A tuple with one element. The trailing comma distinguishes a one-element tuple from a parenthesized n.


Used as a dimension entry, -1 instructs NumPy to choose the length that will keep the total number of elements the same.


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

Axes are defined for arrays with more than one dimension. A 2-dimensional array has two corresponding axes: the first running vertically downwards across rows (axis 0), and the second running horizontally across columns (axis 1).

Many operations can take place along one of these axes. For example, we can sum each row of an array, in which case we operate along columns, or axis 1:

>>> x = np.arange(12).reshape((3,4))

>>> x
array([[ 0,  1,  2,  3],
       [ 4,  5,  6,  7],
       [ 8,  9, 10, 11]])

>>> x.sum(axis=1)
array([ 6, 22, 38])

A homogeneous container of numerical elements. Each element in the array occupies a fixed amount of memory (hence homogeneous), and can be a numerical element of a single type (such as float, int or complex) or a combination (such as (float, int, float)). Each array has an associated data-type (or dtype), which describes the numerical type of its elements:

>>> x = np.array([1, 2, 3], float)

>>> x
array([ 1.,  2.,  3.])

>>> x.dtype # floating point number, 64 bits of memory per element

# More complicated data type: each array element is a combination of
# and integer and a floating point number
>>> np.array([(1, 2.0), (3, 4.0)], dtype=[('x', np.int64), ('y', float)])
array([(1, 2.), (3, 4.)], dtype=[('x', '<i8'), ('y', '<f8')])

Fast element-wise operations, called a ufunc, operate on arrays.


Any sequence that can be interpreted as an ndarray. This includes nested lists, tuples, scalars and existing arrays.

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 borrowing the memory from still another object, so the owning object may be a.base.base.base.... Despite advice to the contrary, testing base is not a surefire way to determine if two arrays are views.


When storing a multi-byte value in memory as a sequence of bytes, the sequence addresses/sends/stores the most significant byte first (lowest address) and the least significant byte last (highest address). Common in micro-processors and used for transmission of data over network protocols.


Basic Linear Algebra Subprograms


NumPy can do operations on arrays whose shapes are mismatched:

>>> x = np.array([1, 2])
>>> y = np.array([[3], [4]])

>>> x
array([1, 2])

>>> y

>>> x + y
array([[4, 5],
       [5, 6]])

See basics.broadcasting for more information.

C order

See row-major


A way to represent items in a N-dimensional array in the 1-dimensional computer memory. In column-major order, the leftmost index “varies the fastest”: for example the array:

[[1, 2, 3],
 [4, 5, 6]]

is represented in the column-major order as:

[1, 4, 2, 5, 3, 6]

Column-major order is also known as the Fortran order, as the Fortran programming language uses it.


See view.


An operator that transforms a function. For example, a log decorator may be defined to print debugging information upon function execution:

>>> def log(f):
...     def new_logging_func(*args, **kwargs):
...         print("Logging call with parameters:", args, kwargs)
...         return f(*args, **kwargs)
...     return new_logging_func

Now, when we define a function, we can “decorate” it using log:

>>> @log
... def add(a, b):
...     return a + b

Calling add then yields:

>>> add(1, 2)
Logging call with parameters: (1, 2) {}

Resembling a language dictionary, which provides a mapping between words and descriptions thereof, a Python dictionary is a mapping between two objects:

>>> x = {1: 'one', 'two': [1, 2]}

Here, x is a dictionary mapping keys to values, in this case the integer 1 to the string “one”, and the string “two” to the list [1, 2]. The values may be accessed using their corresponding keys:

>>> x[1]

>>> x['two']
[1, 2]

Note that dictionaries are not stored in any specific order. Also, most mutable (see immutable below) objects, such as lists, may not be used as keys.

For more information on dictionaries, read the Python tutorial.


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 sub-type 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

See column-major


Collapsed to a one-dimensional array. See numpy.ndarray.flatten for details.


Describes a block of memory comprised of blocks, each block comprised of items and of the same size, and blocks are interpreted in exactly the same way. In the simplest case each block contains a single item, for instance int32 or float64.


An object that cannot be modified after execution is called immutable. Two common examples are strings and tuples.


The size of the dtype element in bytes.


A Python container that can hold any number of objects or items. The items do not have to be of the same type, and can even be lists themselves:

>>> x = [2, 2.0, "two", [2, 2.0]]

The list x contains 4 items, each which can be accessed individually:

>>> x[2] # the string 'two'

>>> x[3] # a list, containing an integer 2 and a float 2.0
[2, 2.0]

It is also possible to select more than one item at a time, using slicing:

>>> x[0:2] # or, equivalently, x[:2]
[2, 2.0]

In code, arrays are often conveniently expressed as nested lists:

>>> np.array([[1, 2], [3, 4]])
array([[1, 2],
       [3, 4]])

For more information, read the section on lists in the Python tutorial. For a mapping type (key-value), see dictionary.


When storing a multi-byte value in memory as a sequence of bytes, the sequence addresses/sends/stores the least significant byte first (lowest address) and the most significant byte last (highest address). Common in x86 processors.


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

Array that suppressed values indicated by a mask:

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

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

Masked arrays are often used when operating on arrays containing missing or invalid entries.


A 2-dimensional ndarray that preserves its two-dimensional nature throughout operations. It has certain special operations, such as * (matrix multiplication) and ** (matrix power), defined:

>>> x = np.mat([[1, 2], [3, 4]])
>>> x
matrix([[1, 2],
        [3, 4]])

>>> x**2
matrix([[ 7, 10],
      [15, 22]])

See array.

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

Flattening collapses a multi-dimensional 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

An ndarray with structured data type which has been subclassed as np.recarray and whose dtype is of type np.record, making the fields of its data type to be accessible by attribute.


If a is a reference to b, then (a is b) == True. Therefore, a and b are different names for the same Python object.


A way to represent items in a N-dimensional array in the 1-dimensional computer memory. In row-major order, the rightmost index “varies the fastest”: for example the array:

[[1, 2, 3],
 [4, 5, 6]]

is represented in the row-major order as:

[1, 2, 3, 4, 5, 6]

Row-major order is also known as the C order, as the C programming language uses it. New NumPy arrays are by default in row-major order.


Used to select only certain elements from a sequence:

>>> x = range(5)
>>> x
[0, 1, 2, 3, 4]
>>> x[1:3] # slice from 1 to 3 (excluding 3 itself)
[1, 2]
>>> x[1:5:2] # slice from 1 to 5, but skipping every second element
[1, 3]
>>> x[::-1] # slice a sequence in reverse
[4, 3, 2, 1, 0]

Arrays may have more than one dimension, each which can be sliced individually:

>>> x = np.array([[1, 2], [3, 4]])
>>> x
array([[1, 2],
       [3, 4]])
>>> x[:, 1]
array([2, 4])

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.


See structured data type

structured array

Array whose dtype is a structured data type.

structured data type

A data type composed of other datatypes

subarray data type

A structured data type may contain a ndarray with its own dtype and shape:

>>> 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,))])

In addition to field names, structured array fields may have an associated title which is an alias to the name and is commonly used for plotting.


Universal function. A fast element-wise, vectorized array operation. Examples include add, sin and logical_or.


Optimizing a looping block by specialized code. In a traditional sense, vectorization performs the same operation on multiple elements with fixed strides between them via specialized hardware. Compilers know how to take advantage of well-constructed loops to implement such optimizations. NumPy uses vectorization to mean any optimization via specialized code performing the same operations on multiple elements, typically achieving speedups by avoiding some of the overhead in looking up and converting the elements.


An array that does not own its data, but refers to another array’s data instead. For example, we may create a view that only shows every second element of another array:

>>> 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])

Python is a high-level (highly abstracted, or English-like) language. This abstraction comes at a price in execution speed, and sometimes it becomes necessary to use lower level languages to do fast computations. A wrapper is code that provides a bridge between high and the low level languages, allowing, e.g., Python to execute code written in C or Fortran.

Examples include ctypes, SWIG and Cython (which wraps C and C++) and f2py (which wraps Fortran).