title: "Mathematical functions" date: 2026-05-24T13:52:06Z

Mathematical functions

1import numpy as np

1np.__version__

'1.11.2'

1__author__ = "kyubyong. kbpark.linguist@gmail.com. https://github.com/kyubyong"

Trigonometric functions

Q1. Calculate sine, cosine, and tangent of x, element-wise.

1x = np.array([0., 1., 30, 90])

sine: [ 0.          0.84147098 -0.98803162  0.89399666]
cosine: [ 1.          0.54030231  0.15425145 -0.44807362]
tangent: [ 0.          1.55740772 -6.4053312  -1.99520041]

Q2. Calculate inverse sine, inverse cosine, and inverse tangent of x, element-wise.

1x = np.array([-1., 0, 1.])

inverse sine: [ 1.57079633  0.          1.57079633]
inverse cosine: [ 0.          1.57079633  0.        ]
inverse tangent: [ 0.78539816  0.          0.78539816]

Q3. Convert angles from radians to degrees.

1x = np.array([-np.pi, -np.pi/2, np.pi/2, np.pi])

[-180.  -90.   90.  180.]

Q4. Convert angles from degrees to radians.

1x = np.array([-180.,  -90.,   90.,  180.])

[-3.14159265 -1.57079633  1.57079633  3.14159265]

Hyperbolic functions

Q5. Calculate hyperbolic sine, hyperbolic cosine, and hyperbolic tangent of x, element-wise.

1x = np.array([-1., 0, 1.])

[-1.17520119  0.          1.17520119]
[ 1.54308063  1.          1.54308063]
[-0.76159416  0.          0.76159416]

Rounding

Q6. Predict the results of these, paying attention to the difference among the family functions.

 1x = np.array([2.1, 1.5, 2.5, 2.9, -2.1, -2.5, -2.9])
 2
 3out1 = np.around(x)
 4out2 = np.floor(x)
 5out3 = np.ceil(x)
 6out4 = np.trunc(x)
 7out5 = [round(elem) for elem in x]
 8
 9#print out1
10#print out2
11#print out3
12#print out4
13#print out5

Q7. Implement out5 in the above question using numpy.

[ 2.  2.  3.  3. -2. -3. -3.]

Sums, products, differences

Q8. Predict the results of these.

 1x = np.array(
 2    [[1, 2, 3, 4],
 3     [5, 6, 7, 8]])
 4
 5outs = [np.sum(x),
 6        np.sum(x, axis=0),
 7        np.sum(x, axis=1, keepdims=True),
 8        "",
 9        np.prod(x),
10        np.prod(x, axis=0),
11        np.prod(x, axis=1, keepdims=True),
12        "",
13        np.cumsum(x),
14        np.cumsum(x, axis=0),
15        np.cumsum(x, axis=1),
16        "",
17        np.cumprod(x),
18        np.cumprod(x, axis=0),
19        np.cumprod(x, axis=1),
20        "",
21        np.min(x),
22        np.min(x, axis=0),
23        np.min(x, axis=1, keepdims=True),
24        "",
25        np.max(x),
26        np.max(x, axis=0),
27        np.max(x, axis=1, keepdims=True),
28        "",
29        np.mean(x),
30        np.mean(x, axis=0),
31        np.mean(x, axis=1, keepdims=True)]
32           
33for out in outs:
34    if out == "":
35        pass
36        #print
37    else:
38        pass
39        #print("->", out)

/usr/local/lib/python2.7/dist-packages/ipykernel/__main__.py:34: FutureWarning: elementwise comparison failed; returning scalar instead, but in the future will perform elementwise comparison

Q9. Calculate the difference between neighboring elements, element-wise.

1x = np.array([1, 2, 4, 7, 0])

[ 1  2  3 -7]

Q10. Calculate the difference between neighboring elements, element-wise, and prepend [0, 0] and append[100] to it.

1x = np.array([1, 2, 4, 7, 0])

[  0   0   1   2   3  -7 100]

Q11. Return the cross product of x and y.

1x = np.array([1, 2, 3])
2y = np.array([4, 5, 6])

[-3  6 -3]

Exponents and logarithms

Q12. Compute $e^x$, element-wise.

1x = np.array([1., 2., 3.], np.float32)

[  2.71828175   7.38905621  20.08553696]

Q13. Calculate exp(x) - 1 for all elements in x.

1x = np.array([1., 2., 3.], np.float32)

[  1.71828175   6.38905621  19.08553696]

Q14. Calculate $2^p$ for all p in x.

1x = np.array([1., 2., 3.], np.float32)

[ 2.  4.  8.]

Q15. Compute natural, base 10, and base 2 logarithms of x element-wise.

1x = np.array([1, np.e, np.e**2])

natural log = [ 0.  1.  2.]
common log = [ 0.          0.43429448  0.86858896]
base 2 log = [ 0.          1.44269504  2.88539008]

Q16. Compute the natural logarithm of one plus each element in x in floating-point accuracy.

1x = np.array([1e-99, 1e-100])

[  1.00000000e-099   1.00000000e-100]

Floating point routines

Q17. Return element-wise True where signbit is set.

1x = np.array([-3, -2, -1, 0, 1, 2, 3])

[ True  True  True False False False False]

Q18. Change the sign of x to that of y, element-wise.

1x = np.array([-1, 0, 1])
2y = -1.1

[-1. -0. -1.]

Arithmetic operations

Q19. Add x and y element-wise.

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

[0 0 0]

Q20. Subtract y from x element-wise.

1x = np.array([3, 4, 5])
2y = np.array(3)

[0 1 2]

Q21. Multiply x by y element-wise.

1x = np.array([3, 4, 5])
2y = np.array([1, 0, -1])

[ 3  0 -5]

Q22. Divide x by y element-wise in two different ways.

1x = np.array([3., 4., 5.])
2y = np.array([1., 2., 3.])

[ 3.          2.          1.66666667]
[ 3.  2.  1.]

Q23. Compute numerical negative value of x, element-wise.

1x = np.array([1, -1])

[-1  1]

Q24. Compute the reciprocal of x, element-wise.

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

[ 1.   0.5  5. ]

Q25. Compute $x^y$, element-wise.

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

[[ 1  4]
 [ 3 16]]

Q26. Compute the remainder of x / y element-wise in two different ways.

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

[1 0 1 1 0 1]
[-1  0 -1  1  0  1]

Miscellaneous

Q27. If an element of x is smaller than 3, replace it with 3. And if an element of x is bigger than 7, replace it with 7.

1x = np.arange(10)

[3 3 3 3 4 5 6 7 7 7]

Q28. Compute the square of x, element-wise.

1x = np.array([1, 2, -1])

[1 4 1]

Q29. Compute square root of x element-wise.

1x = np.array([1., 4., 9.])

[ 1.  2.  3.]

Q30. Compute the absolute value of x.

1x = np.array([[1, -1], [3, -3]])

[[1 1]
 [3 3]]

Q31. Compute an element-wise indication of the sign of x, element-wise.

1x = np.array([1, 3, 0, -1, -3])

[ 1  1  0 -1 -1]