How to Solve Any Math By Hand and Some Algebra
Everything on these pages are in radian mode.
page 1
Floyd Nelson
August 31, 2009
sin-1(x) is pronounced sine inverse of x.
arcsin(x) or asin(x) is one of the infinite number of answers from sin-1(x).
sinh(x) is pronounced hyperbolic sine of x.
ln(x) stands for log natural of x.
sign(7) = 1
sign(0) = sign(0)
sign(-7) = -1
Degrees = π*Radians/180 = 10*Gradients/9
e ≈ 2.7182818284590452354
e ≈10.101101111110000101010001 Binary Code
π ≈ 3.14159265358979323846264338327950288419716939937511 = 4*atan(1)
π ≈ 11.00100100001111110110101 Binary Code
cos(θ) = x/r = 1/sec(θ) =
arcsin(x) = arctan(x/√(-x2+1))
cot(θ)/csc(θ) = sin(θ)/tan(θ) =
2
arccos(x) = arctan(x/√(-x +1))+π/2
sin(θ)*cot(θ)
arcsec(x) = arctan(x/√(x2-1))+sign(sign(x)-1)*π/2
sin(θ) = y/r = 1/csc(θ) =
arccsc(x) = arctan(x/√(x2-1))+sign(x)-1)*π/2
tan(θ)/sec(θ) = cos(θ)/cot(θ) =
arccot(x) = arctan(x)+π/2
cos(θ)*tan(θ)
sinh(x) = (ex-e-x)/2 = (ex-1/ex)/2 = (e2x+1)/(2*ex) = ex/2-1/(2*ex)
tan(θ) = y/x = 1/cot(θ) =
cosh(x) = (ex+e-x)/2 = (e2x+1)/(2*ex) = ex/2+1/(2*ex)
sin(θ)/cos(θ) = sec(θ)*sin(θ)
x -x
x
-x
2x
2x
2x
tanh(x) = (e -e )/(e +e ) = (e -1)/( e +1) = 1-2/( e +1)
sech(x) = 2/(ex+e-x) = 2/(ex+1/ex)
sec(θ) = r/x = 1/cos(θ) =
csc(θ)/cot(θ) = tan(θ)/sin(θ) =
csch(x) = 2/(ex-e-x) = 2/(ex-1/ex)
sec(θ)*cot(θ)
-x
x -x
2*x
coth(x) = (e /(e -e ))*2+1 = 2/(e -1)+1
arcsinh(x) = ln(x/√(x2+1))
csc(θ) = r/y = 1/sin(θ) =
sec(θ)/tan(θ) = cot(θ)/cos(θ) =
arccosh(x) = ln(x+√(x2-1))
csc(θ)*tan(θ)
2
arctanh(x) = ln((1+x)/(1-x))/2 = (ln(1+x)-ln(1-x))/2 = ln(1-x )/2
arccsch(x) = ln(sign(x)*√(x2+1)+1)/x
cot(θ) = x/y = 1/tan(θ) =
2
cos(θ)/sin(θ) = csc(θ)*cos(θ)
arcsech(x) = ln(√(-x +1)+1)/x
2
arccoth(x) = ln((x+1)/(x-1))/2 = (ln(x+1)-ln(x-1))/2 = ln(x -1)/2
function
continuous equation
calculus equation
domain of x
sin(x)
x-x3/3!+x5/5!-x7/7!+...
∑((-1)nx2n+1/(2n+1)!,n=0,∞)
all real x
cos(x)
1-x /2!+x /4!-x /6!+...
∑((-1) x /(2n)!,n=0,∞)
all real x
arctan(x)
x-x3/3+x5/5-x7/7+...
∑((-1)nx2n+1/(2n+1),n=0,∞)
-1<x≤1
ex
1+x/1!+x2/2!+x3/3!+x4/4!+...
∑(xn/n!,n=0,∞)
all real x
2
4
2
3
6
4
ln(1+x)
x-x /2+x /3-x /4+...
π
4*(1/1-1/3+1/5-1/7+1/9-...)
For ln(x+1) = ln(a) and -1<x≤1,
if a≤0 then x = unidentified
if 0<a≤1 then x = a-1
if 1<a then x = error
sin(-x) = -sin(x)
cos(-x) = cos(x)
tan(-x) = -tan(x)
csc(-x) = -csc(x)
sec(-x) = sec(x)
cot(-x) = -cot(x)
((-1)x/π+(-1)-x/π)/2 = cos(x)
((-1)x/π+(-1)-x/π)/(2i) = sin(x)
atan(-x) = -atan(x)
atan(|x|) = 90°-atan(1/|x|)
n 2n
∑((-1)
n+1
*x /n,n=1,∞)
n
-1<x≤1
4*∑((-1)n/(2*n+1),n=0,∞)
For -ln(x+1) = ln(a) and -1<x≤1,
if a≤0 then x = unidentified
if 0<a then x = (1/a)-1
cos(θ) = cos(360°-θ) = cos(θ±360°) = cos(-θ) = sin(90°-θ)
sin(θ) = sin(360°-θ) = sin(θ±360°) = sin(180°-θ) = cos(90°-θ)
tan(θ) = tan(θ±180°) = cot(±90°-θ)
sec(θ) = sec(360°-θ) = sec(θ±360°) = sec(-θ) = csc(90°-θ)
csc(θ) = csc(360°-θ) = csc(θ±360°) = csc(180°-θ) = sec(90°-θ)
cot(θ) = cot(θ±180°) = tan(±90°-θ)
How to Solve Any Math By Hand and Some Algebra
Everything on these pages are in radian mode.
∏(n,n=1,x) = x!
x
decimal x!
0
1
1
1
2
2
3
6
4
24
5
120
6
720
7
5040
8
40320
9
362880
10
3628800
11
39916800
12
479001600
13
6227020800
14
87178291200
15
1307674368000
16
20922789888000
17
355687428096000
18
6402373705728000
19 121645100408832000
20 2432902008176640000
page 2
8! = 1*2*3*4*5*6*7*8
binary x!
Floyd Nelson
August 31, 2009
0! = 1
1
1
10
110
11000
1111000
1011010000
1001110110000
1001110110000000
1011000100110000000
1101110101111100000000
10011000010001010100000000
11100100011001111110000000000
101110011001010001100110000000000
1010001001100001110110010100000000000
10011000001110111011101110101100000000000
100110000011101110111011101011000000000000000
1010000110111111011101110110011011000000000000000
10110101111101110110011001010011100110000000000000000
110110000001010111001001100000110100010010000000000000000
10000111000011011001110111110010000010101101000000000000000000
nPr(x,y) = x!/(x-y)! nCr(x,y) = nPr(x,y)/y! = x!/((x-y)!*y!)
aloga(a)*b = ab
logb(a) = ln(a)/ln(b)
√(a*b) = √(a)* √(b)
√(12) = √(4*3) = √(4)*√(3) = 2*√(3)
ba=c ↔ a√(c)=b ↔ logb(c)=a
1/logb(a) = loga(b)
-logb(a) = logb(1/a) = log1/b(a)
loga(b) = logc(b)/logc(a)
loga(b) + loga(c) = loga(b*c) loga(b) - loga(c) = loga(b/c)
loga(bc) = c*loga(b)
loge(a) = ln(a)
1/√(a) = √(a)/a
a/√(a) = √(a)
b/c
c
b
-1
√(a*b) = √(a)*√(b) √(a/b) = √(a)/√(b)
a = √(a ) a = 1/a
a-b = 1/ab
b c
b+c
b c
b-c
b d f
b*f d*f
b d f
b*f d*f
a *a = a
a /a = a
(a *c ) = a *c
(a /c ) = a /c
a1/b+c = b√(a)*ac
↑ Algebra ↑
ab+c = ab ac
a0.5 = √(a)
a0.25 = √(√(a)) a0.125 = √(√(√(a)))
a0.0625 = √(√(√(√(a))))
1/2
1/4
1/8
a1/b+c = b√(a)*ac a = √(a)
a = √(√(a)) a = √(√(√(a)))
a1/16 = √(√(√(√(a))))
In binary code: a0.1 = √(a)
a0.01 = √(√(a)) a0.001 = √(√(√(a)))
a0.0001 = √(√(√(√(a))))
2*2*2*2
16
√(√(√(√(a)))) =
√(a) = √(a)
Here are examples of how to solve for fractional exponents by hand:
a5 = a*a*a*a*a
a5.78125 = a5*a0.78125 = a5*a0.5+0.28125 = a5*√(a)*a0.28125 = a5*√(a)*a.25+0.03125 =
a5*√(a)*√(√(a))*a0.03125 = a5*√(a)*√(√(a))*√(√(√(√(a))))
In binary code a11.110101 = a*a*a*√(a)*√(√(a))*√(√(√(√(a))))*√(√(√(√(√(√(a))))))
To solve a√(b) by hand, use the formula b1/a. How to solve exponents by hand is shown
above.
There is an algorithm (method) to solve square roots by hand as shown on the next page.
How to Solve Any Math By Hand and Some Algebra
Everything on these pages are in radian mode.
Floyd Nelson
August 31, 2009
page 3
This shows how to find the square root of 28594.81 in decimal code. Each maximum x integer
found is a part of the answer to the square root. The x in x2 has already been solved for. The
other x’s have to be solved for. Each x integer, besides the first, is written in three places.
1
6
9.
1
√
2 85 94. 81
x2 ≤ 2
12 = -1 ↓
↓
↓
1 85 ↓
↓
2*10*x ≤ 185
2*10*6 = -1 20 ↓
↓
65 ↓
↓
x2
62 =
-36 ↓
↓
29 94 ↓
2*160*x ≤ 2994
2*160*9 =
-28 80 ↓
1 14 ↓
x2
92 =
-81 ↓
33 81
2*1690*x ≤ 3381
2*1690*1 =
-33 80
01
x2
12 =
-1
0
To find the square root of 456872.651, organize the digits like this: 45 68 72. 65 10
This shows a simplified way of how to solve square roots in binary code. Arithmetic is always
the easiest and most simplified when done in binary code. The red/blue numbers are crossed out
if they are larger than the black number above them. If that is the case do not subtract and place a
0 for the answer of the square root, otherwise, if the numbers are capable of subtracting, place a 1
in the answer column, do not cross out the red/blue number, and subtract.
1
√10
- 01
1
- 1
1
01
↓
01
01
0
-11
-11
0
00
↓
↓
↓
00
01
0
0
-11
-1
0.
11.
↓
↓
↓
↓
↓
11
01
11
00
11
10
11
-11
-1
0
10
↓
↓
↓
↓
↓
↓
↓
10
01
10
00
10
00
1
10
1
-11
1
-1
0
00
↓
↓
↓
↓
↓
↓
↓
↓
↓
00
01
00
00
11
00
11
00
11
10
1
-11
1
-1
1
00
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
00
01
11
01
11
00
11
00
10
00
10
10
-11
0
00
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
00
01
00
10
00
01
10
00
10
00
10
00
0
00
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
00
01
00
00
11
10
11
01
10
00
1
00
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
00
01
11
01
11
00
10
10
0
00
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
00
01
00
10
01
01
1
00
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
00
01
11
01
0
00
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
00
01
How to Solve Any Math By Hand and Some Algebra
Everything on these pages are in radian mode.
Decimal
Base 10
Binary
Base 2
Hexadecimal
Base 16
.0078125
.015625
.03125
.0625
.125
.25
.5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
.0000001
.000001
.00001
.0001
.001
.01
.1
0
1
10
11
100
101
110
111
1000
1001
1010
1011
1100
1101
1110
1111
10000
10001
10010
10011
10100
.02
.04
.08
.1
.2
.4
.8
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
10
11
12
13
14
21
22
23
24
25
26
27
28
29
30
31
32
33
10101
10110
10111
11000
11001
11010
11011
11100
11101
11110
11111
100000
100001
15
16
17
18
19
1A
1B
1C
1D
1E
1F
20
21
123589
11110001011000101
1E2C5
Floyd Nelson
August 31, 2009
page 4
t
Carry
A
B
Carry+A+B = Z
t
Barrow
A
B
Barr+A-B = Z
7
-1
1
0
0
7
1
0
0
1
6
0
1
1
0
6
-1
0
1
0
5
-1
0
0
1
5
0
1
0
1
4
1
0
0
1
4
0
0
1
1
3
1
0
1
0
3
-1
1
0
0
2
1
1
1
1
2
-1
1
1
1
312 = 28+25+24+23 = Binary: 100111000
985 = 3*162+13*161+9*160 = Hexadecimal: 3D9
Binary: 1101 0011 0101 1001 = Hexadecimal: D359
D
3
5
9
51042 = 5*104+103+4*101+2*100
Binary Subtraction Help for each digit of A-B=Z:
Key: Barrowt + At - Bt = (Barrowt+1/2)+ Zt
If the Barrow output is -2, then the next Barrow input is -1, because the next digit is 2*
the value of the current digit.
0+0-0 = 0+0
0+0-1 = -2+1
0+1-0 = 0+1
0+1-1 = 0+0
-1+0-0 = -2+1
-1+0-1 = -2+0
-1+1-0 = 0+0
-1+1-1 = -2+0
Golden Ratio: 1/x = x-1 ≡ x = (1±√(5))/2 ≈ 1.618033988749895
1
0
1
1
0
1
0
0
1
1
0
0
0
1
1
0
0
1
0
1
How to Solve Any Math By Hand and Some Algebra
Everything on these pages are in radian mode.
Wikipedia
page 5
Floyd Nelson
August 31, 2009