Algebraic Numbers is in Real Numbers.

Real Numbers can be either Algebraic Numbers or Transcendental Number.

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Irrational Number. An Irrational Number is a number that cannot be expressed as a Fraction.

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Constructable Numbers are numbers that can be derived using a straight edge and a compass.

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The Golden Ratio, also known as the Golden Mean and Golden Section, is 1.618033... It is the solution to the equation x^2 - x - 1 = 0, or ( a + b ) / a = a / b.

The Golden Ratio is usually represented by the Greek Letter phi φ.

The Fibaonacci Series converges on the Golden Ratio.

The formula ( 1 + SQRT(5) ) / 2 is the Golden Ratio.

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Square Root of 2, or 2^(1/2) being the length of the diagonal of a square with sides of length 1. It is 1.4142135623...

99/70 = 1.4142857 approximates to the Square Root of 2.

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Natural Logarithm of 2. Is the logarithm of 2 to the base of e. It is usually written as ln2. It is a Transcendental Number.

The Alternating Harmonic Series: SUM(1,INFINITY) ( (-1)**(N+1) ) / N = 1 - (1/2) + (1/3) - (1/4) + (1/5) - (1/6) + ....

Or Binary Rising Constant Factorial: SUM(1,INFINITY) 1 / ( ( 2**N ) *N) = 1/2 + 1/8 + 1/24 + 1/64 + 1/160 + 1/384 + 1/896 + ....

Or SUM(0,INFINITY) 1 / ( ( 2N + 1 ) * ( 2N + 2 ) ) = 1 / ( 4N**2 + 6N + 2 ) = 1/2 + 1/12 + 1/30 + 1/56 + 1/90 + 1/132 + 1/182 + 1/240 + ....

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Non-Constructable Numbers are the solution to algebraic equations with a cube root of higher eg 2^(1/3).

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PI is the ratio of diameter and circumference of a circle. It is a Transcendental Number.

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e. Euler's Number is 2.718254. It is usually written as 'e'. It is a Transcendental Number.

Euler's Number is the number to which compounding, as in compount interest, converges.

For example, if interest is paid at 100% then after one year £1 becomes £2.

If the interest is charged at 50% twice then the interest will be ( 1 * 1.5 ) * 1.5 = 1.5**2 = 2.25

If the interest is charged at 25% four times then the interest will be ( ( ( ( 1 * 1.25 ) * 1.25 ) * 1.25 ) * 1.25 )= 1.25**4 = 2.44

If the interest is charged at 8.33% twelve times then the interest will be 1.0833**12 = 2.612

If the interest is charged at 4.2% twenty-four times year then the interest will be ( 1.042**4 ) = 2.67

If the interest is charged at 0.1% one thousand times then the interest will be ( 1.001**1000 ) = 2.72

If the interest is charged at an infinite number of times then the interest will be ( 1.00/∞ )**∞ = 2.718254... which is Euler's number.

Euler's Number is the sum of the infinite series ( 1 / n! ) ie 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + 1/720 + 1/5040 .... = 2.718281828.

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Fractions are an Integer divided by an Integer eg 1/2, 5/13, 241/98.

Rational Number. A Rational Number is a number that can be expressed as a Fraction of two Integers. Integers are Fractions with a divisor of 1.

Irrational Number. An Irrational Number is a number that cannot be expressed as a Fraction.

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Integers aka whole numbers. Fractions with a divisor of 1.

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Prime Number. A Prime Number is an Integer that is only divisible by 1 and itself with the Remainder ie. 1, 2, 3, 5, 7, 11, 13, 17, 19, ...

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Fermat Prime. A Prime Number that is a solution to 2^2^N + 1 eg 3, 5, 17, 65537, 4294967297, 18446744073709551617

2 ^ 2 ^ 0 + 1 = 3

2 ^ 2 ^ 1 + 1 = 5

2 ^ 2 ^ 2 + 1 = 17

With the exeception of the first and second terms Germat Primes always end in 7.

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Mersenne Prime. A Mersenne Prime is a prime number that is one less than a power of two. Mersenne Primes do not include all numbers that are one less than a power of two eg 16 - 1 = 15 which is divisible by 1, 3 and 5.