Powerful Patterns 5: Fraction Freakiness!

Look at the list in the virtual chalkboard below. You should notice a pattern...

1/2 × 1/2 = 1/4

(1 + 1/2) × 1/2 = 3/4

(2 + 1/2) × 1/2 = 1 + 1/4

(3 + 1/2) × 1/2 = 1 + 3/4

(4 + 1/2) × 1/2 = 2 + 1/4

(5 + 1/2) × 1/2 = 2 + 3/4

(6 + 1/2) × 1/2 = 3 + 1/4

(7 + 1/2) × 1/2 = 3 + 3/4

(8 + 1/2) × 1/2 = 4 + 1/4

(9 + 1/2) × 1/2 = 4 + 3/4

(10 + 1/2) × 1/2 = 5 + 1/4

(11 + 1/2) × 1/2 = 5 + 3/4

(12 + 1/2) × 1/2 = 6 + 1/4

(13 + 1/2) × 1/2 = 6 + 3/4

When the fractions on the left are each multipled by ½, the fractions on the right side of the equal sign form this checkered pattern!

If you look at each product closely, you'll notice how the fractions in the mixed numbers alternate between 1/4 & 3/4.

And about the beginning of the pattern: Don't forget that zero(0) is the number of additive identity. So 0 + 1/2 = 1/2.

Look out, because here comes another list with a similar pattern just below!

1/3 × 1/2 = 1/6

(1 + 1/3) × 1/2 = 2/3

(2 + 1/3) × 1/2 = 1 + 1/6

(3 + 1/3) × 1/2 = 1 + 2/3

(4 + 1/3) × 1/2 = 2 + 1/6

(5 + 1/3) × 1/2 = 2 + 2/3

(6 + 1/3) × 1/2 = 3 + 1/6

In this list, the denominator in each fraction in the left column is 3. If you look at each product closely, you'll notice how the fractions in the mixed numbers alternate between 1/6 & 2/3.

Again, Dottie Doll asks you this question:

These checkered patterns in the math equations look neat & interesting, don't they? The fraction in the mixed number of the product also depends on whether the integer in the mixed number on the left side of the equal sign is even or odd.

Maybe just 1 more list with a checkered pattern like this...

2/5 × 1/2 = 1/5

(1 + 2/5) × 1/2 = 7/10

(2 + 2/5) × 1/2 = 1 + 1/5

(3 + 2/5) × 1/2 = 1 + 7/10

(4 + 2/5) × 1/2 = 2 + 1/5

(5 + 2/5) × 1/2 = 2 + 7/10

(6 + 2/5) × 1/2 = 3 + 1/5

(7 + 2/5) × 1/2 = 3 + 7/10

(8 + 2/5) × 1/2 = 4 + 1/5

In this list, the numerator in each fraction in the left column is 2 instead of 1 & the denominator in each fraction in the left column is 5. If you look at each product closely, you'll notice how the fractions in the mixed numbers alternate between 1/5 & 7/10.

The multiplier in each list is specifically equal to ½ for the sake of this math trick so that you can get checkered patterns like these! But you're allowed to pick any starting fraction you want in the multiplicands of your list! With an even number added to the fraction, you'll get a specific fraction in the product & another with an odd number added to your starting fraction!

Speaking of patterns, maybe you already seen some other numerical math patterns that other mathematicians already discovered & published! (Besides me or you; any open-minded person can be a great mathematician, like me!)

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