Sunday Afternoon Maths VL
Integer part
Let \(\lfloor x\rfloor \) denote the integer part of \(x\) (eg. \(\lfloor 7.8\rfloor =7\)).
When are the following true:
a) \(\lfloor x+1\rfloor = \lfloor x\rfloor + 1\)
b) \(\lfloor nx\rfloor = n\lfloor x\rfloor\) (where \(n\) is an integer)
c) \(\lfloor x+y\rfloor = \lfloor x\rfloor +\lfloor y\rfloor \)
d) \(\lfloor xy\rfloor = \lfloor x\rfloor \lfloor y\rfloor \)
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a) Always
b) When \(n<\frac{1}{f_x}\), where \(f_x\) is the fractional part of \(x\).
c) When the fractional parts of \(x\) and \(y\) add up to less than one.
d) Let \(f_x\) and \(f_y\) be the fractional parts of \(x\) and \(y\) (respectively).
$$\lfloor xy\rfloor = \lfloor (\lfloor x\rfloor +f_x)(\lfloor y\rfloor +f_x)\rfloor $$
$$=\lfloor \lfloor x\rfloor \lfloor y\rfloor +f_x\lfloor y\rfloor +f_y\lfloor x\rfloor +f_yf_x\rfloor $$
$$=\lfloor x\rfloor \lfloor y\rfloor +\lfloor f_x\lfloor y\rfloor +f_y\lfloor x\rfloor +f_yf_x\rfloor $$
This will be equal to \(\lfloor x\rfloor \lfloor y\rfloor \) when \(\lfloor f_x\lfloor y\rfloor +f_y\lfloor x\rfloor +f_yf_x\rfloor =0\).
For this to be true, it is necessary (but not sufficient) that \(f_y<\frac{1}{x}\) and \(f_x<\frac{1}{y}\).
Extension
Show that
$$\lfloor x\rfloor +\left\lfloor x+\frac{1}{n}\right\rfloor +\left\lfloor x+\frac{2}{n}\right\rfloor +...+\left\lfloor x+\frac{n-1}{n}\right\rfloor =\lfloor nx\rfloor.$$
Make the sums
Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums reading across and down are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, \(4+3\times2\) is 14, not 10.
| + | | - | | = 4 |
| + | | - | | × | |
| - | | × | | = 27 |
| - | | × | | ÷ | |
| × | | ÷ | | = 16 |
=
2 | | =
8 | | =
6 | |
Show answer & extension
Hide answer & extension
| 1 | + | 5 | - | 2 | = 4 |
| + | | - | | × | |
| 7 | - | 4 | × | 9 | = 27 |
| - | | × | | × | |
| 6 | × | 8 | - | 3 | = 16 |
=
2 | | =
8 | | =
6 | |
Extension
Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums reading across and down are correct.
| + | | - | | = 5 |
| - | | - | | - | |
| + | | ÷ | | = 5 |
| + | | ÷ | | × | |
| + | | × | | = 99 |
=
0 | | =
1 | | =
18 | |
