Let’s visit some mathematical expressions typically formed due to the distributive laws of addition and multiplication. For the purposes of this tutorial, let’s restrict our domain to two-digit numbers.
(a + b) squared = a squared + b squared + 2 * a * b.
(ab) squared = a squared + b squared – 2 * a * b.
(a+b)*(c+d)=ac+ad+bc+bd.
(a + b) * (c – d) = ac – ad + bc – bd.
(ab) *(c+ d) = ac + ad -bc -bd.
(ab)(cd) = ac – ad – bc + bd.
Let’s rewrite a number K as a sum or difference of two numbers in the form
(a +b) or (ab), where a is equal to the nearest multiple of 10 less than or slightly greater than the number K.
For example, 63 would be rewritten as (60 + 3) and 68 would be rewritten as
(70-2). 73 would be rewritten as 70 + 3, and 79 would be rewritten as 80 – 1.
First let’s calculate 47*47
Let’s use the formula (ab) * (ab) = a * a – 2 * a * b + b *b where a = 50 and b = 3.
So 47 * 47 = 2500 – 300 + 9 = 2209.
Let’s use this formula (a+b) * (a + b) = a*a + 2 * a * b + b*b and let’s calculate 73 * 73
73 * 73 = (70 + 3) ( 70 + 3 ) = 4900 + 420 + 9 = 5329.
26 * 26 = (30 – 4)(30 – 4) = 900 – 240 + 16 = 676.
Use (a +b)(c +d) to calculate 93 * 57 as
(90 + 3) * ( 50 + 7 ) = 4500 + 630 + 150 + 21, simplify this further
writing this as 4500 + 600 + 30+100 + 50 + 20 + 1, 4500 + 700 + 101 which can be easily calculated as 5301
Let’s similarly calculate 47*69 using the expression for (ab)(cd). Let’s rewrite the product as (50-3) (70-1). Expanding individual terms like
3500 – 50 – 210 + 3, rewriting this as 3500 – 200 – 60 + 3; rewriting this as
3500 – 200 – 60 + 3; So 47 * 63 = 3243.
Using the expression (ab)(c+d) let’s calculate 43 * 88, rewrite the original expression as (40 +3)*(90 – 2). Calculate individual sums such as 3600 – 80 + 270 – 6 = 3870 – 86 = 3870 – 70 – 16 = 3800 – 16 = 3784.
Let’s use the expression (ab)(cd) = ac – ad -bc + bd to calculate 57 * 79 quickly. Rewriting this as (60 – 3) * (80 -1) = 4800 – 60 – 240 + 3 = 4800 – 300 + 3 = 4503.
Extending the analysis to 3-digit numbers, these simple sums of products can easily be formulated using the distributive laws of addition and multiplication, respectively.
Let’s calculate 153 * 94 quickly, rewriting 153 as 100 + 50 +3 and 94 as 90 + 4. Let’s write the original product as (100+ 50 + 3) (90 + 4); the sum of the individual terms is 9000 + 400 + 4500 + 200 + 270 + 12. Sorting is by thousands, hundredths, and Units. 13000+1000+300+82; quickly calculate the sum as 14382.