Browsed by
Category: Derivative

Sum from 1 to n

Sum from 1 to n

Input interpretation
Total[arithmetic progression | 1 to n | step size | 1]
Result
1/2 n (n + 1)
Values
n | 1 | 2 | 3 | 4 | 5
1/2 n (n + 1) | 1 | 3 | 6 | 10 | 15
Expanded form
n^2/2 + n/2
2/3=

2/3=

Input
2/3
Exact result
2/3 (irreducible)
Decimal approximation
0.666666666666666666666666666666666666666666666666666666666…
Repeating decimal
0.6^_ (period 1)
Percentage
66.67%
Prime factorization
2×3^(-1)
Continued fraction
[0; 1, 2]
Egyptian fraction expansion
1/2 + 1/6
2/3 + 2/3

2/3 + 2/3

Input
2/3 + 2/3
Exact result
4/3
Decimal approximation
1.333333333333333333333333333333333333333333333333333333333…
Repeating decimal
1.3^_ (period 1)
Mixed fraction
1 1/3
Continued fraction
[1; 3]
Egyptian fraction expansion
1 + 1/3
3 % 2

3 % 2

Input
3 mod 2
Result
1
Number name
one
Visual representation
*
Integers congruent to 1 mod 2
3, 5, 7, 9, 11, 13, 15, 17, 19, 21, …
2x2matrix

2x2matrix

Input
Array[a, {2, 2}]
Exact result
{{a[1, 1], a[1, 2]}, {a[2, 1], a[2, 2]}}
Characteristic polynomial
-lambda a(2, 2) – lambda a(1, 1) + a(1, 1) a(2, 2) – a(1, 2) a(2, 1) + lambda^2
Diagonalization
M = S.J.S^(-1)
where
M = (a(1, 1) | a(1, 2)
a(2, 1) | a(2, 2))
S = (-(-a(1, 1) + a(2, 2) + sqrt(a(1, 1)^2 + 4 a(1, 2) a(2, 1) – 2 a(1, 1) a(2, 2) + a(2, 2)^2))/(2 a(2, 1)) | (a(1, 1) – a(2, 2) + sqrt(a(1, 1)^2 + 4 a(1, 2) a(2, 1) – 2 a(1, 1) a(2, 2) + a(2, 2)^2))/(2 a(2, 1))
1 | 1)
J = (1/2 (a(1, 1) + a(2, 2) – sqrt(a(1, 1)^2 + 4 a(1, 2) a(2, 1) – 2 a(1, 1) a(2, 2) + a(2, 2)^2)) | 0
0 | 1/2 (a(1, 1) + a(2, 2) + sqrt(a(1, 1)^2 + 4 a(1, 2) a(2, 1) – 2 a(1, 1) a(2, 2) + a(2, 2)^2)))
S^(-1) = (-(a(2, 1))/sqrt(a(1, 1)^2 + 4 a(1, 2) a(2, 1) – 2 a(1, 1) a(2, 2) + a(2, 2)^2) | (a(1, 1) – a(2, 2) + sqrt(a(1, 1)^2 + 4 a(1, 2) a(2, 1) – 2 a(1, 1) a(2, 2) + a(2, 2)^2))/(2 sqrt(a(1, 1)^2 + 4 a(1, 2) a(2, 1) – 2 a(1, 1) a(2, 2) + a(2, 2)^2))
(a(2, 1))/sqrt(a(1, 1)^2 + 4 a(1, 2) a(2, 1) – 2 a(1, 1) a(2, 2) + a(2, 2)^2) | (-a(1, 1) + a(2, 2) + sqrt(a(1, 1)^2 + 4 a(1, 2) a(2, 1) – 2 a(1, 1) a(2, 2) + a(2, 2)^2))/(2 sqrt(a(1, 1)^2 + 4 a(1, 2) a(2, 1) – 2 a(1, 1) a(2, 2) + a(2, 2)^2)))
2500/100

2500/100

Input
2500/100
Exact result
25
Quotient and remainder
25×100 + 0
Number name
twenty-five
Visual representation
* * * * * * * * * * * * * * * * * * * * * * * * *