factor pairs

831 and Level 3

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Print the puzzles or type the solution on this excel file: 12 factors 829-834

Before I write a blog post, I look to see how the post number is expressed in different bases. Today I noticed that 831 is 30333 in BASE 4. I was intrigued by all those 3’s because I knew that 831 = 3 × 277. It seems logical that 277 would be 10111 in BASE 4, it turns out that it is! I looked at 831 in all the bases up to BASE 36. Did any others have only multiples of 3 as its digits? Yes, a few did, so I’ve made a chart of 277 and 831 in those five bases to make comparing them easy. I also used only base 10 numbers and not letters of the alphabet to represent the digits in the other bases. As you look at this chart, remember 3 × 277 = 831.

Why are those the ONLY bases for which 3 times the digits of 277 equals the digits for 831? Because in every other base, at least one of the digits times 3 will be greater than or equal to the base and some complicated carrying will have to take place to determine the digits for 831 in that base.

For example, 277 is palindrome 1 11 1 in BASE 12. Obviously 3 times 1 11 1 is 3 33 3. Since 33 is bigger than 12, we somehow end up with non-palindrome 5 9 3 in BASE 12 for 831. This is how that somehow happened: 33÷12 = 2R9. The 9 becomes the middle digit while the 2 is added to the original 3 to make the first digit, 5.

Here’s a little more about the number 831:

Because 277 is one of its factors, 831 is the hypotenuse of a Pythagorean triple: 345-756-831, which is 3 times primitive (115-252-277).

  • 831 is a composite number.
  • Prime factorization: 831 = 3 × 277
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 831 has exactly 4 factors.
  • Factors of 831: 1, 3, 277, 831
  • Factor pairs: 831 = 1 × 831 or 3 × 277
  • 831 has no square factors that allow its square root to be simplified. √831 ≈ 8270706

 

 

830 I Can Divide These Polynomials By (x – 2) Without Even Looking at Them

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Print the puzzles or type the solution on this excel file: 12 factors 829-834

(x – 2) is a factor of an infinite number of polynomials. I am listing only a small, but very special subset of them here. First look for the pattern that allows us to generate a polynomial from a given number in base 2. Then look for another pattern when the polynomial is divided by (x – 2).

Do you see the patterns? I do.

From the first pattern I know there is a similar special polynomial that ends with -830. AND I know from the second pattern what I will get if I divide THAT polynomial by (x – 2). Now get this: Even though I haven’t seen the polynomial yet, I know what the quotient will be! When the polynomial ending in -830 is divided by (x – 2), it will be. . . .

  • x⁸ + 3x⁷ + 6x⁶ + 12x⁵ + 25x⁴ + 51x³ + 103x² + 207x + 415

And guess what, I’m right! I found the quotient without showing any steps or even looking at what I was dividing.

How did I know what that quotient would be without writing down the problem and doing some division first? Well, not only is this polynomial special, but the quotient is special, too!

All I needed to know was that the last term was -830. I then divided 830 repeatedly by 2. Any time my quotient was an odd number, I subtracted one from it before I divided it again by 2. I repeated the process until I reached zero. That is how I got all my coefficients. Even though I could do this problem without showing any work, I made a gif so you and anyone else can quickly see how I did it, but you’ll have to look sideways at it to see it. Showing steps is ALWAYS a good thing.
Find 830 in BASE 2

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As a side benefit, this is another way to find out what 830 is in BASE 2.

Here’s a little more about the number 830:

  • 830 is a composite number.
  • Prime factorization: 830 = 2 × 5 × 83
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 830 has exactly 8 factors.
  • Factors of 830: 1, 2, 5, 10, 83, 166, 415, 830
  • Factor pairs: 830 = 1 × 830, 2 × 415, 5 × 166, or 10 × 83
  • 830 has no square factors that allow its square root to be simplified. √830 ≈ 28.80972

830 is the sum of four consecutive prime numbers:

  • 197 + 199 + 211 + 223 = 830

Because 5 is one of its factors, 830 is the hypotenuse of a Pythagorean triple:

  • 498-664-830; that’s 166 times (3-4-5)

 

828 Try Synthetic Division on These Special Polynomials

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828 is the sum of consecutive prime numbers 409 and 419.

828 has a lot of factors so I decided to use it in my examples of synthetic division. What are the factors of 828?

  • 828 is a composite number.
  • Prime factorization: 828 = 2 × 2 × 3 × 3 × 23, which can be written 828 = 2² × 3² × 23
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 828 has exactly 18 factors.
  • Factors of 828: 1, 2, 3, 4, 6, 9, 12, 18, 23, 36, 46, 69, 92, 138, 207, 276, 414, 828
  • Factor pairs: 828 = 1 × 828, 2 × 414, 3 × 276, 4 × 207, 6 × 138, 9 × 92, 12 × 69, 18 × 46 or 23 × 36
  • Taking the factor pair with the largest square number factor, we get √828 = (√36)(√23) = 6√23 ≈ 28.774989.

Synthetic division is taught in many schools in the United States, but in other places in the world it typically is not taught at all. Although I prefer the area model for dividing polynomials, I still like synthetic division. I disagree with those few people who describe it as a mostly useless trick that isn’t worth learning. Yes, its usefulness is limited, but when it can be used, it can be absolutely wonderful. Personally, for many years I always use synthetic division when dividing polynomials by (x-a) or (x+a) where a is any whole number. (If a is a fraction, synthetic division can still be done, but it might not be much fun.)

What are some of the advantages of using synthetic division?

  • If you had a polynomial where x is raised to several different powers, such as x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x – 8, you would only have to write 1 1 1 1 1 1 1 1 1 -8 to perform the algorithm. That could prevent writer’s cramp if the polynomial is quite long. Ha ha. Seriously, less writing often means fewer chances for mistakes.
  • Instead of needing 9×2 lines to do long division for the problem, only three total lines are needed. That saves paper.
  • Using a instead of (x-a) or -a instead of (x+a) in the algorithm means we use addition instead of subtraction to find the quotient. Most people make fewer mistakes adding numbers than they do subtracting. Fewer mistakes means less frustration and less erasing.

Before we can do synthetic division we need to write some polynomials. Since this is my 828th post, I will write some polynomials based on the following chart, and they will be very special polynomials!

The numbers in bold print end in a zero because the corresponding base number is a factor of 828. For base 11 or greater, sometimes a digit is represented by a letter of the alphabet. The key to translating those letters to the corresponding number in base 10 is A = 10, B = 11, C = 12, D = 13, E = 14, F = 15, G = 16, H = 17, I = 18, J = 19, K = 20, L = 21, M = 22. This chart goes to BASE 28 because √828 ≈ 28.77.

We can write a special polynomial using the digits given from a desired base. The last digit for these special polynomials will be replaced with -828, but as you will see, that original last digit will not be forgotten.

Because 828 is 30330 in BASE 4, let’s use that information as our first example:

  • The digits 30330 make the polynomial 3x⁴ + 0x³ + 3x² + 3x -828.
  • The digits 3 0 3 3 -828 will be used as the coefficients in our synthetic division algorithm.
  • BASE 4 will be seen in the divisor (x – 4) and as “4” in the algorithm.

Now watch as this gif uses synthetic division to find the quotient.

 

828 Synthetic Division

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The remainder is zero, and the last digit of 30330 is zero. From the remainder theorem we also know that 3(4⁴) + 3(4²) + 3(4) -828 = 0.

It turns out we can know what the remainder is for each of these special polynomials BEFORE we do any dividing! The remainder will be the last digit times negative one. That does not usually happen when we use synthetic division on a polynomial, but it will always happen on these special polynomials!

Here are a four more examples of writing one of these special polynomials and dividing it using synthetic division. Try writing the rest of the problems using some of the other bases and doing the division yourself, too.

Now for today’s Find the Factors puzzle:

Print the puzzles or type the solution on this excel file: 10-factors-822-828

It’s not an easy puzzle! If you get stumped, here is the logic I used to solve it:

  1. Clue 27 will use a 3, so clue 9 cannot be 3×3. Thus, clues  9 and 18 will put 9 in the first column and 1 and 2 in the top row.
  2. Can both 40’s be 4×10? No, because that would use both 10’s, and make the 8 and the 18 use both 2’s. That would mean that clue 10 could not be 10×1 or 2×5.
  3. So 56 and one of the 40’s will use both 8’s. That means 24 has to use 4 and 6. Thus 24 and 42 will use both 6’s, so 30 will be 10×3.
  4. We know one of the 40’s is 4×10, but we don’t know which one. Nevertheless, we know that its 4 will be in the first column because its 10 cannot be. Since 24 must use 4 and 6, its 4 must be in the top row above the 24.

This table shows the rest of the logic I used:

That was pretty complicated, so here’s where all the factors go, too. 🙂

 

 

How to Find Consecutive Numbers That Sum to 826

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Most numbers greater than 2 can be written as the sum of consecutive numbers. How can you know what those consecutive numbers are? By factoring, of course! Let’s take 826 as an example.

To find consecutive numbers that add up to 826, we are only interested in its odd factors that are less than or equal to 40 AND its factor that is the greatest power of 2. (We arrived at the number 40 because the largest triangular number less than 826 is the 40th triangular number, 820. We could also find the number 40 if we round √(1 + 826×2) – 1 to the nearest whole number.)

The factor of 820 that is the greatest power of 2 is 2. When we double that greatest power of 2, we get 4. The odd factors of 826 that are less than or equal to 40 are 1 and 7. Now we don’t ever count a number being the sum of just 1 consecutive number. So for 826, we are interested in just three numbers, all of which are less than or equal to the maximum number allowable, 40. Those numbers are 7, 1×4, and 7×4.

Thus, we can conclude that 826 can be written as the sum of 4 consecutive numbers, 7 consecutive numbers, and 28 consecutive numbers. Can you figure out what all those consecutive numbers are? How are the consecutive number sums derived from an odd factor different from the sums derived from an even number?

I’ve written out the sum of 4 consecutive numbers as an example and given some hints to help you figure out or check your answer to those two questions:

  • 826÷4 = 206.5, and that number lies right smack in the middle of the 4 consecutive numbers that make this sum: 205 + 206 + 207 + 208 = 826. Note that 205 + 208 and 206 + 207 both add up to 413, a factor of 826.
  • 826÷7 = 118, which is the exact middle number of the 7 consecutive numbers that sum up to 826. Note that 7 × 118 = 826.
  • 826÷28 = 29.5 so the 14 numbers from 16 to 29 plus the 14 numbers from 30 to 43 make the 28 numbers from 16 to 43 that add up to 826. Note that 16 + 43 = 59, a factor of 826.

Here’s today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors-822-828

In order for (821, 823, 827, 829) to be the 4th prime decade, 825 or one of the even numbers between 820 and 830 has to be divisible by 7.

826 is the one that answered that call. Here is the 7 divisibility trick applied to 826:

  • 82-2(6) = 70, a number divisible by 7, so 826 is divisible by 7.

Why does 826 become the palindrome 181 in BASE 25? Because 1(25²) + 8(25¹) + 1(25º) = 826

Let’s begin with 826’s factoring information:

  • 826 is a composite number.
  • Prime factorization: 826 = 2 × 7 × 59
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 826 has exactly 8 factors.
  • Factors of 826: 1, 2, 7, 14, 59, 118, 413, 826
  • Factor pairs: 826 = 1 × 826, 2 × 413, 7 × 118, or 14 × 59
  • 826 has no square factors that allow its square root to be simplified. √826 ≈ 28.7402

 

825 Quarters Make Dividing by 25 Easy

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Numbers ending in 00, 25, 50, and 75 can be divided evenly by 25. How much is 825 divided by 25? That quotient is the same as the answer to “how many quarters are in $8.25?” (A quarter is ¼ of a dollar and is written .25 or 25¢.)

You probably could visualize the answer in your head even if I hadn’t included a picture! That’s why I often ask kids the how-many-quarters question when they are stumped dividing something by 25 . It seems that kids are always able to give the quotient after that question. Notice that “8.25 ÷ .25 =” and  “8 ¼ ÷ ¼ =” have the same answer, too. You can also ask that how-many-quarters question to find the answer when something is divided by .25 or ¼.

It would almost be as easy to divide $8.26 or $8.39 by 25. The quotient would be the same as the problem above but with some loose change becoming the remainder. Using money for division problems could even help kids better understand dividends, divisors, quotients, and remainders.

Here’s an example of a how-many-quarters type question that will help you divide by 75, .75 or ¾.

We can count and see that there are 11 sets of 3 quarters in $8.25. That means that $8.25 ÷ .75 is 11. It also means that 8¼ ÷ ¾ = 11.

Dividing by fractions can be a very abstract concept for students, and they may ask questions like, “What does 8¼ ÷ 1¼ even mean?” Again quarters come to the rescue! 5 quarters can be so much more friendly than 1¼ is. Shorthand for 5 quarters is the fraction, 5/4. Since they have the same denominators, dividing 8¼ by 1¼ is the same as dividing 33 by 5:

Kids think money is more fun than math, but money is just a subset of mathematics which is full of lots of other fun topics, too. Here are a couple of ways other educators have used money to teach a math topic.

Jen of Beyond Tradit’l Math shared her way to teach subtracting decimals using money. Her method will surely captivate any child who tries it and even make regrouping fun:

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Robert Kaplinsky uses several very short videos to keep students engaged without them actually touching any money. Check out the replies, too. Paula Beardell Krieg’s excellent $1.00 art project is there, and that would be fun for anyone 2nd grade or older to do:

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Now back to the number 825.

Clearly 825 has to be divisible by both 5 AND 3 in order for (821, 823, 827, 829) to be the fourth prime decade, which it is.

  • The last digit of 825 is 5, so it is divisible by 5.
  • 8 + 2 + 5 = 15, a multiple of 3, so 825 is divisible by 3.
  • 8 – 2 + 5 = 11, so 825 is divisible by 11.

All numbers ending in 00, 25, 50, or 75 can have their square roots simplified. If you were trying to simplify √825, you could visualize quarters in your mind to easily divide 825 by 25. Then √825 = (√25)(√33) = 5√33

Here is 825’s factoring information:

  • 825 is a composite number.
  • Prime factorization: 825 = 3 × 5 × 5 × 11, which can be written 825 = 3 × 5² × 11
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 825 has exactly 12 factors.
  • Factors of 825: 1, 3, 5, 11, 15, 25, 33, 55, 75, 165, 275, 825
  • Factor pairs: 825 = 1 × 825, 3 × 275, 5 × 165, 11 × 75, 15 × 55, or 25 × 33
  • Taking the factor pair with the largest square number factor, we get √825 = (√25)(√33) = 5√33 ≈ 28.72281

 

 

824 and Level 3

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824 is the sum of all the prime numbers from 61 all the way to 103, which just happens to be one of its prime factors!

Print the puzzles or type the solution on this excel file: 10-factors-822-828

824 is a leg in a few Pythagorean triples:

  • 618-824-1030 because that is 206 times (3-4-5)
  • 824-1545-1751 because that is 103 times (8-15-17)
  • 824-10593-10625 because 2(103)(4) = 824
  • 824-21210-21226 because 105² – 101² = 824
  • 824-42432-42440 because 2(206)(2) = 824
  • 824-84870-84874 because 207² – 205² = 824
  • Primitive 824-169743-169745 because 2(412)(1) = 824

Five of those triples were derived directly from 824’s factor pairs.

Two of the triples were derived indirectly:

  • What is (105+101)/2, (105-101)/2?
  • Also, what is (207+205)/2, (207+205)/2?

The answer to both questions is a factor pair of 824.

You can read more about finding Pythagorean triples for numbers that are divisible by 4 here.

  • 824 is a composite number.
  • Prime factorization: 824 = 2 × 2 × 2 × 103, which can be written 824 = 2³ × 103
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 824 has exactly 8 factors.
  • Factors of 824: 1, 2, 4, 8, 103, 206, 412, 824
  • Factor pairs: 824 = 1 × 824, 2 × 412, 4 × 206, or 8 × 103
  • Taking the factor pair with the largest square number factor, we get √824 = (√4)(√206) = 2√206 ≈ 28.7054

 

 

823 and Level 2

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All of the odd numbers between 820 and 830, except 825, are prime numbers. That makes (821, 823, 827, 829) the fourth prime decade.

Print the puzzles or type the solution on this excel file: 10-factors-822-828

 

  • 823 is a prime number.
  • Prime factorization: 823 is prime.
  • The exponent of prime number 823 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 823 has exactly 2 factors.
  • Factors of 823: 1, 823
  • Factor pairs: 823 = 1 x 823
  • 823 has no square factors that allow its square root to be simplified. √823 ≈ 28.687977

How do we know that 823 is a prime number? If 823 were not a prime number, then it would be divisible by at least one prime number less than or equal to √823 ≈ 28.7. Since 823 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 823 is a prime number.

822 and Level 1

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The sum of 822’s digits is 12, a number divisible by 3. That means that even number 822 can be evenly divided by 2, 3, and 6.

822 is the sum of the 12 prime numbers from 43 to 97.

822 is palindrome 212 in base 20 because 2(20²) + 1(20¹) + 2(20º) = 822.

Print the puzzles or type the solution on this excel file: 10-factors-822-828

  • 822 is a composite number.
  • Prime factorization: 822 = 2 x 3 x 137
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 822 has exactly 8 factors.
  • Factors of 822: 1, 2, 3, 6, 137, 274, 411, 822
  • Factor pairs: 822 = 1 x 822, 2 x 411, 3 x 274, or 6 x 137
  • 822 has no square factors that allow its square root to be simplified. √822 ≈ 28.67054

Numbers up to 820 with Exactly 12 Factors

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Let’s begin with today’s puzzle. Afterwards I’ll tell you a little about the number 820 and why I decided to make a list of all the numbers up to 820 with exactly 12 factors.

Print the puzzles or type the solution on this excel file: 12 factors 815-820

 

Now here’s some information about the number 820:

  • 820 is a composite number.
  • Prime factorization: 820 = 2 × 2 × 5 × 41, which can be written 820 = 2² × 5 × 41
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 820 has exactly 12 factors.
  • Factors of 820: 1, 2, 4, 5, 10, 20, 41, 82, 164, 205, 410, 820
  • Factor pairs: 820 = 1 × 820, 2 × 410, 4 × 205, 5 × 164, 10 × 82, or 20 × 41
  • Taking the factor pair with the largest square number factor, we get √820 = (√4)(√205) = 2√205 ≈ 28.635642

820 is the sum of two squares two ways:

  • 28² + 6² = 820
  • 26² + 12² = 820

Because 5 and 41 are two of its prime factors, 820 is the hypotenuse of FOUR Pythagorean triples:

  • 180-800-820 which is 20 times 9-40-41
  • 336-748-820 which is 4 times 84-187-205, but it could also be calculated from 2(28)(6), 28² – 6², 28² + 6²
  • 492-656-820 which is 164 times 3-4-5
  • 532-624-820 which is 4 times 133-156-205, but it could also be calculated from 26² – 12², 2(26)(12),26² + 12²

Since 820 = (40×41)/2, we know that 820 is the 40th triangular number, thus

  • 1 + 2 + 3 + . . . + 38 + 39 + 40 = 820

820 is also a palindrome in four other bases:

  • 1010101 BASE 3 because 3⁶+3⁴+3²+3⁰=820
  • 1111 BASE 9 because 9³+9²+9¹+9⁰=820
  • 868 BASE 11 because 8(11²)+6(11¹)+8(11⁰)=820
  • 1I1 BASE 21 (I is 18 base 10) because 21²+18(21¹)+21⁰=820

Below is a chart of the numbers up to 820 with exactly 12 factors. Notice that two sets of consecutive numbers, (735, 736) and (819, 820), are on the list. Look at their prime factorizations:

  • 735=3×5×7², 736=2⁵×23
  • 819=3²×7×13, 820=2²×5×41

Those prime factorizations mean that while 735 and 736 are the smallest consecutive numbers with exactly 12 factors, 819 and 820 are the smallest consecutive numbers whose prime factorizations consist of one prime number squared and exactly two other primes. Thanks to OEIS.org for alerting me to that fact. Here’s something interesting about the chart: of the 77 numbers listed, only six are odd numbers.

So, how did I know what numbers to put on the list?

In order to determine how many numbers up to 820 have exactly 12 factors, we must first factor 12. We know that 12=12, 6×2, 4×3, and 3×2×2.

Next we subtract 1 from each of those factors to determine the exponents we need to use:

12 gives us 12-1=11. For prime number a, when is a¹¹ not larger than 820? Never, because 2¹¹>820.

6×2 gives us 6-1=5 and 2-1=1. For prime numbers a and b, with a≠b, when is a⁵b¹ less than or equal to 820? These nine times:

  • 2⁵×3=96, 2⁵×5=160, 2⁵×7=224, 2⁵×11=352,
  • 2⁵×13=416, 2⁵×17=544, 2⁵×19=608, 2⁵×23=736
  • 3⁵×2=486

4×3 gives us 4-1=3 and 3-1=2. For prime numbers a and b, with a≠b, when is a³b² not larger than 820? These six times:

  • 2³×3²=72, 2³×5²=200, 2³×7²=392
  • 3³×2²=108, 3³×5²=675
  • 5³×2²=500

3×2×2 gives us 3-1=2, 2-1=1, and 2-1=1. For distinct prime numbers a, b, and c, when is a²bc not larger than 820? 52 times. Here’s the breakdown: It happens 35 times when 2²=4 is the square number:

And it happens another 27 times when a prime number other than 2 is squared:

 

That’s a lot of numbers with exactly 12 factors! After I sorted all the numbers that I found in numerical order, I was able to make that chart of numbers up to 820 with exactly 12 factors, and yes 819 and 820 are the smallest two consecutive numbers whose prime factorization consists exactly of one prime number squared and two other prime numbers.

819 How to Type eˣᵖᵒⁿᵉⁿᵗˢ in WordPress

/ ivasallay / 4 Comments

Exponents, ⁰¹²³⁴⁵⁶⁷⁸⁹, are written to the right of their base numbers and a little higher. They are about half the height and about half the width of the base number, too.

Exponents are important to me. They and other special characters allow me to include factoring information and interesting number facts in every post I write. For example …

  • 819 is a composite number.
  • Prime factorization: 819 = 3 × 3 × 7 × 13, which can be written 819 = 3² × 7 × 13
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 819 has exactly 12 factors.
  • Factors of 819: 1, 3, 7, 9, 13, 21, 39, 63, 91, 117, 273, 819
  • Factor pairs: 819 = 1 × 819, 3 × 273, 7 × 117, 9 × 91, 13 × 63, or 21 × 39
  • Taking the factor pair with the largest square number factor, we get √819 = (√9)(√91) = 3√91 ≈ 28.618176

1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² + 11² + 12² + 13² = 819, making 819 the 13th square pyramidal number.

315² + 756² = 819² so 819 is the hypotenuse of a Pythagorean triple.

2⁹ + 2⁸ + 2⁵ +  2⁴ + 2¹ + 2⁰  = 819 because 819 is 1100110011 in BASE 2.

I like that pattern of 1’s and 0’s. Here are a few more of 819’s cool number patterns:

  • 3⁶ + 3⁴ + 3² = 819 because 819 is 101010 in BASE 3.
  • 3·4⁴ + 3·4² + 3·4⁰ = 819 because 819 is 30303 in BASE 4.
  • 3·16² + 3·16¹ + 3·16º = 819 because 819 is 333 in BASE 16.

———————————

Okay. Enough about 819. HOW do we type exponents when we write a blog?

Option #1: WordPress gives us some special characters in the editor. I’ve put red boxes around the exponents so you can find them faster:

As you can see, the WordPress’s editor only offers us º ¹ ² ³ ª as exponents, and they MIGHT fill all your needs. (Who am I kidding?) You can get to any of the symbols shown above by clicking on the Ω symbol in YOUR WordPress editor. I’ve put a red box around the Ω special character symbol in the PICTURE of the editor below.

Those symbols are good if you’re writing x³ or even 8¹º³². But what if you want to write an expression with a 4, 5, 6, 7, 8, or 9 as part of the exponent? Do you really have to settle for (2^7)×(3^5) when you really want to type 2⁷×3⁵? That carrot ^ symbol can look needlessly intimidating to people even if they are familiar with exponents.

So how do we type all those other exponents in WordPress? That is something I have been frustrated about and have googled about many times. I’ve read about and tried a couple more options: Superscripts and LaTeX.

Option #2 Superscripts: When I followed the superscripts’ directions for WordPress, and typed e<sup>xponents</sup> in the text editor as instructed, it made beautiful eˣᵖᵒⁿᵉⁿᵗˢ in the visual editor, but look at all these exponents marked in red, they fell down when I published this post (until I upgraded my WordPress theme). That was not acceptable. Thus depending on the theme used, some people may be able to get those superscripts to stay up, while others may not. On a related note: While writing this post I learned something useful about Microsoft Word. If you push down the Shift, Ctrl, and = keys at the same time, you can type in superscript in a Microsoft Word document. (You press the same keys to get out of superscript mode). Unfortunately, if you copy and paste that superscript writing into WordPress, they don’t stay up.

Option #3. LaTeX can be a great looking option. Still, when the exponents from the WordPress editor are typed alongside those in LaTeX, they can look a little wobbly: 2³ + 2^4 + 2^5 + 7² + 3^4. In addition, LaTeX looked like LaTeX notation instead of exponents when I tried to use it in the title of this post.  The biggest drawback: LaTeX looks good when it’s published, but it is practically unreadable when it’s being typed. For example, without spaces, you must type [ latex ]2^4[ /latex ], just to get 2⁴. (If I took out the spaces it would read 2⁴ instead of showing you what LaTeX notation looks like.)

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This week I found a good 4th option: Microsoft Word has quite a few exponents, and WordPress liked them!

I’ve gathered the superscripts of the English alphabet and numbers from Microsoft Word in one place and included them here for the convenience of all other bloggers, making this post a great 5th option. Copy what you need from here, or copy and paste the whole list into a handy document of your own. True, not every letter of the English alphabet is available as an exponent in Word, but most of them are. This is the method I used to include eˣᵖᵒⁿᵉⁿᵗˢ in the title of this post.

x⁰¹²³⁴⁵⁶⁷⁸⁹ᴬᵃᵅᴮᵇᶜᴰᵈᴱᵉᶠᴳᵍᴴʰᴵⁱᴶʲᴷᵏᴸᴹᵐᴺⁿᴼᵒᴾᵖᴿʳˢᵀᵗᵁᵘⱽᵛᵂʷˣʸᶻ ⁺ ⁻ ⁼ ⁽  ⁾ Those superscripts or exponents stay up! And…these subscripts stay down! ₉₈₇₆₅₄₃₂₁₀ ₊ ₋ ₌ ₍ ₎ aₐ eₑ  jⱼ oₒ xₓ. Curiously,  hₕ kₖ lₗ mₘ nₙ pₚ sₛ tₜ seem to stay down on home computers but disappear on smartphones so you might not want to use them.

Subscripts are often used in notation for Permutations like ₆P₃ or Combinations like ₆C₃. (Those links will take you to some useful online calculators.)  Subscripts used with “⅟ ” can write infinitely many unit fractions like ⅟₃₂₁. Subscripts can also be used to write the base of logarithms such as log₂4=2.

Here’s a bonus, the Greek letters: Some of the Greek letters have superscripts and/or subscripts next to them, while others do not. For some reason unknown to me, Microsoft Word didn’t give π either one. (I could not have written this part of the post without zooming to 175% first. You might want to do that before using any of these, too.):

Ααᵅ, Ββᵝᵦ, Γ⸀γᵞᵧ, Δδᵟ, Εεᵋ, Ζᶻζ, Ηη, Θᶱθᶿ, Ιᶦιᶥ, Κκ, Λᶺλ, Μμ, Νᶰν, Ξξ, Οο, Ππ, Ρρ, Σσ, Ττ, Υυᶹ, Φᶲφᵠᵩ, Χχᵡᵪ, Ψψ, Ωω

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Those exponents from Microsoft Word will allow you to write important identities like the following without using awkward LaTeX notation:

  • sin t = (eⁱᵗ – e־ⁱᵗ)/2i
  • cos t = (eⁱᵗ + e־ⁱᵗ)/2

Back to the 4th option, Microsoft Word does include some other incomplete alphabets from other languages that are not included in my lists above. Here’s what you’ll need to do to get subscripts or superscripts from Microsoft Word:

  1. In Microsoft Word click on “insert”,
  2. click on “symbol”,
  3. click on “symbol” (NOT “equation” because WordPress won’t copy anything you type there),
  4. click on “more symbols”.
  5. Next LOOK for the desired superscript or subscript on the chart. You may have to look for a while. Some of the them are listed together, while others seem to be randomly placed by themselves. For the alphabet, only use a letter that is in the top CENTER of its box. If you use a letter that is in the top LEFT of its box, you might end up typing something like 3 ͩͪ  or 7ͪͫ.
  6. Type your expression in Word, then copy and paste it onto your blog.

So now you have been saved countless hours of frustration trying to type a few simple exponents or subscripts. Perhaps, now you can chance getting frustrated trying to solve this Level 5 puzzle?!

Print the puzzles or type the solution in this excel file: 12 factors 815-820