural numbers.

They are natural numbers greater than 1 with only two factors, 1 and the number itself.

$2, 3, 5, 7, 11, ...$

Note: A good estimation of the number of primes $<N$ is $logN$. This is useful for complexity analysis.

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Modulus operator

The modulus operation between two integers, a % b, returns the remainder when a is divided by b. For example:

$5 \% 2 = 1$

$100 \% 89 = 11$

$2 \% 5 = 2$

$9 \% 3 = 0$

$21 \% 21 = 0$

The result of the modulus operator is $0$ if a is a multiple of b.

We can easily check if a number is even with if if (x % 2 == 0).

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In the next lesson, we’ll go over set theory, definitions and basic operations.