Alolan_Apples
“Assorted” Collector
This is another fact entry to my blog. I apologize for the lack of organization, but that big organization was for the original StarFall Press. I did want to talk about many things throughout the entire time. But I had to let the AC stuff come in first. In other news, I am no longer advertising my blog in my signature anymore. I'm going full on Oliver & Company on my forum profile. You can still see my old sig inside the gray bar, but I'm done now. You can check my blog at any time you like. I already written 100+ entries.
Now onto the subject. As you should know, I excel at math very well. I can find patterns very easily. Math isn't that hard for me, even on the higher levels. So if you need help on math, you can PM me. This blog is a math blog, discussing the domain restrictions and why they exist.
As you should know, a domain restriction is where you are limited to some numbers in a function. Addition and subtraction both have no domain restrictions since they're open to all numbers. But for other functions like sine, cosine, tangent, exponents, and even differentiation and integration, there are certain sets of numbers that cannot be used. For example, when you use an arcsine (reverse sine function), you are only limited to using numbers between -1 to 1 (including these ugly decimal numbers. If it's any higher than 1 or lower than -1, you're going to get no solution.
But there are three domain restrictions in math (besides my arcsine/arccosine example I used) that are base restrictions.
So there we have it. That's all three restrictions. Tomorrow's entry (in case I write one for tomorrow) will be about some weird changes you'll never see in chemistry.
Now onto the subject. As you should know, I excel at math very well. I can find patterns very easily. Math isn't that hard for me, even on the higher levels. So if you need help on math, you can PM me. This blog is a math blog, discussing the domain restrictions and why they exist.
As you should know, a domain restriction is where you are limited to some numbers in a function. Addition and subtraction both have no domain restrictions since they're open to all numbers. But for other functions like sine, cosine, tangent, exponents, and even differentiation and integration, there are certain sets of numbers that cannot be used. For example, when you use an arcsine (reverse sine function), you are only limited to using numbers between -1 to 1 (including these ugly decimal numbers. If it's any higher than 1 or lower than -1, you're going to get no solution.
But there are three domain restrictions in math (besides my arcsine/arccosine example I used) that are base restrictions.
The first domain restriction to go over is the most infamous one - dividing by zero. The reason why these restrictions exist is because math keeps piling up. So we know that adding and subtracting are basic, right? An extension to addition is multiplication, where we keep adding the same number over and over again. But every function has an inverse. Subtraction is to addition as division is to multiplication. Exponents have two inverses. Roots are to finding the base as logarithms are to finding the exponents. As math continues piling up, we discover operations that are impossible to perform. And zero division is one of them.
Here is some proof behind zero division being impossible:
So here we have it. You can't have zero groups of one thing to equal a number, you can't keep adding zero to get the number you want, and you can't have zero as the whole when you have some as the part.
Other related restrictions:
However, zero division isn't always ruled out. It may equal infinity because if you notice, a fraction gets bigger when the denominator gets smaller. When it gets infinitely small, the number is infinitely huge. But don't take it as fact that any number divided by zero is always infinity. It can also equal negative infinity. A positive can't be equal to its negative counterpart, so division by zero will always be a restriction.
Here is some proof behind zero division being impossible:
- Groups - one way to divide is to put everything into groups. The divisor can either be the number of groups you have or the number per group.
- Let's say you have 24 marbles. You want to put them into six groups. So you take six marbles out, and separate them as you add more marbles to each group. You'll end up having four per group. What about six marbles per group? You draw out six marbles at a time, but you don't separate them. You keep doing this until you have no more marbles. That's four groups.
- Now let's try doing this with zero. The truth here is that it's literally impossible to have no groups of one number to get a certain number. If 24 is the dividend, and you want to divide that into 0 groups, that's impossble because any number time zero equals zero. If you have zero per group, that could be easier, but a group with nothing is hardly a geoup at all. But you can try. Each time when you keep adding zero, you get the same number. You can have as many groups of zero as you want. But the sum will always be zero. How you can get to 24 if you only have nothing in each group? It's impossible to keep adding zero to get 24.
- Adding and subtracting divisor - you can try division by counting how many times you can add the divisor to get to the dividend, or subtracting the divisor from the dividend until you get to zero too.
- You have 20 as your dividend, and two is your divisor. How many times can you add two to zero to get to 20? The answer, 10 times. The same is true if you want to get to 0 by subtracting two from 20.
- Replace two with zero. You want to divide 20 by zero. Remember, adding zero will always have the same result. Same with subtracting zero. It's impossible to get to 20 by adding only zero to zero. It's also impossible to get to zero from 20 by only subtracting zero. As a result, zero division is undefined.
- Fractions - fractions can be read the same way as division, where the numerator is the dividend, as the denominator is the divisor. In general, the denominator is the whole thing as the numerator is only part of it. If the numerator exceeds the denominator, it means you got extra from what the denominator has for you. If the numerator is twice as high, that's two times as much. If it's thrice as high, that's three times as much. In this case, you may result in division. But can zero be the denominator? No. How can you have anything from the pile if there's nothing in the pile to begin with?
So here we have it. You can't have zero groups of one thing to equal a number, you can't keep adding zero to get the number you want, and you can't have zero as the whole when you have some as the part.
Other related restrictions:
- Tangent of 90° (degrees) or pi/2 (radians) - if you're using a triangle, you have to realize that the opposite side of the right angle is the hypotenuse. The tangent formula is OL/AL (opposite leg over adjacent leg). Both legs are adjacent, so we may never know which one is correct. By using trigonometric identities, the tangent is the same as the sine over the cosine (sinx/cosx=tanx). The sine of 90° is 1, but the cosine of 90° is 0. This results in a zero division. Not to mention, but the cycle of tangents repeat itself every 180° or half rotation in radians. Using the trig identities, a function is undefined when any of the following is true:
- Degree measures are 0°, 180°, or 360° when the sine is in the denominator. Radian measures are 0, pi, or 2pi. This implies to cosecants and cotangents.
- Degree measures are 90° or 270° when the cosine is in the denominator. Radian measures are pi/2 and 3pi/2. This implies to tangents and secants.
- 0 base when exponent is 0 or negative - when you raise any number to the 0th power, the result will always be 1. When you raise 0 to any power, the result will be 0. But when you raise 0 to the 0th power, you get a fight between the two answers, and this is a lose/lose situation. 0^0 can't equal 1 because 0 raised to any power is 0. It also can't equal 0, so this is an undefined result. When raising it to a negative exponent, we have to take the reciprocal of 0, but 0 can't have a reciprocal because that means you have to divide by 0.
- 0th root of a number - as you should know, roots are an inverse to exponents. Any number raised the the 0th power equals 1. So if we take the 0th root of 1, you can get any number, so this is indeterminate. You can't take the 0th root of any number but 1 since it's impossible. Also, roots are the same as rational exponents. A square root has an exponent of 1/2. A cube root has an exponent of 1/3. So if we have a 0th root, we have an exponent of 1/0, which is undefined.
However, zero division isn't always ruled out. It may equal infinity because if you notice, a fraction gets bigger when the denominator gets smaller. When it gets infinitely small, the number is infinitely huge. But don't take it as fact that any number divided by zero is always infinity. It can also equal negative infinity. A positive can't be equal to its negative counterpart, so division by zero will always be a restriction.
Another major restriction is taking the square root (or other even root) of a negative number. You see, there are very good reasons why an even root of a negative number has no real solution.
As a result, the square root of a negative number is undefined. However, there are such things as imaginary numbers. Those come from square roots of negative numbers. So after all, you can find the square root of a negative number, but you won't get a real number that way. This is why it's a domain restriction.
- A number times itself results in an exponent.
- A negative times a negative equals a positive. This is true even with even exponents.
- A number can't equal its own negative.
As a result, the square root of a negative number is undefined. However, there are such things as imaginary numbers. Those come from square roots of negative numbers. So after all, you can find the square root of a negative number, but you won't get a real number that way. This is why it's a domain restriction.
The last restriction to go over is finding the logarithm of a number. The logarithm always grabs for the exponent that connects the base to the number you're "logging". For a base greater than one, when the number you're logging is greater than one, then you get a positive result. If it's 1, the answer is 0. If it's less than 1, the answer is negative. Basically, the answer has to the the path from the base number to the number you're logging.
If the number turns out to be 0, the result is undefined. Why? Because what power do you need to raise the base to get to 0? It's not possible, unless you use infinity. As for negative numbers, it's not possible to raise a positive base to get a negative number.
If the number turns out to be 0, the result is undefined. Why? Because what power do you need to raise the base to get to 0? It's not possible, unless you use infinity. As for negative numbers, it's not possible to raise a positive base to get a negative number.
So there we have it. That's all three restrictions. Tomorrow's entry (in case I write one for tomorrow) will be about some weird changes you'll never see in chemistry.
