Arithmetic Functions.ipynb

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   "source": [
    "The arithmetic functions are all **scalar** functions; they penetrate into nested arrays all the way down to the scalars. The standard functions `+`, `-`, `×`, `÷` all act as you might expect:"
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   "source": [
    "8+9"
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      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">1 ¯3 ¯2\n",
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   "source": [
    "4 3 2-3 6 4"
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     "execution_count": 3,
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   "source": [
    "length←20.5\n",
    "height←13.1\n",
    "area←length×height\n",
    "area"
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   "execution_count": 4,
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       "<span style=\"white-space:pre; font-family: monospace\">┌───────┬───────┬───────┐\n",
       "│4 4 0.8│2 0.4 2│0.6 3 3│\n",
       "└───────┴───────┴───────┘\n",
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     "execution_count": 4,
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   "source": [
    "(4 2 3)÷(1 1 5)(1 5 1)(5 1 1)"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All of those functions can also be applied monadically. For example `-x` is negative `x` (or `0-x`). `÷x` is the reciprocal of `x` (or `1÷x`)"
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     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">¯3 1 ¯4 1 ¯5\n",
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   "source": [
    "-3 ¯1 4 ¯1 5"
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   "source": [
    "÷3"
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   "source": [
    "Monadic `+` and `×` are designed for complex numbers. `+` gives the complex conjugate. `×` gives a complex number with magnitude 1 (unless it is applied to `0`) in the direction of its argument. For real numbers, `+` has no effect, and `×` gives the sign of its argument."
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   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">1J¯2 0J1 4 3J¯3\n",
       "</span>"
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     "execution_count": 7,
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   ],
   "source": [
    "+1j2 0j¯1 4 3j3"
   ]
  },
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   "execution_count": 8,
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       "<span style=\"white-space:pre; font-family: monospace\">1 0.707107J0.707107 0.707107J0.707107 0.341743J¯0.939793 ¯1 0\n",
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   "source": [
    "×10 3j3 4j4 8j¯22 ¯3 0"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`*` is the power function. `X*Y` is `X` to the power of `Y`"
   ]
  },
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   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">2.197\n",
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     "execution_count": 9,
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   "source": [
    "1.3×1.3×1.3"
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   "execution_count": 10,
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       "<span style=\"white-space:pre; font-family: monospace\">2.197\n",
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   "source": [
    "1.3*3"
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   "execution_count": 11,
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     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">3\n",
       "</span>"
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     "execution_count": 11,
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   ],
   "source": [
    "9*0.5  ⍝ square root"
   ]
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   "execution_count": 12,
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     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">0.05\n",
       "</span>"
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     "execution_count": 12,
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   ],
   "source": [
    "20*¯1  ⍝ reciprocal"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Monadically, `*x` is *e* to the power of `x`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">24.5325 24.5325\n",
       "</span>"
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     "execution_count": 13,
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   ],
   "source": [
    "e←2.71828182845\n",
    "(e*3.2) (*3.2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A close relative of `*` is `⍟` which is logarithm. `a⍟b` is the base-`a` logarithm of `b`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
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       "<span style=\"white-space:pre; font-family: monospace\">6\n",
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     "execution_count": 14,
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   ],
   "source": [
    "2⍟64"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a monad, `⍟x` is the natural logarithm of `x` (or `e⍟x`)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
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     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">0\n",
       "</span>"
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     "execution_count": 15,
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   "source": [
    "⍟1"
   ]
  },
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   "execution_count": 16,
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     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">23.14\n",
       "</span>"
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     "execution_count": 16,
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   "source": [
    "⍟(*23.14)"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`a⌈b`, gives the maximum of `a` and `b`. While `⌈x` is the ceiling of `x`, that is, the smallest integer larger than `x`."
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   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">4 4 2 0 0 0\n",
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     "execution_count": 17,
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   ],
   "source": [
    "4 4 2 ¯4 0 ¯10 ⌈ 0"
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   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
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    {
     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">2 3 0 6 ¯2\n",
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     "execution_count": 18,
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   ],
   "source": [
    "⌈1.5 3 ¯0.3 5.0001 ¯2.7"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After seeing what `⌈` does, it is not hard to guess what `⌊` does. It is of course *minimum* and *floor*"
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   "cell_type": "code",
   "execution_count": 19,
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   "outputs": [
    {
     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">¯2 2 5\n",
       "</span>"
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     "execution_count": 19,
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   "source": [
    "3 2 10⌊¯2 5 5"
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   "execution_count": 20,
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    {
     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">1 ¯7 5\n",
       "</span>"
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   "source": [
    "⌊1.4 ¯6.02 5"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Numbers can be rounded to the nearest integer with `⌊0.5+`. You can see it highlighted below because it is an idiom, recognised and optimised by the interpreter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">3 4 7 1\n",
       "</span>"
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     "execution_count": 21,
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   "source": [
    "⌊0.5+3.3 3.7 6.5 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Monadically, `|x` gives the magnitude of `x`. For real numbers, this is the absolute value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">2 3 4 0 10.5\n",
       "</span>"
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     },
     "execution_count": 22,
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   ],
   "source": [
    "| 2 ¯3 4 0 ¯10.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">5\n",
       "</span>"
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     },
     "execution_count": 23,
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   ],
   "source": [
    "| 3j4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dyadically, `|` is residue. `x|y` is the remainder when `y` is divided by `x`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">0 1 2 0 1 2 0 1\n",
       "</span>"
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     "execution_count": 24,
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   ],
   "source": [
    "3|0 1 2 3 4 5 6 7"
   ]
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   "execution_count": 25,
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     "data": {
      "text/html": [
       "<span style=\"white-space:pre; font-family: monospace\">0 0.4 0.31 0.2\n",
       "</span>"
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     "execution_count": 25,
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   ],
   "source": [
    "1|2 3.4 0.31 5.2   ⍝ The fractional parts of each number"
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   "category": "Closer Looks at Some Functions",
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