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 - √4)·√4 = 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} 383 384 385 {3}({120} + {5}) + {10}

(1) By Andre Gustavo dos Santos, Brasil

(2) By Richard Tschumpel, Vienna, Austria

(3) By Don Gosiewski

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