A parenthesized number followed by a comma denotes a tuple with one element. The trailing comma distinguishes a one-element tuple from a parenthesized n.

  • In a dimension entry, instructs NumPy to choose the length that will keep the total number of array elements the same.

    >>> np.arange(12).reshape(4, -1).shape
    (4, 3)
  • In an index, any negative value denotes indexing from the right.

An Ellipsis.

  • When indexing an array, shorthand that the missing axes, if they exist, are full slices.

    >>> a = np.arange(24).reshape(2,3,4)
    >>> a[...].shape
    (2, 3, 4)
    >>> a[...,0].shape
    (2, 3)
    >>> a[0,...].shape
    (3, 4)
    >>> a[0,...,0].shape

    It can be used at most once; a[...,0,...] raises an IndexError.

  • In printouts, NumPy substitutes ... for the middle elements of large arrays. To see the entire array, use numpy.printoptions


The Python slice operator. In ndarrays, slicing can be applied to every axis:

>>> a = np.arange(24).reshape(2,3,4)
>>> a
array([[[ 0,  1,  2,  3],
        [ 4,  5,  6,  7],
        [ 8,  9, 10, 11]],

       [[12, 13, 14, 15],
        [16, 17, 18, 19],
        [20, 21, 22, 23]]])

>>> a[1:,-2:,:-1]
array([[[16, 17, 18],
        [20, 21, 22]]])

Trailing slices can be omitted:

>>> a[1] == a[1,:,:]
array([[ True,  True,  True,  True],
       [ True,  True,  True,  True],
       [ True,  True,  True,  True]])

In contrast to Python, where slicing creates a copy, in NumPy slicing creates a view.

For details, see Combining advanced and basic indexing.


In a dtype declaration, indicates that the data is little-endian (the bracket is big on the right).

>>> dt = np.dtype('<f')  # little-endian single-precision float

In a dtype declaration, indicates that the data is big-endian (the bracket is big on the left).

>>> dt = np.dtype('>H')  # big-endian unsigned short
advanced indexing

Rather than using a scalar or slice as an index, an axis can be indexed with an array, providing fine-grained selection. This is known as advanced indexing or “fancy indexing”.

along an axis

An operation along axis n of array a behaves as if its argument were an array of slices of a where each slice has a successive index of axis n.

For example, if a is a 3 x N array, an operation along axis 0 behaves as if its argument were an array containing slices of each row:

>>> np.array((a[0,:], a[1,:], a[2,:])) 

To make it concrete, we can pick the operation to be the array-reversal function numpy.flip, which accepts an axis argument. We construct a 3 x 4 array a:

>>> a = np.arange(12).reshape(3,4)
>>> a
array([[ 0,  1,  2,  3],
       [ 4,  5,  6,  7],
       [ 8,  9, 10, 11]])

Reversing along axis 0 (the row axis) yields

>>> np.flip(a,axis=0)
array([[ 8,  9, 10, 11],
       [ 4,  5,  6,  7],
       [ 0,  1,  2,  3]])

Recalling the definition of along an axis, flip along axis 0 is treating its argument as if it were

>>> np.array((a[0,:], a[1,:], a[2,:]))
array([[ 0,  1,  2,  3],
       [ 4,  5,  6,  7],
       [ 8,  9, 10, 11]])

and the result of np.flip(a,axis=0) is to reverse the slices:

>>> np.array((a[2,:],a[1,:],a[0,:]))
array([[ 8,  9, 10, 11],
       [ 4,  5,  6,  7],
       [ 0,  1,  2,  3]])

Used synonymously in the NumPy docs with ndarray.


Any scalar or sequence that can be interpreted as an ndarray. In addition to ndarrays and scalars this category includes lists (possibly nested and with different element types) and tuples. Any argument accepted by numpy.array is array_like.

>>> a = np.array([[1, 2.0], [0, 0], (1+1j, 3.)])

>>> a
array([[1.+0.j, 2.+0.j],
       [0.+0.j, 0.+0.j],
       [1.+1.j, 3.+0.j]])
array scalar

For uniformity in handling operands, NumPy treats a scalar as an array of zero dimension.


Another term for an array dimension. Axes are numbered left to right; axis 0 is the first element in the shape tuple.

In a two-dimensional vector, the elements of axis 0 are rows and the elements of axis 1 are columns.

In higher dimensions, the picture changes. NumPy prints higher-dimensional vectors as replications of row-by-column building blocks, as in this three-dimensional vector:

>>> a = np.arange(12).reshape(2,2,3)
>>> a
array([[[ 0,  1,  2],
        [ 3,  4,  5]],
       [[ 6,  7,  8],
        [ 9, 10, 11]]])

a is depicted as a two-element array whose elements are 2x3 vectors. From this point of view, rows and columns are the final two axes, respectively, in any shape.

This rule helps you anticipate how a vector will be printed, and conversely how to find the index of any of the printed elements. For instance, in the example, the last two values of 8’s index must be 0 and 2. Since 8 appears in the second of the two 2x3’s, the first index must be 1:

>>> a[1,0,2]

A convenient way to count dimensions in a printed vector is to count [ symbols after the open-parenthesis. This is useful in distinguishing, say, a (1,2,3) shape from a (2,3) shape:

>>> a = np.arange(6).reshape(2,3)
>>> a.ndim
>>> a
array([[0, 1, 2],
       [3, 4, 5]])
>>> a = np.arange(6).reshape(1,2,3)
>>> a.ndim
>>> a
array([[[0, 1, 2],
        [3, 4, 5]]])

If an array does not own its memory, then its base attribute returns the object whose memory the array is referencing. That object may be referencing the memory from still another object, so the owning object may be a.base.base.base.... Some writers erroneously claim that testing base determines if arrays are views. For the correct way, see numpy.shares_memory.


See Endianness.


Basic Linear Algebra Subprograms


broadcasting is NumPy’s ability to process ndarrays of different sizes as if all were the same size.

It permits an elegant do-what-I-mean behavior where, for instance, adding a scalar to a vector adds the scalar value to every element.

>>> a = np.arange(3)
>>> a
array([0, 1, 2])
>>> a + [3, 3, 3]
array([3, 4, 5])
>>> a + 3
array([3, 4, 5])

Ordinarly, vector operands must all be the same size, because NumPy works element by element – for instance, c = a * b is

 c[0,0,0] = a[0,0,0] * b[0,0,0]
 c[0,0,1] = a[0,0,1] * b[0,0,1]

But in certain useful cases, NumPy can duplicate data along “missing” axes or “too-short” dimensions so shapes will match. The duplication costs no memory or time. For details, see Broadcasting.

C order

Same as row-major.


See Row- and column-major order.

An array is contiguous if
  • it occupies an unbroken block of memory, and

  • array elements with higher indexes occupy higher addresses (that is, no stride is negative).


See view.


See axis.


The datatype describing the (identically typed) elements in an ndarray. It can be changed to reinterpret the array contents. For details, see Data type objects (dtype).

fancy indexing

Another term for advanced indexing.


In a structured data type, each subtype is called a field. The field has a name (a string), a type (any valid dtype), and an optional title. See Data type objects (dtype).

Fortran order

Same as column-major.


See ravel.


All elements of a homogeneous array have the same type. ndarrays, in contrast to Python lists, are homogeneous. The type can be complicated, as in a structured array, but all elements have that type.

NumPy object arrays, which contain references to Python objects, fill the role of heterogeneous arrays.


The size of the dtype element in bytes.


See Endianness.


A boolean array used to select only certain elements for an operation:

>>> x = np.arange(5)
>>> x
array([0, 1, 2, 3, 4])
>>> mask = (x > 2)
>>> mask
array([False, False, False, True,  True])
>>> x[mask] = -1
>>> x
array([ 0,  1,  2,  -1, -1])
masked array

Bad or missing data can be cleanly ignored by putting it in a masked array, which has an internal boolean array indicating invalid entries. Operations with masked arrays ignore these entries.

>>> a =[np.nan, 2, np.nan], [True, False, True])
>>> a
masked_array(data=[--, 2.0, --],
             mask=[ True, False,  True],

>>> a + [1, 2, 3]
masked_array(data=[--, 4.0, --],
             mask=[ True, False,  True],

For details, see Masked arrays.


NumPy’s two-dimensional matrix class should no longer be used; use regular ndarrays.


NumPy’s basic structure.

object array

An array whose dtype is object; that is, it contains references to Python objects. Indexing the array dereferences the Python objects, so unlike other ndarrays, an object array has the ability to hold heterogeneous objects.


numpy.ravel and numpy.flatten both flatten an ndarray. ravel will return a view if possible; flatten always returns a copy.

Flattening collapses a multimdimensional array to a single dimension; details of how this is done (for instance, whether a[n+1] should be the next row or next column) are parameters.

record array

A structured array with allowing access in an attribute style (a.field) in addition to a['field']. For details, see numpy.recarray.


See Row- and column-major order. NumPy creates arrays in row-major order by default.


In NumPy, usually a synonym for array scalar.


A tuple showing the length of each dimension of an ndarray. The length of the tuple itself is the number of dimensions (numpy.ndim). The product of the tuple elements is the number of elements in the array. For details, see numpy.ndarray.shape.


Physical memory is one-dimensional; strides provide a mechanism to map a given index to an address in memory. For an N-dimensional array, its strides attribute is an N-element tuple; advancing from index i to index i+1 on axis n means adding a.strides[n] bytes to the address.

Strides are computed automatically from an array’s dtype and shape, but can be directly specified using as_strided.

For details, see numpy.ndarray.strides.

To see how striding underlies the power of NumPy views, see The NumPy array: a structure for efficient numerical computation.

structured array

Array whose dtype is a structured data type.

structured data type

Users can create arbitrarily complex dtypes that can include other arrays and dtypes. These composite dtypes are called structured data types.


An array nested in a structured data type, as b is here:

>>> dt = np.dtype([('a', np.int32), ('b', np.float32, (3,))])
>>> np.zeros(3, dtype=dt)
array([(0, [0., 0., 0.]), (0, [0., 0., 0.]), (0, [0., 0., 0.])],
      dtype=[('a', '<i4'), ('b', '<f4', (3,))])
subarray data type

An element of a structured datatype that behaves like an ndarray.


An alias for a field name in a structured datatype.


In NumPy, usually a synonym for dtype. For the more general Python meaning, see here.


NumPy’s fast element-by-element computation (vectorization) gives a choice which function gets applied. The general term for the function is ufunc, short for universal function. NumPy routines have built-in ufuncs, but users can also write their own.


NumPy hands off array processing to C, where looping and computation are much faster than in Python. To exploit this, programmers using NumPy eliminate Python loops in favor of array-to-array operations. vectorization can refer both to the C offloading and to structuring NumPy code to leverage it.


Without touching underlying data, NumPy can make one array appear to change its datatype and shape.

An array created this way is a view, and NumPy often exploits the performance gain of using a view versus making a new array.

A potential drawback is that writing to a view can alter the original as well. If this is a problem, NumPy instead needs to create a physically distinct array – a copy.

Some NumPy routines always return views, some always return copies, some may return one or the other, and for some the choice can be specified. Responsiblity for managing views and copies falls to the programmer. numpy.shares_memory will check whether b is a view of a, but an exact answer isn’t always feasible, as the documentation page explains.

>>> x = np.arange(5)
>>> x
array([0, 1, 2, 3, 4])
>>> y = x[::2]
>>> y
array([0, 2, 4])
>>> x[0] = 3 # changing x changes y as well, since y is a view on x
>>> y
array([3, 2, 4])