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Beutiful eplanation Alan!

Convert to binary........

convert 3.141592 to base 2 with steps please, if possible.

Thank you for any help.

[pre-decimal point position]:

\(\begin{array}{|rcrcrr|} \hline && && & \text{Remainder} \\ \hline 3 &:& 2 &=& \color{blue}{1} & \color{red}{1} \\ \color{blue}{1} &:& 2 &=& 0 & \color{red}{1} \\ \hline \end{array}\)

decimals:

\(\begin{array}{|rcrcrr|} \hline && && & \text{Carry} \\ \hline 0.141592 &*& 2 &=& {\color{red}0}.283184 & \color{red}{0} \\ 0.283184 &*& 2 &=& {\color{red}0}.566368 & \color{red}{0} \\ 0.566368 &*& 2 &=& {\color{red}1}.132736 & \color{red}{1} \\ 0.132736 &*& 2 &=& {\color{red}0}.265472 & \color{red}{0} \\ 0.265472 &*& 2 &=& {\color{red}0}.530944 & \color{red}{0} \\ 0.530944 &*& 2 &=& {\color{red}1}.061888 & \color{red}{1} \\ 0.061888 &*& 2 &=& {\color{red}0}.123776 & \color{red}{0} \\ 0.123776 &*& 2 &=& {\color{red}0}.247552 & \color{red}{0} \\ 0.247552 &*& 2 &=& {\color{red}0}.495104 & \color{red}{0} \\ 0.495104 &*& 2 &=& {\color{red}0}.990208 & \color{red}{0} \\ 0.990208 &*& 2 &=& {\color{red}1}.980416 & \color{red}{1} \\ 0.980416 &*& 2 &=& {\color{red}1}.960832 & \color{red}{1} \\ 0.960832 &*& 2 &=& {\color{red}1}.921664 & \color{red}{1} \\ 0.921664 &*& 2 &=& {\color{red}1}.843328 & \color{red}{1} \\ 0.843328 &*& 2 &=& {\color{red}1}.686656 & \color{red}{1} \\ 0.686656 &*& 2 &=& {\color{red}1}.373312 & \color{red}{1} \\ 0.373312 &*& 2 &=& {\color{red}0}.746624 & \color{red}{0} \\ 0.746624 &*& 2 &=& {\color{red}1}.493248 & \color{red}{1} \\ 0.493248 &*& 2 &=& {\color{red}0}.986496 & \color{red}{0} \\ 0.986496 &*& 2 &=& {\color{red}1}.972992 & \color{red}{1} \\ \cdots \\ \hline \end{array}\)

\(3.141592_{10} = 11.00100100001111110101\ldots_2\)

Convert to binary........

convert 3.141592 to base 2 with steps please, if possible.

Thank you for any help.

[pre-decimal point position]:

\(\begin{array}{|rcrcrr|} \hline && && & \text{Remainder} \\ \hline 3 &:& 2 &=& 1 & \color{red}{1} \\ 1 &:& 2 &=& 0 & \color{red}{1} \\ \hline \end{array}\)

decimals:

\(\begin{array}{|rcrcrr|} \hline && && & \text{Carry} \\ \hline 0.141592 &*& 2 &=& {\color{red}0}.283184 & \color{red}{0} \\ 0.283184 &*& 2 &=& {\color{red}0}.566368 & \color{red}{0} \\ 0.566368 &*& 2 &=& {\color{red}1}.132736 & \color{red}{1} \\ 0.132736 &*& 2 &=& {\color{red}0}.265472 & \color{red}{0} \\ 0.265472 &*& 2 &=& {\color{red}0}.530944 & \color{red}{0} \\ 0.530944 &*& 2 &=& {\color{red}1}.061888 & \color{red}{1} \\ 0.061888 &*& 2 &=& {\color{red}0}.123776 & \color{red}{0} \\ 0.123776 &*& 2 &=& {\color{red}0}.247552 & \color{red}{0} \\ 0.247552 &*& 2 &=& {\color{red}0}.495104 & \color{red}{0} \\ 0.495104 &*& 2 &=& {\color{red}0}.990208 & \color{red}{0} \\ 0.990208 &*& 2 &=& {\color{red}1}.980416 & \color{red}{1} \\ 0.980416 &*& 2 &=& {\color{red}1}.960832 & \color{red}{1} \\ 0.960832 &*& 2 &=& {\color{red}1}.921664 & \color{red}{1} \\ 0.921664 &*& 2 &=& {\color{red}1}.843328 & \color{red}{1} \\ 0.843328 &*& 2 &=& {\color{red}1}.686656 & \color{red}{1} \\ 0.686656 &*& 2 &=& {\color{red}1}.373312 & \color{red}{1} \\ 0.373312 &*& 2 &=& {\color{red}0}.746624 & \color{red}{0} \\ 0.746624 &*& 2 &=& {\color{red}1}.493248 & \color{red}{1} \\ 0.493248 &*& 2 &=& {\color{red}0}.986496 & \color{red}{0} \\ 0.986496 &*& 2 &=& {\color{red}1}.972992 & \color{red}{1} \\ \cdots \\ \hline \end{array}\)

\(3.141592_{10} = 11.00100100001111110101\ldots_2\)

Now, I believe you know how bases work-

suppose we are writing using base n. That means we have n digits that represent numbers:

0, 1, 2, ......n-1

for example, if n=2. we have 2 digits- 0 and 1.

Lets call the first digit the main digit (for example, the main digit in 11001010110 be 0, in 345 it will be 5 and so on)

If the digit d is K places to the left of the main digit, its value will be the base multiplied by itself K times, then multiplied by the digit. if the base is N it means the value will be d*N^K. It also means that if the digit is K places to the right of the main digit, its value will be d*N^-K=d/(N^K)

for example, if our base is 10, then the value of the marked digit in 0.007 will be 7*10^-3=7/(10³).

If our base is 2, it means the value of the marked digit in 0.01 will be 1*2^-2=1/(2²)

So, if we want to convert 3.141592 to base 2, we have to use those rules. 3.141592 (base 10)~~11.0011

Unfortunately for you, pi is irrational, meaning we cant write it with a finite number of digits (i know you didnt want me to do that but i just had to say that)

you wanted me to write 3.141592 in base 2, and im afraid to tell you we cant write this rational number with a finite number of digits either! but we CAN write it as a Repeating decimal. why? because if you dont want a fraction to be a repeating decimal (lets say the fraction is m₁/m₂) then after reducing the fraction you need all of m₂'s factors to be factors of the base N as well. It doesnt work with 141592/1000000, but we can write it as a repeating decimal.

Can you to that?

complete _x5=3x10

\({\color{red} 6}\times5=3\times10\)

Did you cook this one up yourself??

(all values of x are solutions).

what is the area of a circle with a radius of 8 centimeters

\(\begin{array}{|rcll|} \hline A &=& \pi r^2 \quad & | \quad r = 8\ cm \\ &=& \pi 8^2 \ cm^2 \\ &=& 64 \pi \ cm^2 \quad & | \quad \pi = 3.14159265359\ldots \\ &=& 201.061929830 \ cm^2 \\ \hline \end{array}\)

If you have a right-angled triangle and the opposite side length (= b, say) and the length of the hypotenuse (= h, say)

then the angle ( = B) is given by 2nd asin (b/h) enter

If you have the adjacent side length (= a, say) and the hypotenuse 

then B is given by 2nd acos (a/h) enter

If you have the opposite and adjacent side lengths

then B is given by 2nd atan (b/a) enter

"sinh" is what is known as a hyperbolic function.  It is defined as follows:

sinh(x) = (e^x - e^-x)/2

We also have cosh(x) = (e^x + e^-x)/2

and tanh(x) = sinh(x)/cosh(x)

..

Click on the "Draw Graph" command at the bottom right of the calculator.  Enter your function in one of the input lines and press enter.  You may well have to alter the ymin, ymax, xmin and xmax values below the graph to get a good view (press enter after each one).

e.g.

.

"I am trying to work out standard deviations and have to square numbers I am getting two different answers if I use the X2 square function it comes out as a negative if I just times the number by itself it comes out positive which one is right"

-2² = -4  because exponentiation takes priority over negation

(-2)² = +4  because parentheses take highest priority

So -2 x -2 = +4  because -2 x -2 is the same as (-2)²

.

p(x-3)=15C^3(0.5)3(0.5)15-3

\(\begin{array}{|rcll|} \hline && 15C_{3}(0.5)^{3}(0.5)^{15-3} \quad & | \quad 15C_{3} = \frac{15}{3}\cdot \frac{14}{2}\cdot \frac{13}{1} = 5\cdot 7 \cdot 13 = 455 \\ &=& 455 \cdot (0.5)^{3}(0.5)^{15-3} \\ &=& 455 \cdot(0.5)^{3+15-3} \\ &=& 455 \cdot(0.5)^{15} \\ &=& 455 \cdot(\frac12 )^{15} \\ &=& 455 \cdot \frac{1}{2^{15}} \\ &=& \frac{455}{2^{15}} \\ &=& \frac{455}{32768} \\ &=& 0.01388549805 \\ \hline \end{array}\)

We want to choose any 5 of 15 men and any 7 of 11 women....so...

C(15,5) * C(11,7)   = 990,990 different juries

Your're smart! Thanks!

Very nice!

-4z^2 + 20z - 6

The z that will maxmize this occurs at  

-20 / [2 * -4]  =   -20/-8  =  5/2 = 2.5

Subbing this back into the function, the max value is

-4(2.5)^2 + 20(2.5) - 6  =

19

4x -5,,,,,,,,,,,,,,,,,,,, yes!

0.......................

3B9D₁₆   

Using the method....multiply by the base, add the next digit, multiply by the base, add the next digit, etc., we have :

3*16 + 11  =  59

59 * 16 + 9  = 953

953 * 16 +  13 = 15261

uhhhhhhhhhhhhhhh

Very nice Cphill!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!