If:
$$a = \sqrt{ b^2 - b }$$
The problem I have is that for values of:
$0 < b < 1$
the result of:
$b^2 - b$
Is a negative number which gives rise to an error on Excel and my calculator.
I understand that negative numbers don't have square roots (I read it on Wikipedia at least), so how do I solve this for values of $b$ less than 1?
Thanks! :)
Mathematicians have defined a new number, called $i$, such that $i^2=-1$ (it's not really that new). Commonly $i$ is called an imaginary number. If you're familiar with coordinate geometry like the Cartesian plane, the complex numbers are very similar. Every complex number has the form $a+bi$ for some $a,b\in\mathbb{R}$ so we can plot complex numbers (that is, numbers that have a real part and an imaginary part) as pairs (a,b) where we view the typical $x$-axis as the real part and the $y$-axis as the imaginary axis. If you have a number like $\sqrt{-64}$, you can simplify it by pulling out the $-1$ as an $i$. That is, $$ \sqrt{-64}=i\sqrt{64}=\pm8i $$ Complex numbers have lots of interesting properties. I recommend checking out the wikipedia page on complex numbers for more information.
Specifically to answer your question, if $b^2-b<0$, there are no solutions over the real numbers. You need to use complex numbers in order to find solutions.
If $b^2-b<0$, then $b-b^2>0$ and $$a = \pm i\sqrt{ b - b^2 },$$ where $i$ is the imaginary unit, which by definition is the unique complex number that satisfies $$i^2=-1\Leftrightarrow i=\pm\sqrt{-1}.$$
The complex numbers are numbers of the form $a+bi$, where $a$ and $b$ are real numbers. They appear e.g. in the solution of a quadratic equation with negative discriminant, such as this one $$x^2+x+1=0,$$ whose solutions are $$x=\dfrac{-1\pm\sqrt{1-4}}{2}=\dfrac{-1\pm\sqrt{-3}}{2}=\dfrac{-1\pm\sqrt{3}\ i}{2}.$$
Example: For $b=1/2$, we have $b^2-b=1/4-1/2=-1/4$ and $$a = \pm i\sqrt{ \frac{1}{2} - \frac{1}{4 }}=\pm i\sqrt{ \frac{1}{4} }=\pm \frac{1}{2}i .$$ We could have computed as follows
$$a = \sqrt{ \frac{1}{4 }-\frac{1}{2} }=\sqrt{ -\frac{1}{4} }=\sqrt{ -1}\sqrt{ \frac{1}{4} }=\sqrt{ -1}\frac{1}{2}=\pm i \frac{1}{2}=\pm \frac{1}{2}i .$$
Aside from the use if i, one cannot take the square root of negative numbers.
Svenkat's and Google answers: "one cannot take the square root of negative numbers" "Negative numbers don't have real square roots" "you can't take the square root of a negative number"
It is very difficult to break 16th century scholastic dogma, e.g. imaginary numbers, used in taking the square root of negative numbers. Nobody wants to discuss or in other words - taboo. If we are consistent, we should accept to take square roots only from positive numbers. The result of multiplication of the same numbers is always a positive number. If we take the square root of a negative number, we should place the negative sign before the radical sign/radix -√. Details: https://globid.eu/
