Representation of numbers with four 4's

Remark

There is a convention I applied to identities below, especially for the numbers beyond 100. As the table grew, some awkward formulas became more common. In order to simplify the entries and shorten the file that contains this page, I introduced several shorthands as listed below. These are used to express numbers with a single 4:

  • [4] = 1
  • [4!!] = [24!1/16] = 30
  • [30] = 5
  • 5! = 120
  • [120] = 10
  • [10] = 3
  • 3! = 6
  • [(6!)] = 26

where brackets denote the whole part function. ([x] is the largest integer not exceeding x.)

Now, let's agree to use {1},{30},{5},{120},{3} as substitutes for the corresponding formulas. Thus for example, {3} means 3 expressed as [[[4!!]!]]. Similarly, {5} and {120} stand for 5 and 120, respectively, in terms of the just described expressions.

1 4·4/4/4 = (4 + 4)/(4 + 4) = 44/44(1)
2 4/4 + 4/4 = 4/4 + 4 - 4
3 (4 + 4 + 4)/4 = 4!!/4 + 4/4 = 4!/4!!·4/4
4 4 + 4·(4 - 4) = 4!!/(4 + 4)·4
5 (4 + 4·4)/4 = 4 + (4 + 4)/4
6 4 + (4 + 4)/4 = 4!/4 + 4 - 4
7 4 + 4 - 4/4 = 4!/4 + 4/4
8 4 + 4 - 4 + 4 = 4!! + (4 - 4)/4 = (4 + 4)·4/4
9 4 + 4 + 4/4 = 4/.4 - 4/4 = 4/.(4) + 4 - 4 = (4!/4/4)4(1)
10 (44 - 4)/4 = 4/.4 - 4 + 4 = 44/4.4(1)
11 44/(4 + 4) = 4/.4 + 4/4
12 (44 + 4)/4 = 4/.4 + 4/4
13 44/4 + 4 = 4!! + 4 + 4/4
14 4!/4 + 4 + 4 = 4!! + 4 + 4/4
15 4·4 - 4/4 = 44/4 + 4(1) = ((4)4! - 4)/4(2)
16 4 + 4 + 4 + 4 = 4·4·4/4 = 44/4/4 = 4! - 4 - 4
17 4·4 + 4/4 = ((4)4! + 4)/4(2)
18 4·4 + 4 - 4
19 4! - 4 - 4/4 = 4!!/.4 - 4/4
20 (4! - 4)·4/4 = (4! - 4) + 4 - 4 = (4 + 4/4)·4
21 4! - 4 + 4/4 = 4!!/.4 + 4/4
22 4·4 + 4 + 4 = 4/.(4)·4 + 4
23 4! - 4 + 4/4 = 4!!/.4 + 4/4
24 4·4 + 4 + 4 = 4/.4·4 + 4
25 4! + (4 + 4)/4
26 4! + (4 + 4)/4
27 4! + 4 - 4/4 = (4·4 - 4)/.4(1)
28 4! + 4 + 4 - 4 = (4! + 4)·4/4
29 4! + 4 + 4/4
30 4·4·4 - 4 = ((4)4! - 4)/4(2)
31 4!!·4 - 4/4 = ((4)4! - 4)/4(2)
32 4!!·4 + 4 - 4
33 4!!·4 + 4/4 = ((4)4! + 4)/4(2)
34 4/.(4)·4 - 4 = 4·4·4 + 4 = ((4)4! + 4)/4(2)
35 4!!/.(4) - 4/4 = 4! + 44/4
36 4·(4 + 4) + 4 = 4·4·4 + 4
37 4!!/.(4) + 4/4
38 4!!/.(4) + 4/4
39 4!!/.4 - 4/4
40 4·4/.(4) + 4 = 4!!·(4 + 4/4) = 4!!/.4 + 4 - 4
41 4!!/.4 + 4/4
42 44 - 4/4
43 44 - 4/4
44 44 + 4 - 4
45 44 + 4/4
46 44 + 4/4
47 4!·4 - 4/4
48 44 + 4 + 4
49 4!·4 + 4/4
50 4!·4 + 4/4
51 4!·4 + 4 - {1} = (4! - 4 + .4)/.4(1)
52 (4! + 4)·4/4
53 4!·4 + 4 + {1}
54 4!·4 + 4 + 4
55 44/4/.4(1)
56 4!·4 + 4 + 4 = 4! + 4! + 4 + 4(1)
57 44 + [(4!!·4!)]
58 4!/.4 - 4 + 4(1)
59 4!/.4 - 4/4(1)
60 4·4·4 - 4(1)
61 4!/.4 + 4/4(1)
62 4·4·4 - 4(1) = 4!!4 - 4/4
63 4!!4 - 4/4
64 4! + 44 - 4(1)
65 4!!4 + 4/4
66 4!/.4 + 4!/4(1) = 4·4·4 + 4(1)
67 4!!4 + 4 - {1}
68 4·4·4 + 4(1)
69 4!!4 + 4 + {1}
70 44 + 4! + 4(1)
71 (4! + 4 + .4 )/.4(1)
72 (4!·4!)/(4 + 4)(1)
73 4!!/.4 + [(4!!·4!)]
74 4!·4 - 4! - 4(1)
75 (4! + 4 + 4)/.4(1)
76 4!·4 - 4! + 4(1) = (4! - 4)·4 - 4 = 4!!/.4·4 - 4
77 (4!!)4 + [(4!!·4!)]
78 (4! - 4)·4 - 4(1)
79 (4! - 4)·4 - {1}
80 (44 - 4!)·4(1) = 4!!/.4·4·{1}
81 (4 - 4/4)4(1)
82 (4! - 4)·4 + 4(1)
83 4!·4 - [(4!!·4!)]
84 44·4 - 4(1) = (4! - 4)·4 -4(1) = (4! - 4)·4 + 4 = 4!!/.4·4 + 4
85 (4! + 4/.4)/.4(1)
86 (4! - 4)·4 - 4(1)
87 (4! - 4)·4 - {1}
88 44·(4 - 4)(1) = 4·4·4 + 4!(1)
89 (4! - 4)·4 + {1}
90 (4! - 4)·4 + 4(1)
91 4!·4 - 4/.4
92 (4! - 4/4)·4(1)
93 4!·4 - [4/4]
94 4!·4 - 4 + 4(1)
95 4!·4 - 4/4(1)
96 (4 + 4/4)! - 4!(1)
97 4!·4 + 4/4(1)
98 4!·4 + 4 - 4(1)
99 4!·4 + 4 - {1} = 44/.(44)
100 (4! + 4/4)·4(1) = 44/.44(1)
101 4!·4 + 4/.4
102 4!·4 + 4 + 4
103 4!·4 + 4!! - {1}
104 4!·4 + 4 + 4
105 4!·4 + 4!! + {1}
106 4!·4 + 4!! + 4 = 44/.4 - 4
107 44/.(4) + 4!!
108 4!·4 + 4!! + 4 = 44/.4 - 4
109 44/.4 - {1}
110 44/.4·{1}
111 44/.4 + {1}
112 44/.4 + 4
113 44/.4 + {3}
114 44/.4 + 4
115 44/.4 + {5}
116 4!/.4·4 - 4
117 4!/.4·4 - {3}
118 4!/.4·4 - 4
119 4!/.4·4 - {1}
120 4!/.4·4·{1}
121 4!/.4·4 + {1}
122 4!/.4·4 + 4
123 4!/.4·4 + {3}
124 4!/.4·4 + 4 = (4!!4 - 44 = 4!!4·4 - 4
125 4!!4·4 - {3}
126 4!!4·4 - 4
127 4!!4·4 - {1}
128 4!!4·4·{1}
129 4!!4·4 + {1}
130 4!!4·4 + 4
131 4!!4·4 + {3}
132 4!!4·4 + 4
133 4!!4·4 + {5}
134 {120} + 4/.4 + 4 = {120} + 4·4 - 4
135 {120} + 4·4 - [4]
136 {120} + (4 + 4)·4 = 4!!4·4 + 4!!
137 {120} + 4/.(4) + 4!!
138 {120} + 4·4 + 4
139 {120} + [4·4!/4!]
140 {120} + 4!! + 4!! + 4 = {120} + 4·4 + 4
141 {120} + 4! - 4 + [4]
142 {120} + 4! - 4/4
143 {120} + 4! - 4/4
144 {120} + 4! + 4 - 4 = {120} + 4!! + 4!! + 4!!
145 {120} + 4! + 4/4
146 {120} + 4! + 4/4
147 {120} + 4! + 4 - [4]
148 {120} + 4! + (4·4)
149 {120} + 4! + 4 + [4]
150 {120} + 4! + 4 + 4
151 {120} + 4!!·4 - [4]
152 {120} + 4! + 4 + 4
153 {120} + 4!!·4 + [4] = {120} + 4! + 4/.(4)
154 {120} + 4! + 4/.4
155 {120} + {30} + 4 + {1}
156 {120} + {30} + 4 + 4
157 {120} + {30} + 4 + {3}
158 {120} + {30} + 4 + 4
159 {120} + {30} + {5} + 4
160 {120} + {30} + {10}·{1}
161 {120} + {30} + {10} + {1}
162 {120} + {30} + {10} + 4
163 {120} + {30} + {10} + {3}
164 {120} + {30} + {10} + 4
165 {120} + {30} + {10} + {5}
166 {120} + {30} + 4!! + 4!!
167 {120} + 4! + 4! - {1}
168 {120} + 4! + 4!·{1}
169 {120} + 4! + 4! + {1}
170 {120} + 4! + 4! + 4
171 {120} + 4! + 4! + {3}
172 {120} + 4! + 4! + 4
173 {120} + 4! + 4! + {5}
174 {120} + 4! + 4! + {3}!
175 {120} + {30} + 4! + {1}
176 {120} + 4! + 4! + 4!!
177 {120} + {30} + 4! + {3}
178 {120} + {30} + 4! + 4
179 {120} + {30} + 4! + {5}
180 {120} + {30} + {30}·{1} = {120} + {120} - {30} - {30} = {30}·{10} - {120}·{1}
181 {120} + {30} + {30} + {1}
182 {120} + {30} + {30} + 4
183 {120} + {30} + {30} + {3}
184 {120} + {30} + {30} + 4
185 {120} + {30} + {30} + {5}
186 {120} + {30} + {30} + {3}!
187 {120} + 4! · {3} - {5}
188 {120} + 4! · {3} - 4
189 {120} + 4! · {3} - {3}
190 {120} + 4! · {3} - 4
191 {120} + 4! · {3} - {1}
192 {120} + 4! + 4! + 4!
193 {120} + {26} · {3} - {5}
194 {120} + {26} · {3} - 4
195 {120} + {26} · {3} - {3}
196 {120} + {26} · {3} - 4
197 {120} + {26} · {3} - {1}
198 {120} + {26} · {3} · {1}
199 {120} + {26} · {3} + {1}
200 {120} + {26} · {3} + 4
201 {120} + {26} · {3} + {3}
202 {120} + {26} · {3} + 4
203 {120} + {26} · {3} + {5}
204 {120} + {26} · {3} + {3}!
205 {120} + {30} · {3} - {5}
206 {120} + {30} · {3} - 4
207 {120} + {30} · {3} - {3}
208 {120} + {30} · {3} - 4
209 {120} + {30} · {3} - {1}
210 {120} + {30} · {3} · {1}
211 {120} + {30} · {3} + {1}
212 {120} + {30} · {3} + 4
213 {120} + {30} · {3} + {3}
214 {120} + {30} · {3} + 4
215 {120} + {30} · {3} + {5}
216 {120} + {30} · {3} + {3}!
217 {120} + 4! · 4 + {1}
218 {120} + 4! · 4 + 4
219 {120} + 4! · 4 + {3}
220 {120} + {30} · {3} + {10} = {120} + 4! · 4 + 4
221 {120} + 4! · 4 + {5}
222 {120} + 4! · 4 + {3}!
223 {120} + {26} ·4 - {1}
224 {120} + {120} - 4*4
225 {120} + {120} - {5} · {3}
226 {120} + {120} - 4! + {10}
227 {120} + {26} ·4 + {3}
228 {120} + {26} ·4 + 4
229 {120} + {26} ·4 + {5}
230 {120} + {26} ·4 + {3}!
231 {120} + {120} - {3} - {3}!
232 {120} + {120} - {5} - {3}
233 {120} + {120} - 4 - {3}
234 {120} + {120} - {3} - {3}
235 {120} + {120} - 4 - {1}
236 {120} + {120} - 4 · {1}
237 {120} + {120} - 4 + {1}
238 {120} + {120} - {1} - {1}
239 {120} + {120} - {1} · {1}
240 {120} · {1} + {120} · {1}
241 {120} + {120} + {1} · {1}
242 {120} + {120} + {1} + {1}
243 {120} + {120} + 4 - {1}
244 {120} + {120} + 4 · {1}
245 {120} + {120} + 4 + {1}
246 {120} + {120} + {3} + {3}
247 {120} + {120} + 4 + {3}
248 {120} + {120} + {5} + {3}
249 {120} + {120} + {3} + {3}!
250 {120} + {120} + {5} + {5}
251 {120} + {120} + {5} + {3}!
252 {120} + {120} + {3} · 4
253 {26} · {5} + {120} + {3}
254 {120} + {120} + 4! - {10}
255 {120} + {120} + {3} · {5}
256 {120} + {120} + 4 · 4
257 {26} · {5} · {2} - {3}
258 {120} + {120} + 4! - {3}!
259 {120} + {120} + 4! - {5}
260 {120} + {120} + {10} + {10}
261 {26} · {10} + {1} · {1}
262 {26} · {10} + {1} + {1}
263 {26} · {10} + {3} · {1}
264 {26} · {10} + 4 · {1}
265 {26} · {10} + {5} · {1}
266 {26} · {10} + {3}! · {1}
267 {26} · {10} + {3} + 4
268 {26} · {10} + 4 + 4
269 {26} · {10} + {5} + 4
270 {26} · {10} + {10} · {1}
271 {26} · {10} + {10} + {1}
272 {26} · {10} + {3}! + {3}!
273 {26} · {10} + {5} + 4!!
274 {26} · {10} + {10} + 4
275 {26} · {10} + {10} + {5}
276 {26} · {10} + {10} + {3}!
278 {26} · {10} + 4! - {3}!
279 {26} · {10} + 4! - {5}
280 {26} · {10} + {10} + {10}
281 {26} · {10} + 4! - {3}
282 {26} · {10} + 4! - 4
283 {26} · {10} + 4! - {1}
284 {26} · {10} + 4! · {1}
285 {26} · {10} + 4! + {1}
286 {26} · {10} + 4! + 4
287 {26} · {10} + 4! + {3}
288 {26} · {10} + 4! + 4
289 {26} · {10} + 4! + {5}
290 {26} · {10} + 4! + {3}!
291 {26} · {10} + {30} + {1}
292 {26} · {10} + 4! + 4!!
293 {26} · {10} + {30} + {3}
294 {26} · {10} + {30} + 4
295 {26} · {10} + {30} + {5}
296 {26} · {10} + {30} + {3}!
297 {30} · {10} - {3} · {1}
298 {26} · {10} + {30} + 4!!
299 {30} · {10} - {1} · {1}
300 {30} · {10} · {1} · {1}
301 {30} · {10} + {1} · {1}
302 {30} · {10} + {1} + {1}
303 {30} · {10} + {3} · {1}
304 {30} · {10} + {3} + {1}
305 {30} · {10} + {5} · {1}
306 {30} · {10} + {5} + {1}
307 {30} · {10} + {10} - {3}
308 {30} · {10} + 4 + 4(1)
309 {30} · {10} + {10} - {1}
310 {30} · {10} + {10} · {1}
311 {30} · {10} + {10} + {1}(1)
312 {30} · {10} + {3}! + {3}!
313 {30} · {10} + {10} + {3}
314 {30} · {10} + {3}! + 4!!
315 {30} · {10} + {3}·{5}
316 {30} · {10} + 4·4
317 4! · {26} / 4 + {5}(1)
318 {30} · {10} + {3}·{3}! = ({3}·{6})4 - {6}
319 {3}4 · 4 - {5}(1) = ({3}·{6})4 - {5}
320 {30} · {10} + {10} + {10} = ({3}·{6})4 - 4
321 {3}4 · 4 - {3}(1) = ({3}·{6})4 - {3}
322 {3}4 · 4 - 4(1) = ({3}·{6})4 - 4
323 {3}4 · 4 - {1}(1) = ({3}·{6})4 - {1}
324 (4 - {1})4 · 4(1) = {3}·{6}·{3}·{6} = {3} · {120} - {30} - {6}
325 {3}4 · 4 + {1}(1) = ({3}·{6})4 + {1}
326 {3}4 · 4 + 4(1) = ({3}·{6})4 + 4
327 {3}4 · 4 + {3}(1) = ({3}·{6})4 + {3}
328 {3}4 · 4 + 4(1) = ({3}·{6})4 + 4
329 {3}4 · 4 + {5}(1) = ({3}·{6})4 + {5}
330 {3}4 · 4 + {6}(1) = ({3}·{6})4 + {6} = {3}·{120} - {30}·{1}
331{3}·{120} - {30} + {1}
332{3}·{120} - {30} + 4
333{3}·{120} - {30} + {3}
334 {3}·{120} - {30} + 4 = ({3}·{6})4 + {10}
335{3}·{120} - {30} + {5}
336{3}·{120} - {30} + {6}
337{3}·{120} - {26} + {3}
338{3}·{120} - {26} + 4
339{3}·{120} - {26} + {5}
340{3}·{120} - {30} + {10} = {3}·{120} - {26} + {6}
341{3}·{120} - 4! + {5}
342{3}·{120} - 4! + {6}
343({6} + {1}){3}·{1}
344{3}·{120} - {26} + {10}
345({6} + {1}){3} + 4
346({6} + {1}){3} + {3}
347({6} + {1}){3} + 4
348({6} + {1}){3} + {5}
349({6} + {1}){3} + {6}
350{3}·{120} - {10}·{1}
351{3}·{120} - {10} + {1}
352{3}·{120} - {10} + 4
353{3}·{120} - {10} + {3}
354{3}·{120} - {10} + 4
355{3}·{120} - {10} + {5}
356{3}·{120} - {10} + {6}
357{3}·{120} - {3}·{1}
358{3}·{120} - {1} - {1}
359{3}·{120} - {1} · {1}
360{3}·{120} - {1} + {1}
361{3}·{120} + {1} · {1}
362{3}·{120} + {1} + {1}
363{3}·{120} + {3} · {1}
364{3}·{120} + 4 · {1}
365{3}·{120} + {5} · {1}
366{3}·{120} + {6} · {1}
366{3}·{120} + {6} · {1}
367{3}·{120} + {6} + {1} = ({6} + {1}){3} + 4!
368{3}·{120} + {6} + 4
369{3}·{120} + {6} + {3} = ({6} + {1}){3} + {26}
370{3}·{120} + {10}·{1}
371{3}·{120} + {10} + {1}
372{3}·{120} + {10} + 4
373{3}·{120} + {10} + {3}
374{3}·{120} + {10} + 4
375{3}·{120} + {10} + {5}
376{3}·{120} + {10} + {6}
377{3}({120} + 4) + {5}(3)
378{3}·{120} + 4! - {6} = {3}({120} + 4) + {6}
379{3}·{120} + 4! - {5}
380{3}·{120} + {10} + {10}
381{3}{120} + {26}- {5}
382{3}({120} + 4) + {10}
3834!·4·4 - [4](4)
384{3}({120} + {30}) - {6}
385{3}({120} + {5}) + {10}  =  4!·4·4 + [4](4)
3864!·4·4 + 4(4)

(1) By Andre Gustavo dos Santos, Brasil

(2) By Richard Tschumpel, Vienna, Austria

(3) By Don Gosiewski

(4) By Byron Igoe

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