n-MULTIPLES, BONUSES, BASALS, USES
n-MULTIPLES (n = 8, 7, 6, 13, etc.)

The POWER2 and POWER5 PATTERNS are easy to see in a 10s-GRID, because 2n is a factor of 10n which enumerates the decimal notation; similarly, with 5n.

The 9-PATTERN and the 11-PATTERN are easy to see in a 10s-GRID because NINE and ELEVEN are THE IMMEDIATE NEIGHBOR OF THE BASE TEN used for the decimal system and the GRID.

However, less "immediate neighbors" also "cast their patterns" in a 10s-GRID. But these are less obvious than those for NINE and ELEVEN.

Consider number EIGHT. In decimal notation, 8 = 10 - 2. Then MULTIPLES OF EIGHT "cast a pattern" which

Actually, as part of the POWERTWO PATTERNS, you''ve already seen the EIGHT PATTERN.

You may guess from this what happens in the SEVEN PATTERN. In decimal notation, 7 = 10 - 3, so MULTIPLES OF SEVEN "cast a pattern" which

As you can see.

And guessing about the SIX PATTERN should be easy. In decimal notation, 6 = 10 - 4, so MULTIPLES OF SIX "cast a pattern" which

As may be seen.

Actually, as you learn elsewhere, in the SIEVING MATHTIVITY, you NEED ONLY BE CONCERNED ABOUT PRIME NUMBER PATTERNS in DECIMAL NOTATION. The first few PRIMES less than 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. You've already, earlier, gone through the TWO PATTERN and the FIVE PATTERN and the ELEVEN PATTERN; and I just showed you the SEVEN PATTERN.

And the (prime) THREE PATTERN. Ah! You implicitly learned about this in studying the NINE PATTERN. Remember the RULE? Add decimal digits repeatedly until a single digit is obtainained: the DIGITAL ROOT. If the DIGITAL ROOT of a number is 9, THEN THE NUMBER IS A MULTIPLE OF 9. The same RULE applies to THREE: repeatedly add the decimal digits until a single digit is obtaineed. IF THE DIGITAL ROOT IS THREE OR A MULTIPLE OF THREE -- namely, 3, 6, or 9 -- THEN THE NUMBER IS A MULTIPLE OF THREE!. Consider the prime 123 when multiplied by 3: 131 x 3 = 393. Add its digits: 3 + 9 + 3 = 15, and 1 + 5 = 6, hence, a MULTIPLE of 3. Conversely, confronted with 393, to test for PRIMALITY or NONPRIMALITY, you elicit the THREE FACTOR AND DIVIDE IT OUT: 393/3 = 131. We need only test 131 for PRIMES 2, 3, 5, 7, 11. We have easy tests which eliminate 2, 3, 5, 11, and 7 can be easily tested and rejected by division. (This exercise illustrates the uses of SIEVING and COLORED PATTERNING, combined.)

Just to see one more PATTERN, consider THIRTEEN. In decimal notation, 13 = 10 + 3. We say that, for 11 = 10 + 1, YOU DROP DOWN ONE ROW IN THE GRID (for the 10 part), and SHIFT ONE COLUMN TO THE RIGHT (FOR THE + 1 PART). Here, sine 13 = 10 + 3, YOU DROP DOWN ONE ROW, AND SHIFT THREE COLUMNS TO THE RIGHT, as may be seen.

I've carried the discussion this far to emphasize the EFFECT OF THE ACTIVITHM IN THE FORM OF AN IMPOSITION OF BASE TEN ON THE NUMBERS. "The ACTIVITHM-IN-DECIMAL FORM CASTS PATTERNS!

ALGEBRAIC BONUSES

This is a discussion about NUMERICAL ALGEBRA. I make this distinction because mathematicians speak of many kinds of algebra: ALGEBRA OF STATEMENTS, BOOLEAN ALGEBRA, ALGEBRA OF SETS, ALGEBRA OF VECTORS, ALGEBRA OF LIMITS ("calculus"), etc.

Starting with ELEMENTARY SCHOOL, kids should be shown that decimal notation such as 543210 is SIMPLY A SHORTCUT for the POLYNOMIAL FORM: 5·105 + 4·104 + 3·103 + 2·102 + 1·101 + 1·100. (Here, for example, 103 = 10·10·10; and 101 is SHORTCUT SIMPLY to 10; and 100 = 1. Also, in taking up algebra, where we use the term "x" for "the unknown", you shift from using "x" or "X" for the multiplication sign and use the medial dot, ., to avoid confusion.) Another reason for translating positional decimals into the polynomial form is that students (too often ignored today) who are afflicted by dyscalia or aculia -- and have trouble lining up numerals to add, subtract, or multiply -- are given the EQUALOPPORTUNITY ALTERNATIVE OF USING THE POLYNOMIAL FORMS WHICH MAKE THE LINING UP DIGITS A MORE OBVIOUS TASK.

In NUMERICAL ALGEBRA, an EQUATION such as x5 - x4 + x3 - x + x - 1 = 0 CAN BE THOUGHT OF AS A STRUCTURE WRITTEN IN THE BASE x) the form x - 1 is HOMOLOGOUS TO 9 IN THE BASE 10. That is, we have THE HOMOLOGY:


		x: x - 1 :: 10 : 9
Hey! That CONNECTION TELLS US SOMETHING. THE NINES RULE (BASE TEN) HAS A FORM IN BASEx.

RULE: ADD THE COEFFICIENTS OF AN ALGEBRAIC EQUATION IN x; IF THE SUM IS ZERO, THEN THE EQUATION HAS A FACTOR x - 1, that is:


	x5 - x4 + x3 - x + x - 1 =
        (x - 1)(x4 + x3 + x + x + 1).
So, "casting out nines" in ARITHMETIC prepares the student for "casting out x - 1" in NUMERICAL ALGEBRA.

But, hey! We saw evidence of a CODIGITAL ROOT in connection with the 11 = 10 + 1 PATTERN. That suggests a HOMOLOGY of the form, x: x + 1 :: 10 : 11. CORRECT? YES!!! To show this, let's multiply that equation cited above by x + 1, so we know it has this factor, then see what happens.


(x + 1)(x5 - x4 + x3 - x + x - 1) = 0, simplifying to x6 - 1 = 0
. Clearly, the COEFFICIENT SUM is 1 - 1 = 0, so it has the x - 1 factor.

Testing this same equation for the x + 1 factor is a little more tedious, since we must put zeros for the powers in between these two explicit terms to work out the ALTERNATION.

From left to right the coefficients are: 1, 0, 0, 0, 0, 0, -1. Since the alternation proceeds from right to left, let's rewrite that: -1, 0, 0, 0, 0, 0, 1.

Now. we alternate, left to right, with the positive-one and negative-one multipliers:


(+1)(-1), (-1)(0), (+1)(0), (-1)(0), (+1)(0), (-1)(0), (=1)(0) -> -1, 0, 0, 0, 0, 0, 1.
Clearly, the numbers on the right sum as: -1 + 0 + 0 + 0 + 0 + 0 + 1 = -1 + 1 = 0. And what does that denote? That the cited equation not only has a factor x - 1, but it also has a factor x + 1, or combining, (x - 1)(x + 1) = x2 - 1. Wherefore, knowing this, the student can divide that equation by the know factor x2 - 1 for any easier equation to try to solve! Dig?

Thus, something learned in ARITHMETIC sometimes pays off big in NUMERICAL ALGEBRA.

OTHER BASES

Besides the DECIMAL or DENARY BASE, two other BASES are important in our COMUTER AGE:

The interesting point is that, IN THE OCTAL SYSTEM:

You can see this in the SEVEN-DIAGONAL PATTERN.

And in the NINE-DIAGONAL PATTERN.

But this last result is SIMPLY ANOTHER RESULT OF THE ACTIVITM STRATEGY. IMPOSE A DIFFERENT PATTERN (not TENS, but EIGHTS) UPON THE NUMERS AND DIFFERENT SUBPATTERNS APPEAR!
USES: WHAT "GOOD" IS ALL THIS PATTERNING FOR KIDS LEARNING ARITHMETIC?

Kids go on to FACTORING in ARITHMETIC and, later as teens, to FACTORING in NUMERICAL ALGEBRA. The PATTERNS teach them many useful FACTORING ALGORITHMS. To see this click here.