# Tensor Ranks, Shapes, and Types

TensorFlow programs use a tensor data structure to represent all data. You can
think of a TensorFlow tensor as an n-dimensional array or list.
A tensor has a static type and dynamic dimensions. Only tensors may be passed
between nodes in the computation graph.

## Rank

In the TensorFlow system, tensors are described by a unit of dimensionality
known as *rank*. Tensor rank is not the same as matrix rank. Tensor rank
(sometimes referred to as *order* or *degree* or *n-dimension*) is the number
of dimensions of the tensor. For example, the following tensor (defined as a
Python list) has a rank of 2:

    t = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

A rank two tensor is what we typically think of as a matrix, a rank one tensor
is a vector. For a rank two tensor you can access any element with the syntax
`t[i, j]`.  For a rank three tensor you would need to address an element with
`t[i, j, k]`.

Rank | Math entity | Python example
--- | --- | ---
0 | Scalar (magnitude only) | `s = 483`
1 | Vector (magnitude and direction) | `v = [1.1, 2.2, 3.3]`
2 | Matrix (table of numbers) | `m = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]`
3 | 3-Tensor (cube of numbers) | `t = [[[2], [4], [6]], [[8], [10], [12]], [[14], [16], [18]]]`
n | n-Tensor (you get the idea) | `....`

## Shape

The TensorFlow documentation uses three notational conventions to describe
tensor dimensionality: rank, shape, and dimension number. The following table
shows how these relate to one another:

Rank | Shape | Dimension number | Example
--- | --- | --- | ---
0 | [] | 0-D | A 0-D tensor.  A scalar.
1 | [D0] | 1-D | A 1-D tensor with shape [5].
2 | [D0, D1] | 2-D | A 2-D tensor with shape [3, 4].
3 | [D0, D1, D2] | 3-D | A 3-D tensor with shape [1, 4, 3].
n | [D0, D1, ... Dn-1] | n-D | A tensor with shape [D0, D1, ... Dn-1].

Shapes can be represented via Python lists / tuples of ints, or with the
[`TensorShape` class](../api_docs/python/framework.md#TensorShape).

## Data types

In addition to dimensionality, Tensors have a data type. You can assign any one
of the following data types to a tensor:

Data type | Python type | Description
--- | --- | ---
`DT_FLOAT` | `tf.float32` | 32 bits floating point.
`DT_DOUBLE` | `tf.float64` | 64 bits floating point.
`DT_INT8` | `tf.int8` | 8 bits signed integer.
`DT_INT16` | `tf.int16` | 16 bits signed integer.
`DT_INT32` | `tf.int32` | 32 bits signed integer.
`DT_INT64` | `tf.int64` | 64 bits signed integer.
`DT_UINT8` | `tf.uint8` | 8 bits unsigned integer.
`DT_STRING` | `tf.string` | Variable length byte arrays.  Each element of a Tensor is a byte array.
`DT_BOOL` | `tf.bool` | Boolean.
`DT_COMPLEX64` | `tf.complex64` | Complex number made of two 32 bits floating points: real and imaginary parts.
`DT_COMPLEX128` | `tf.complex128` | Complex number made of two 64 bits floating points: real and imaginary parts.
`DT_QINT8` | `tf.qint8` | 8 bits signed integer used in quantized Ops.
`DT_QINT32` | `tf.qint32` | 32 bits signed integer used in quantized Ops.
`DT_QUINT8` | `tf.quint8` | 8 bits unsigned integer used in quantized Ops.
