Category Archives: Amazing Facts

Facts about the colors

The weekend is over, so I am back to blogging. This is technically my ninth week, but due to the one week hiatus, I will classify this as Week 8.

Today’s entry is going to be about colors. I will cover information on the RYB model and RGB model.

Red-Yellow-Blue

The common model of colors is the red-yellow-blue model, which is used in art, such as painting. In this model, there are five primary colors, the colors that no other color builds up to. These colors are red, yellow, blue, black, and white. Three of these are on the color wheel, which means they have a hue. The other two are neutral colors at the two extreme points. Not including neutral colors, red, yellow, and blue are the primary colors.

There are three properties of colors: Hue, Value, and Intensity (or Saturation in the RGB model):

Hue:

The hue is the location of the color on the color wheel. The three parent colors are red, yellow, and blue. Assuming that they all have the same value (no white or black), blue is the darkest of the hues as yellow is the brightest. The colors with a hue are on the color wheel. The ones without are neutral colors.

Assuming that you only have three colors of paint – red, yellow, and blue. The question is, how are you going to get more colors. By mixing them. Here are the types of colors:

  • Primary – Red, Yellow, and Blue.
  • Secondary – Two primary colors combined where no primary color has more than one amount. In simple English, orange, green, and purple.
  • Tertiary – A combination of a secondary color and a primary color that built up to the color. One primary color is three times as strong as the other.
  • Quaternary – Although this is unofficial, quaternary colors are colors in between primary and tertiary, or secondary and tertiary. This includes all hues in between.
  • Hot – colors of the fire. All colors with more yellow than yellow-green and more red than red-violet are considered hot colors. Red, red-orange, orange, yellow-orange, and yellow are all considered hot. Yellow-green and red-violet are mild.
  • Cold – colors of the water. All colors with more blue than yellow-green and red-violet are considered cold. Green, blue-green, blue, blue-violet, and violet (purple) are all considered cold.
  • Neutral – colors without a hue. Black, white, gray, and brown are all considered neutral.

Let’s say that we have a color mixing lab. Each drop is one fluid ounce. Let’s say that four drops makes a full color.

  • Primary – four drops of one primary color.
  • Secondary – two drops of one primary color, and two drops of another.
  • Tertiary – three drops of one primary color, and one drop of the other.
  • Quaternary (strong) – a color in between a primary color and tertiary color. So if one drop is one fluid ounce, less than one drop, but more than none of one primary color is needed as the other needs more than three drops, but less than four.
  • Quaternary (weak) – a color in between a secondary color and tertiary color. So we need more than one drop of one and less than three drops of the other, but neither should be in equal amounts.

To summarize each arc, when we have two primary colors with a ratio, it is a pure color at 1:0. At a tertiary color, we can add the stronger primary color to become a strong quaternary color. We may get to the point when we have a 1:0 or 0:1, depending on what the former color and latter colors are. If we add the weaker primary color to a tertiary color, we get a weak quaternary color. If we keep it up, we get to a pure secondary color, where the ratio is 1:1. When we add more of a primary color to a secondary color, it moves away to a weak quaternary color, a tertiary color, or a strong quaternary color.

Value:

Assuming that all hues are bases, we get to the second property – value. To increase the value, white needs to be added. You can’t subtract colors once mixed in, and adding black makes a gray mixture to the color. Therefore, only a pure primary color can be used for value. White makes colors lighter as black makes colors darker. A color lighter than another of the same hue has a higher value, as a color darker than another has a lower value.

When value is above normal, we have a tint. When value is below normal, we have a shade.

Going back to the color mixing lab, strength applies to tints and shades too.

  • Weak tint – a color where the hue exceeds white. They are lighter than normal, but still close to the base.
  • Medium tint – a color where the hue and white are balanced.
  • Strong tint – a color where white exceeds the hue. These colors are very light.
  • Weak shade – a color where the hue exceeds black. They are darker than normal, but still close to the base.
  • Medium shade – a color where the hue and black are balanced.
  • Strong shade – a color where black exceeds the hue. These colors are very dark.

You can use brown or gray as the substitute as well.

Intensity:

The intensity is the brightness or dullness of a color. There are two base neutral colors that are in neither extreme: brown and gray. Brown comes from mixing a primary color with the opposite secondary color (or complementary colors). Gray comes from mixing black and white. When a hue has none of the complement, it is 100% bright. You can add gray to weaken the intensity as we get a tone of gray. To go to the brown side, either add brown or a complement.

So once again, we see the color mixing lab.

  • Absolute bright – a color where we see 100% of the hue with no mix of gray, brown, or the complement.
  • Strong tone – a color where the hue exceeds gray, brown, or the complement.
  • Medium tone – a color where the hue is balanced with gray, brown, or the complement.
  • Weak tone – a color where the hue is lesser than gray, brown, or the complement.
  • Neutral – just gray or brown. The hue is completely absent.

Red-Green-Blue

So we are done with the RYB model, so let’s take a look at the colors of the light (including computers, TVs, and other devices that use light. In science, this model is the true color model. There are only three primary colors this time, which are red, green, and blue. The hue, value, and saturation are dependant on how much of red, green, or blue you have. Even black and white aren’t primary colors anymore, but they’re still neutral. Add to that, brown isn’t a neutral color either.

Hue:

Like I said on the RYB model, the hue is the location on the color wheel. The RGB model is different to the RYB. Instead of mixing colors, we have amounts of red, green, and blue in lighting. The primary colors are different.

To change the hue, value, or saturation, each of the three colors have a specific value in the colors of R, G, and B. For example, black is R=0, G=0, B=0. Pure red is R=255, G=0, B=0. Pure green is R=0, G=255, B=0. Pure blue is R=0, G=0, B=255. White is R=255, G=255, B=255.

  • Primary – red, green, and blue. Any color where two primary colors have the same value, but the other primary color is dominant.
    • If saturation is 100%, then both recessive colors must be 0 or the dominant color must be 255.
    • If value is 50% (assuming that white is 100% and black is 0%), both recessive colors should still be the same amount, but the sum of the value of the dominant color and one of the recessive colors must equal 256.
    • If both saturation is 100% and value is 50%, then one color must be 255 as the other must be 0.
  • Secondary – any color where two primary colors have the exact same value, but the other primary color is recessive. Basically yellow, cyan, or magenta.
    • If the saturation is 100%, then both dominant colors must be 255 or the recessive color must be 0.
    • If the value is 50%, both dominant colors must be the same amount, but the sum of the value of one of the dominant colors and the recessive color must equal 256.
    • If both saturation is 100% and value is 50%, then two colors must be 255 as one is 0.
  • Tertiary – any color that is exactly in between the primary and secondary color. In order to be a tertiary color, all three colors must be different and have a common difference (assuming that 255 can be rounded off to 256). At this point, we have a dominant color, an intermediate color, and a recessive color.
    • If the saturation is 100%, then dominant color must be 255 or the recessive color must be 0.
    • If the value is 50%, then the sum of the dominant color and recessive color must be 256 (or 255) while the intermediate color must remain to be 128.
    • If both saturation is 100% and value is 50%, then the dominant color is 255, intermediate color is 128, and recessive color is 0.
  • Quaternary – like I said with the RYB model, a quaternary color is any color between the primary and tertiary colors or the secondary and tertiary colors. Once again, the values of red, green, and blue are different, but only this time, there is no common difference.
    • If the saturation is 100%, then the dominant color must be 255 or the recessive color must be 0.
    • If the value is 50%, then the sum of the dominant color and recessive color must be 256 while the intermediate color must be unequal to 128.
    • If both saturation is 100% and value is 50%, then the dominant color must be 255, the recessive color must be 0, and the intermediate color must not be 128.
  • Hot – All colors with more red than violet (tertiary color between blue and magenta) and lime (tertiary color between yellow and green) are considered hot colors. Magenta, hot pink, red, orange, and yellow are all considered hot. Violet and lime are mild.
  • Cold – all colors with more green than lime or more blue than violet are considered cold. Green, turquoise, cyan, sky blue, and blue are all considered cold.
  • Neutral – all colors where the red, green, and blue values are completely equal.

Let’s re-open the quaternary color strength. Like the RYB model, a strong quaternary color is closer to the primary color as a weak quaternary color is closer to a secondary color. So let’s say that the saturation is 100% and the value is 50%. In order to be a tertiary color, the intermediate color must be 128. If the intermediate color is less than 128, we have a strong quaternary color. As it keeps going down, it may reach a pure primary color. If the intermediate is greater than 128, we have a week quaternary color. As it keeps going up, it may reach a pure secondary color.

The last subject on the hue property on the RGB model is color families. Each color is part of one family based on how dominant or recessive one color is:

  • No pure primary color is part of a secondary color family.
  • No secondary color is part of a prime color family.
  • Red family – colors where red is the dominant color.
  • Yellow family – colors where blue is the recessive color.
  • Green family – colors where green is the dominant color.
  • Cyan family – colors where red is the recessive color.
  • Blue family – colors where blue is the dominant color.
  • Magenta family – colors where magenta is the recessive color.

Value:

The difference between changing value on the RYB model and RGB model is that the colors on the RGB value increase in respect to each other. It is much easier on the HSV model than the RGB model (which are the same colors, but different readings).

To make a tint of a color with saturation of 100%:

  • In a primary color, the dominant color is always 255. The recessive colors always have the same value. A tint is stronger when both recessive colors go up.
  • In a secondary color, both dominant colors are always 255. A tint is stronger when the recessive color goes up.
  • In a tertiary or quaternary color, the dominant color is always 255. The ratio between the intermediate color and recessive color is always the same no matter what the difference is. The intermediate color goes up more slowly than the recessive color during an increase in value, depending on how strong the quaternary color is.
  • The strength of a tint is determined on how much the recessive color has:
    • Weak tint – recessive color is less than 128.
    • Medium tint – recessive color is 128.
    • Strong tint – recessive color is greater than 128.

To make a shade of a color with a saturation of 100%.

  • In a primary color, the recessive colors are always 0. A shade is stronger when the dominant color goes down.
  • In a secondary color, the recessive color is always 0. The dominant colors always have the same value. A shade is stronger when both dominant colors go down.
  • In a tertiary or quaternary color, the recessive color is always 0. The ratio between the intermediate color and dominant color is always the same no matter what the difference is. The intermediate color goes down more slowly than the dominant color during a decrease in value, depending on how strong the quaternary color is.
  • The strength of a shade is determined on how much the dominant color has:
    • Weak shade – dominant color is greater than 128.
    • Medium shade – dominant color is 128.
    • Strong shade – dominant color is less than 128.

Saturation:

The RGB version of intensity is saturation. This time, there’s only one neutral color – gray. The saturation of RGB is dependent on how far the dominant and recessive colors are from each other. A 100% bright color has the dominant color being 255 or the recessive color being 0. When the recessive color(s) get(s) greater than 0 while the dominant color(s) get(s) less than 255, the saturation decreases. The closer the values are, the grayer the color is. It is completely neutral when all three colors have the same value.

Back to the strength of a color again:

  • Absolute bright – saturation is 100%. Either the dominant color is 255 or the recessive color is 0.
  • Strong tone – saturation is greater than 50%. The difference between the dominant and recessive colors is greater than the median.
  • Medium tone – saturation is 50%. The difference between the dominant and recessive colors is the median.
  • Weak tone – saturation is less than 50%. The difference between the dominant color and recessive colors is less than the median.
  • Neutral – saturation is 0%. All three colors are the same.

And that concludes the color property facts.

Facts about the numbers (and the functions)

While I already had two fact entries, this one is more of a true fact entry. The purpose of these entries is to share some amazing facts of the world. I am still not far from the math entries yet. I am going over three different types of facts, the facts about all 10 numbers to each digit, other numerical facts, and some interesting math theories.

Digital Facts

All ten numbers that can be in a digit have some amazing facts about them. It doesn’t matter which one is which, they all have something amazing.

  1. 1 is the only natural number to be neither prime nor composite.
  2. 2 and 5 are the only prime numbers to end with a 2 or a 5.
  3. 3 is the last one’s digit number to be part of a composite number.
  4. 4 is the smallest composite number. Also, it’s the only number to have the same amount of letters in its name as the value in the English language.
  5. 5 is the only odd number to never be prime in numbers with two or more digits.
  6. 6 is the smallest number to have a composite amount of factors.
  7. There are 7 days in the week.
  8. No perfect square ends with a 2, 3, 7, or 8.
  9. 9 is the only odd composite number smaller than 10.
  10. All products ending with a 1, 5, 6, and 0 will always have the same ones digit if both factors multiplied have the same ones digit.

Other Number Facts

  • The sum of the first 10 numbers is 55. The average is the same as the median, which is 5.5.
  • The product of the first 10 numbers is 3,628,800. You can find the product by using factorials (!) too.
  • The least common multiple of the first 10 numbers is 2,520.
  • The palindrome of 89 and 98 uses the most iterations out of all numbers under 10,000. 196 doesn’t have a palindrome as of now.
  • 64 is the smallest number besides 1 that can be both a square and a cube.
  • The perimeter and area of a square is the same number if the side length is 4.
  • The surface area and volume of a cube is the same number if the edge length is 6.
  • 1,000 is the first number to have an “a” in its word name.
  • All numbers from 1 to 10 have 3, 4, or 5 letters in their names.
  • 7 is the smallest multisyllabic number greater than 0. 12 is the largest number to have only one syllable.

Interesting Math Facts

  • The sum of a trigonometric function (take sine for example) and a linear term (either positive or negative) deflects the cycle and creates a trend line that resembles the linear function.
  • Multiplying to or dividing x from a sine or cosine function on the outside causes the amplitude to change as the wave diverges from the center. Multiplication makes the amplitude greater as it moves away as division decreases it.
  • Exponents and roots of the term inside a trigonometric function alters the wavelength as the wave diverges from the center. Exponents increase the frequency away from the center, roots do the same thing, but only decreases.
  • Adding a number ending with a 1 increases the ones digit by 1 every time, then moves from 9 to 0 at the end. Adding a number ending with a 9 does the opposite to the ones digit, but still makes the overall number bigger.
  • Adding a number ending with a 2 makes the ones digit go in this pattern: 1, 3, 5, 7, and 9 for the odds, and 2, 4, 6, 8, and 0 for the evens. Adding a number ending with an 8 makes it go backwards.
  • Adding a number ending with a 3 makes the ones digit go in this pattern: 3, 6, 9, 2, 5, 8, 1, 4, 7, 0. Adding a number ending with a 7 (or going to the next date on the same day of the week) makes the ones digit go the other way.
  • Adding a number ending with a 4 makes the ones digit go in this pattern: 1, 5, 9, 3, and 7 for the odds, and 4, 8, 2, 6, and 0 for the evens. Adding a number ending with a 6 makes it go backwards.
  • Adding a number ending with a 5 makes the ones digit alternate between two different numbers. 1 and 6, 2 and 7, 3 and 8, 4 and 9, and 5 and 0. Adding a number ending with a 0 yields a constant result.
  • Multiplying two numbers with the same base and different exponent will keep the base the same, but the exponents add up. Multiplying two numbers with a different base, but same exponent will keep the exponent the same, but the bases multiply.
  • The discriminant in the quadratic formula will always be 0 when plugging in a perfect square trinomial in the quadratic formula.

Defactorization

One hobby I enjoy is counting things. As I play with things or count things, I discover new things. Today, I’m going over factoring and what I have found while playing with factors. I’ve been doing this for a long time. It’s a high time I explain my theory.

Counting the Factors

Defactorization is the process where you count out the factors and then use the factor count as the number’s answer. There are two ways to count out the factors. First way is the long method. The other way is the short method.

The Long Method:

The long method is where you try to plug in a number to see if it fits into the number. One way to try is by using the divisibility test. Here are the rules:

  1. All natural numbers are divisible by 1.
  2. A number is divisible by 2 if the number is an even number. All even numbers have a 2, 4, 6, 8, or 0 in the one’s digit.
  3. A number is divisible by 3 if the sum of the digits is divisible by 3. Most smaller numbers (like 3 or less digits) that fit this rule have a sum of 3, 6, 9, 12, 15, or 18 if using its digits.
  4. A number is divisible by 4 if the form of the last two digits is divisible by 4. To tell if it’s divisible by 4, it has to be an even number to be divisible by 4. The ten’s digit must be even if the one’s digit is 0, 4, or 8. The ten’s digit must be odd if the one’s digit is 2 or 6.
  5. A number is divisible by 5 if the number ends with a 5 or 0.
  6. A number is divisible by 6 if the sum of the digits is divisible by 3 AND is an even number.
  7. A number is divisible by 7 if the difference between all but the last digit and double the last digit is divisible by 7. Basically, you have to double the last digit and subtract it from the rest of the number. If the result is divisible by 7, then the number is divisible by 7.
  8. A number is divisible by 8 if the form of the last three digits is divisible by 8. In order to divide evenly by 8, it has to be divisible by 4. The hundred’s digit must be even if the last two digits are 00, 08, 16, 24, 32, 40, 48. 56. 64. 72, 80, 88, or 96. The hundred’s digit must be odd if the last two digits are 04, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, or 92.
  9. A number is divisible by 9 if the sum of the digits is divisible by 9.
  10. A number is divisible by 10 if the number ends with a 0.

I can’t go over higher numbers, but if you divide all of these numbers out of the big picture and the result isn’t a prime number, you still have a lot to go.

Let’s try the number 2,520 and see if it passes the divisibility test.

  1. 2,520 is divisible by 1 since it’s a natural number.
  2. 2,520 is divisible by 2 since it’s an even number.
  3. 2,520 is divisible by 3 since the sum of the digits is 9, which is divisible by 3.
  4. 2,520 is divisible by 4 since the form of the last two digits is 20, which is divisible by 4.
  5. 2,520 is divisible by 5 since it ends with a 5 or 0.
  6. 2,520 is divisible by 6 since it’s an even number divisible by 3.
  7. 2,520 is divisible by 7. Without the 0, the number is 252. The difference between the first two numbers and double the last number is 21, which is divisible by 7.
  8. 2,520 is divisible by 8 since the form of the last three digits is 520, which is divisible by 8.
  9. 2,520 is divisible by 9 since the sum of the digits is 9, which is obviously divisible by 9.
  10. 2,520 is divisible by 10 since it ends with a 0.

Fact: 2,520 is the least common multiple of the first 10 numbers.

Now let’s try 541. Remember, if the number isn’t divisible by 2, then it can’t be divisible by other even numbers, including 4, 6, and 8. If the number isn’t divisible by 3, it can’t be divisible by 6, 9 or other multiples of 3. So let’s start.

  1. 541 is divisible by 1 since it’s a natural number.
  2. 541 isn’t divisible by 2 since it’s an odd number.
  3. 541 isn’t divisible by 3 since the sum of the digits is 10, which is not divisible by 3.
  4. 541 isn’t divisible by 4 since it isn’t divisible by 2.
  5. 541 isn’t divisible by 5 since it ends with a 1, which is neither 5 or 0.
  6. 541 isn’t divisible by 6 since it fails both the 2 and 3 divisibility tests.
  7. 541 isn’t divisible by 7 since it the difference between the first two numbers and double the last digit is 52, which isn’t divisible by 7.
  8. 541 isn’t divisible by 8 since it isn’t divisible by 4.
  9. 541 isn’t divisible by 9 since the sum of the digits is 10, which is not divisible by 9.
  10. 541 isn’t divisible by 10 since it doesn’t end with a 0.

So 541 fails the divisibility test of all but one of them. Try plugging in the other prime numbers. It fails on them too. In fact, the closest number after plugging in the other numbers to the midpoint, 23, is not a factor. Therefore, 541 is a prime number.

Our number to factor is 24. To factor out a number, you need to see if it passes the divisibility test. If if passes for that number, then the number is one of the factors. Then you divide the number by that factor, and you put in both the divisor and the quotient in the factor list.

  • 24 is divisible by 1. A rule of thumb is that all numbers have one and itself as two of the factors. So 1 and 24 are factors.
  • 24 is divisible by 2 since it’s an even number. 24 divided by 2 is 12, making 2 and 12 factors.
  • 24 is divisible by 3 since the sum of the digits is 6, which is divisible by 3. 24 divided by 3 is 8, making 3 and 8 factors.
  • 24 is divisible by 4 since the ten’s digit is even while the number ends with a 4. 24 divided by 4 is 6, making both 4 and 6 factors.
  • 24 is not divisible by 5 since the number ends with a 4, which is neither 5 or 0. Therefore, 5 is not a factor.

When we keep factoring until we get to the point where both numbers are at their closest, then we can do no more. So our factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Now let’s count how many factors there are. All I see are 8 factors. So the factor count of 24 is 8.

The Short Method:

The short method is better off for larger numbers since it takes longer to use the long method. The short method is tree factoring. To do this, you divide the lowest divisible number from the big number to branch off. Once one divisor was stripped off from the other, we get two branches, which are the divisor and the quotient. For every iteration you make, the first branch goes down as the same number as the other number breaks down. The whole process continues until all what remains are prime numbers.

Once again, we will be using the number 24. Here is a diagram of the process:

Tree Factoring

To word it out, 24 is our main number. To branch off from the number, I once again use the divisibility test. We can’t use 1 since that doesn’t really do anything. According to the divisibility test, 24 is divisible by 2, so we can “branch off” 2 from the number. 24 divided by 2 gets us 12, so our numbers are 2 and 12. We already taken care of 2, so we should go to 12 now. Once again, we have another number divisible by 2, so we take 2 down again, as we get 2 and 6. The 2 we broke off earlier can move down as 2 again as we got a new branch. 6 is divisible by 2, so we “branch off” one more time, and our results are 2 and 3. Since all we have left are prime numbers, we count how many of each prime number we see. There are three 2’s, but one 3. For every repeat of a number, we give it a superscript that we call “exponents” as the count is the exponent’s number. If there are one of each, the exponent is 1. If there are two of it, the exponent is 2. If there are three of the same number, the exponent is three. Our prime factorization is 2^3*3.

How can we count factors by using prime factorization? To do this, we strip off the exponents from the number. Notice that a number without an exponent really has an exponent. The only reason why it doesn’t have an exponent is because the exponent is 1. 2^3*3 is the same as 2^3*3^1. The exponents in this prime factorization are 3 and 1. The next step is to add 1 to each exponent, which results in 4 and 2. Then we multiply what remains. Our result is 8. Anything familiar? 8 was the factor count of the long method, and we got the same from the short method.

So to count how many factors by using the short method, here is the following:

  1. Put the number into prime factorization. You can do this through tree factoring like in the diagram.
  2. Strip the exponents from the bases. We only want to focus on the exponents. Remember that a number without an exponent has an exponent of 1.
  3. Add 1 to every exponent.
  4. Multiply what remains.

And here we are, the factor count. For every new prime number introduced, we have a new number to multiply to the number. Here are the most common re-occurrences:

  • A simple prime number with an exponent of 1 has 2 factors.
  • A prime number with an exponent of 2 has 3 factors.
  • A prime number with an exponent of 3 has 4 factors.
  • A prime number with an exponent of 4 has 5 factors.
  • Two prime numbers with both exponents being 1 has 4 factors.
  • When one has an exponent of 2, but the other has 1, there are 6 factors.
  • When one has an exponent of 3, but the other has 1, there are 8 factors.
  • When both have an exponent of 2, there are 9 factors.
  • When one has an exponent of 3, but the other is 2, there are 12 factors.
  • When both numbers have an exponent of 3, there are 16 factors.
  • When there are three prime numbers with an exponent of 1, there are 8 factors.
  • When one prime number has an exponent of 2, but the other two have 1, there are 12 factors.
  • When one prime number has an exponent of 3, but the other two have 1, there are 16 factors.
  • When two prime numbers have an exponent of 2, but the other has 1, there are 18 factors.
  • When all three prime numbers have an exponent of 2, there are 27 factors.
  • When there are four prime factors with an exponent of 1, there are 16 factors.

Defactorization

Defactorization is when we use the factor count as the number’s secondary number. The primary number is the case in point. The secondary number is the factor count. And then, we continue defactorizing until we get to the point when we have no new results. This number is 2. Some numbers have more iterations than others. The level of defactorization is the number of iterations it takes to get to 2. Here is a chart of the first 100 numbers color-coded by their levels of defactorization:

Defactorization Chart

I will now explain what the colors mean.

  • Black – there is only one black number on this chart, which is 1. Since 1 is neither prime nor composite, there is no way we can get to 2. The level of defactorization is 0.
  • Gray – prime numbers. A number is considered a prime number if the factors are 1 and itself. Since it takes only one iteration to get to 2 (including 2 itself), the level of defactorization is 1.
  • Red – numbers where the secondary number (or factor count) is a prime number. Notice that all prime numbers except for 2 are odd-numbered. All numbers with an odd-number of factors are perfect squares, but not all perfect squares have a prime number of factors. The only ones with a possibility are the ones with only one prime factor. Because of this, these numbers are the rarest type of numbers. Since the secondary number is a prime number other than 2, the level of defactorization of these numbers is 2.
  • Yellow – numbers where the tertiary number is a prime number. As we continue piling up, we start getting numbers where the secondary number is a composite number. At this point, the prime number (odd-numbered) in the hierarchy is the tertiary number. These numbers are one of the most common numbers. On the chart seen above, there are 34 of these numbers. In order to be this kind of number, the secondary number must have a prime number as its factor count (besides 2). They can have 4 factors, 9 factors, 16 factors, or 25 factors. However, the majority of these numbers are 4-factored numbers. A number has 4 factors either because they are cubes of prime numbers (like 8) or have no composite factors except for itself while there are two prime factors. Next in line are the 16-factored numbers. They are mostly numbers with four prime factors with an exponent of 1 or have three prime factors when one has an exponent of three as the other to have exponents of one. The level of defactorization of these numbers is 3.
  • Green – numbers where the quaternary number is a prime number. As we continue piling up, this is when we begin having quaternary numbers in the hierarchy (besides 2). These numbers take 4 iterations to get to the number 2. There are no numbers of this type from 10 and under, and only four of these numbers under 25. As we start diving into larger numbers, these get increasingly common. We start seeing them when we have 6-factored numbers. Then we see 8-factored numbers, and 10-factored numbers, 14-factored numbers, 15-factored numbers, and so on. But the most common numbers of this type have 6 or 8 factors. A number has 6 factors when there are two prime factors, but one has a maximum exponent of 2 while the other is 1. A number has 8 factors when there are three prime factors with an exponent of 1, two prime factors when one has an exponent of 3 (like 24), or when there is only one prime factor, but has an exponent of 7. The level of defactorization of these numbers is 4.
  • Cyan – numbers where the quaternary number is a composite number (but the quaternary number’s factor count is prime). These are the golden numbers of defactorization. They take 5 iterations (that’s right, 5 iterations) to get to 2. As numbers keep getting bigger, they have more factors. Eventually, we may get to a factor count where the level of defactorization of that number is 4. The smallest number of this kind is 60. Most numbers of this type have 12 factors, either because there are three prime factors as only one has a maximum exponent of 2 while the others are 1, or because there are two prime factors where one has an exponent of 3 while the other’s is 2. The first few numbers of this type have 12 factors. The first number of this type to not have 12 factors is 180. All numbers of this type under 300 are even numbers, and all numbers of this type under 100 are divisible by 3. Notice that there are only five two-digit numbers of this type, which are 60, 72, 84, 90, and 96. They are rare at first, but they begin getting more common when there are three digits or more. The first odd number of this type is 315. The level of defactorization of these numbers is 5.

Coding:

Aside to the color-coding of these numbers, there is another type of coding. The level of defactorization can be simply expressed by having a number next to the letter L. For instance, L3 numbers are numbers where the level of defactorization is 3, also the numbers where the prime number is the tertiary number. Here are the list of levels and their stats:

  • L0: Level of defactorization is 0. 1 is the only number of this level.
  • L1: Level of defactorization is 1. The primary number is a prime number. The first two numbers of this level are 2 and 3.
  • L2: Level of defactorization is 2. The secondary number is a prime number. The first number of this level is 4.
  • L3: Level of defactorization is 3. The tertiary number is a prime number. The first number of this level is 6.
  • L4: Level of defactorization is 4. The quaternary number is a prime number. The first number of this level is 12.
  • L5: Level of defactorization is 5. The quaternary number is an L2 composite number. The first number of this level is 60.

L5 isn’t the highest level, but it is the highest level from numbers 1 to 1,000. There’s only one number under 10,000 to be at an even higher level.

Proof of Defactorization:

If you want to know what I mean by iterations, I mean by factor counting each step until I can do this no more. The reason why 2 is the stopping point is because if I continue doing this, I’m only going to get the same result. It’s all about getting unique results. I’ll begin with this process.

  • L1 numbers: 3 is a good example. The factors are 1 and 3. The factor count is 2. That’s one iteration.
  • L2 numbers: 4 has 3 factors. The factor count is 3. The second factor count is 2. That’s two iterations.
  • L3 numbers: 36 has 9 factors. The factor count is 9. The second factor count is 3. The third factor count is 2. That’s three iterations.
  • L4 numbers: 48 has 10 factors. The factor count is 10. The second factor count is 4. The third factor count is 3. The fourth factor count is 2. That’s four iterations.
  • L5 numbers: 72 has 12 factors. The factor count is 12. The second factor count is 6. The third factor count is 4. The fourth factor count is 3. The fifth factor count is 2. That’s five iterations.

I was able to do this with the first 100 numbers and first 1,000 numbers, but not the first 10,000 numbers. As the factor count is a higher level, the primary number adds one more to its level.

And that’s it about defactorization. You now know a new theory.

Units of Measurement – Positives and Negatives

Last week was the first week for Town of StarFall. Every other Monday, I will have a list of facts and anything else educational as part of my blog. You might want to see some interesting facts about the world, as I’ll share it here.

If you want to know, math is my strong subject in school. I may have some math as part of my blog. Today’s entry will be about the units of measurement, and what are the positives and negatives of measuring. An increase in a measurement is positive, but a decrease is negative.

Length:

Measurement Positive Negative
Distance Longer Shorter
Length Longer Shorter
Width Wider Narrower
Height Taller Shorter
Depth Deeper More Shallow
Altitude Higher Lower

Weight/Mass:

Measurement Positive Negative
Weight Heavier Lighter
Mass Bigger Smaller
Density More Mass Less Mass
Density Smaller Volume Greater Volume

Temperature/Pressure:

Measurement Positive Negative
Temperature Hotter/Warmer Colder/Cooler
Air Pressure More Dense Less Dense
Water Pressure Stronger Weaker
Pressure Greater Weaker

Time/Speed:

Measurement Positive Negative
Time Longer Shorter
Speed Faster Slower
Acceleration Accelerating Decelerating

Volume:

Measurement Positive Negative
Capacity More full More empty
Size Bigger Smaller

Light/Sound:

Measurement Positive Negative
Wavelength Longer Shorter
Wave Frequency More frequent Less frequent
Brightness Brighter Dimmer
Sound Pitch Higher Pitch Lower Pitch
Volume Louder More silent

Of course, there is more than what I stated, but this is all I can name today. If you have any more to say, or if I got some wrong, feel free to comment.