Multiply like a magician
Tonight's idea
The distributive law says \(a(b + c) = ab + ac\). It is the engine behind every multiplication trick, and the easiest way to believe it is as area. A rectangle \(a\) tall and \(b + c\) wide can be cut into two rectangles, \(a \times b\) and \(a \times c\). The area does not change when you cut.
Now the magician's version. Write \(23 \times 17\) as \((20 + 3)(20 - 3)\). Multiply out: \(400 - 60 + 60 - 9 = 400 - 9 = 391\). The middle terms always cancel. In general:
\((a + b)(a - b) = a^2 - b^2\). Any two numbers the same distance either side of a round number multiply in one step.
And a second trick for free. To square a whole number ending in 5, like 35: take all the digits before the final 5 as a number, 3, multiply by the next number up, 4, giving 12, and write 25 after it: \(35^2 = 1225\). For 105, use 10, not the tens digit 0: \(10 \times 11 = 110\), then append 25 to get \(105^2 = 11025\). It works because \((10n + 5)^2 = 100n^2 + 100n + 25 = 100\,n(n + 1) + 25\).
Watch one
\(47 \times 53\). Both are 3 away from 50. So \(50^2 - 3^2 = 2500 - 9 = 2491\).
\(65^2\). The number before the final 5 is 6, next up is 7, \(6 \times 7 = 42\), append 25: \(4225\).
\(7 \times 48\). Use \(48 = 50 - 2\): \(350 - 14 = 336\).
Your turn
- Distributive law, in your head: \(6 \times 199\), \(\;25 \times 36\) (hint: \(36 = 4 \times 9\)), \(\;8 \times 73\), \(\;15 \times 24\).
- Difference of squares: \(19 \times 21\), \(\;28 \times 32\), \(\;98 \times 102\), \(\;45 \times 55\).
- Squares ending in 5: \(45^2\), \(\;95^2\), \(\;105^2\), \(\;2.5^2\) (the trick still works; think about where the decimal point goes).
- \(999 \times 1001\). Then \(999 \times 999\), using \((1000 - 1)^2 = 1000^2 - 2 \times 1000 + 1\).
- Draw the rectangle picture for \(34 \times 26\) split as \((30 + 4)(20 + 6)\) and add the four parts. Then do it the fast way as \((30 + 4)(30 - 4)\) and check the answers agree.
- Square 41 and 39 using \((40 \pm 1)^2 = 40^2 \pm 2 \times 40 + 1\). Then explain why \(41^2 - 39^2 = 160\) without squaring anything.
Stretch: a proof you can cut out
Draw an \(a \times a\) square, with \(a > b > 0\). Cut a \(b \times b\) square out of one corner. The L-shape left over has area \(a^2 - b^2\). Now cut the L-shape into two rectangles and turn one of them a quarter turn and move it so that together they make a single rectangle. What are its length and width? You have just proved the difference-of-squares formula with scissors.
Then a last trick: for numbers near 50, \((50 + n)^2 = 2500 + 100n + n^2\). So \(53^2 = 2500 + 300 + 9 = 2809\). Use it for \(47^2\) (take \(n = -3\)) and \(58^2\).
Answers
- 1194 (\(1200 - 6\)); 900 (\(25 \times 4 = 100\), then \(\times 9\)); 584 (\(560 + 24\)); 360 (\(15 \times 24 = 30 \times 12\)).
- 399; 896 (\(900 - 4\)); 9996 (\(10000 - 4\)); 2475 (\(2500 - 25\)).
- 2025; 9025; 11025 (\(10 \times 11 = 110\), append 25); 6.25.
- \(999{,}999\) (\(1000^2 - 1\)). \(999^2 = 1{,}000{,}000 - 2000 + 1 = 998{,}001\).
- \(600 + 180 + 80 + 24 = 884\). Fast way: \(900 - 16 = 884\).
- \(41^2 = 1600 + 80 + 1 = 1681\); \(39^2 = 1600 - 80 + 1 = 1521\). Difference of squares: \(41^2 - 39^2 = (41 + 39)(41 - 39) = 80 \times 2 = 160\).
Stretch. The two pieces of the L-shape are \(b \times (a - b)\) and \(a \times (a - b)\) (or similar, depending on your cut). Laid end to end they form a rectangle \((a + b)\) long and \((a - b)\) wide, area \((a+b)(a-b)\). Since nothing was thrown away, that equals \(a^2 - b^2\). Near-50 trick: \(47^2 = 2500 - 300 + 9 = 2209\); \(58^2 = 2500 + 800 + 64 = 3364\).