The Sign of One

Make the following numbers using FOUR 1s using any mathematics operators and/or symbols, such as $dfrac{x}{y}$, $sqrt{x}$, decimal dots, $+$, $-$, $times$, $div$, $($ $)$, etc by the Sign of One.

Click the numbers below to see the answers.

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1

( 1 = 1 times 1 times 1 times 1 )

2

( 2 = (1+1) times 1 times 1 )

3

( 3 = (1+1+1) times 1 )

4

( 4 = 1+1+1+1 )

5

( displaystyle 5 = frac{1}{.1} ] div (1+1) )

Note that the dot ius the decimal point such as ( .1 = 0.1 )

6

( (1+1+1)! times 1 )

Note that the exclamation mark ! means factorial such as ( 3! = 3 times 2 times 1 = 6 )

7

( 7 = (1+1+1)! + 1 )

8

( displaystyle 8 = frac{1}{.dot{1}}-1 times 1 )

Note that ( .dot{1} = 0.111 cdots )

9

( displaystyle 9 = frac{1}{.dot{1}} +1-1 )

10

( displaystyle 10 = frac{1}{.dot{1}} + 1 times 1 )

11

( displaystyle 11 = frac{1}{.dot{1}} + 1 + 1 )

12

( 12 = 11 + 1 times 1 )

13

( 13 = 11 + 1 + 1 )

14

( displaystyle 14 = 11 + sqrt{frac{1}{.dot{1}}} )

15

( displaystyle 15 = frac{1}{.dot{1}} + left( sqrt{frac{1}{.dot{1}}} right)! )

16

( displaystyle 16 = frac{1}{.1} + left( sqrt{frac{1}{.dot{1}}} right)! )

17

( displaystyle 17 = 11 + Bigg(sqrt{dfrac{1}{.dot{1}}}Bigg)! )

18

( displaystyle 18 = dfrac{1}{.dot{1}} + dfrac{1}{.dot{1}} )

19

( displaystyle 19 = dfrac{1}{.1} + dfrac{1}{.dot{1}} )

20

( displaystyle 20 = dfrac{1}{.1} + dfrac{1}{.1} )

21

( displaystyle 21 = dfrac{1}{.1} + 11 )

22

( displaystyle 22 = 11 + 11 )

23

( displaystyle 23 = Bigg(sqrt{dfrac{1}{.dot{1}}}+1Bigg)!-1 )

24

( displaystyle 24 = Bigg(sqrt{dfrac{1}{.dot{1}}}+1Bigg)!times 1 )

25

( displaystyle 25 = Bigg(sqrt{dfrac{1}{.dot{1}}}+1Bigg)!+1 )

26

( begin{align} displaystyle
26 &= Biggllceil sqrt{dfrac{1}{.1}} Bigglrceil ! + 1 + 1 \
&= Bigllceil sqrt{10} Biglrceil ! + 2 \
&= bigllceil 3.162 cdots bigrrceil ! + 2 \
&= 4! + 2 \
&= 4 times 3 times 2 times 1 + 2 \
&= 24 + 2 \
&= 26 text{ Boom!}\
text{Note that:} \
bigllfloor x bigrrfloor &= text{the greatest integer less than or equal to } x text{ (floor)} \
bigllfloor 3.4 bigrrfloor &= 3 \
bigllceil x bigrrceil &= text{the smallest integer greater than or equal to } x text{ (ceiling)} \
bigllceil 3.4 bigrrceil &= 4 \
end{align} )

27

( begin{align} displaystyle
27 &= Biggllceil sqrt{dfrac{1}{.1}} Bigglrceil ! + Biggllfloor sqrt{dfrac{1}{.1}} Bigglrfloor \
&= Bigllceil sqrt{10} Biglrceil ! + Bigllfloor sqrt{10} Biglrfloor \
&= bigllceil 3.162 cdots biglrceil ! + bigllfloor 3.162 cdots biglrfloor \
&= 4! + 3 \
&= 4 times 3 times 2 times 1 + 3 \
&= 24 + 3 \
&= 27 \
end{align} )

28

( begin{align} displaystyle
28 &= Biggllceil sqrt{dfrac{1}{.1}} Bigglrceil ! + Biggllceil sqrt{dfrac{1}{.1}} Bigglrceil \
&= bigllceil 3.162 cdots biglrceil ! + bigllceil 3.162 cdots biglrceil \
&= 4! + 4 \
&= 4 times 3 times 2 times 1 + 4 \
&= 24 + 4 \
&= 28 \
end{align} )