# Python: Numbers¶

There are four distinct numeric types:

• int: integers. (Limited precision.)
• long: long integers. (Unlimited precision.) They behave exactly like plain int‘s, but they can assume arbitrary values.
• float: floating point numbers. Also called floats, they represent real numbers. They are written with a decimal point dividing the integer and fractional parts. Floats may also be in scientific notation, with E or e indicating the power of 10.
• bool: Boolean conditions. They can assume only two values: True and False. They are used to represent the result of comparisons.

Example. Let’s create four variables, one for each type, and print them on the screen with print:

x = 10
y = 10L
z = 3.14
c = False

# Print the four variables
print x, y, z, c

# The same, adding some text
print "x =", x, "y =", y, "z =", z, "; the condition is", c


This syntax of print is valid for all variable types, not only numeric types.

## Operations¶

All numeric types support default arithmetic operations:

• sum +, difference -, product *
• division /, integer division //, remainder %.
• power **.

The type of the result of n operation m is the most “complex” type between the type of n and m. The complexity hierarchy is:

bool < int < long < float


Example. Two examples of automatic conversion:

result = 1 * 1L              # int * long
print result
print type(result)           # long

result = 1.0 + 1L            # float * long
print result
print type(result)           # float


Warning

To avoid errors, you should choose the most appropriate type for your variables, so that the type of the result is sufficiently “complex” to represent the value of the result!

See for example the following GC-content exercise.

All numeric types support comparison operations:

• equal == not equal != (or <>)
• strictly less than < strictly greater than >
• less than or equal<= greater than or equal >=

The expression has value True if the condition is satisfied, otherwise False. The result of a comparison is always bool.

Example. Let’s consider:

na, nc, ng, nt = 2, 6, 50, 4

result = (na + nt) > (nc + ng)
print result
print type(result)


We can build complex conditions using Boolean operations:

• a and b is True if both a and b are True.
• a or b is True if at least one between a and b is True.
• not a is True only if a is False.

Warning

In this context, both a and b should be bool, otherwise the result may seem apparently bizarre.

Example. x > 12 and x < 34 both result bool, so we can combine them to obtain:

#     int          int
#      |            |
print (x > 12) and (x < 34)
#     \______/     \______/
#       bool         bool
#     \___________________/
#             bool


or:

#          int          int
#           |            |
print (not (x > 12)) or (x < 34)
#          \______/
#            bool
#     \____________/    \______/
#          bool           bool
#     \________________________/
#                bool


## Examples¶

Example. Let’s calculate the zeros of the quadratic equation $$x^2 - 1 = 0$$:

a, b, c = 1.0, 0.0, -1.0

delta = b**2 - 4*a*c

zero1 = (-b + delta**0.5) / (2 * a)
zero2 = (-b - delta**0.5) / (2 * a)

print zero1, zero2


Here we use x**0.5 to calculate the square root: $$\sqrt{x} = x^\frac{1}{2}$$.

Example. We want to calculate the GC-content of a genomic region. We know that the region:

• Is 1521 nucleotides long.
• Contains 316 cytosines.
• Contains 235 guanines.

The GC-content is defined as $$(g + c) / n$$. To calculate the result, we may write:

n, c, g = 1521, 316, 235

gc_content = (c + g) / n
print gc_content


unfortunately, the result of this code is 0! Why?

The problem is that the resulting proportion, amounting to approximately 0.362, cannot be expressed by an integer: we need a rational number. Since both n and m are integers, also gc_content = n / m will be an integer:

print type(gc_content)


As a consequence, 0.362 is approximated to the nearest integer: 0. Oops!

To solve this problem, gc_content needs to be a float. We can do it in two ways:

• Modifying the type of n, m or g:

n, c, g = 1521.0, 316.0, 235.0

gc_content = (c + g) / n
print gc_content

• Using an explicit conversion to a float:

n, c, g = 1521, 316, 235

gc_content = float(c + g) / float(n)
print gc_content


Example. To check that x falls in the interval A $$= [10,50]$$ let’s write:

x = 17
min_a, max_a = 10, 50

inside_a = (min_a <= x <= max_a)
print inside_a


or:

inside_a = ((x >= min_a) and (x <= max_a))


Assuming that inside_a, inside_b and inside_c indicate that x is contained in interval A, B or C, respectively, we can create more complex conditions:

# x is in at least one interval
inside_at_least_one = inside_a or inside_b or inside_c

# x is in both A and B, but not in C
inside_a_and_b_but_not_c = inside_a and inside_b and (not inside_c)


## Exercises¶

1. Create the following variables, checking that value and type are correct (using print and type):

1. a and b with values 12 and 23 as integers.
2. c and d with values 34 and 45 as long integers.
3. x and y with values 21 and 14 as floats.
2. Using print (once), print:

1. All the above variables in the same line.
2. All the above variables, separated by ;, on the same line.
3. The text “the product of a and b is a * b”, replacing a, b and a * b with the variable values.
3. Find the value and the type of:

1. The product of a and b.
2. The difference between c and d.
3. The division of x by y.
4. The integer division of a by b.
5. The integer division of c by d.
6. The integer division of x by y.
7. The product of a and c.
8. The product of b and y.
9. The product of x and d.
10. 2 to the power 0.
11. 2 to the power 100.
12. 2 to the power 1.2.
13. 2 to the power -2.
14. The square root of 4.
15. The square root of 2.
4. What is the difference between:

1. 10 / 12
2. 10 / 12.0
3. 10 // 12
4. 10 // 12.0
5. What is the difference between:

1. 10 % 3
2. 10 % 3.0
6. Using pi = 3.141592 and given r = 2.5, calculate:

1. The circumference of a circle with radius r: $$2 \pi r$$.
2. The area of a circle with radius r: $$\pi r^2$$.
3. The volume of a sphere with radius r: $$\frac{4}{3} \pi r^3$$.
7. Create 2 variables a = 100 and b = True. Using an adequate number of auxiliary variables (with arbitrary names!), give the value of a to b and viceversa.

(Writing a = True and b = 100 is not sufficient!)

Can it be done with only one auxiliary variable?

8. On the same strand of DNA there are 2 genes. The first includes i nucelotides from position 10 to position 20, the second nucleotides from position 30 to position 40. Let’s write this:

gene1_start, gene1_end = 10, 20
gene2_start, gene2_end = 30, 40


Given a variable pos representing an arbitrary position on the DNA strand, write some comparisons to verify if:

1. pos is in the first gene.
2. pos is in the second gene.
3. pos is between the start of the first gene and the end of the second.
4. pos is between the start of the first gene and the end of the second, but not in any of the genes.
5. pos is before the start of the first gene or after the end of the second.
6. pos is inside one of the genes.
7. pos is distant no more than 10 nucleotides from the beginning of the first gene.
9. Given three Boolean variables t, u, and v, write expressions that are True if and only if:

1. t, u, v are all true.
2. t is true or u is true, but not both.
3. Exactly one of the three variables is false.
4. Exactly one of the three variables is true.
10. Exercise 1.4 from [book2]: Exponential Growth

Given optimal growth conditions, a single E. coli bacterium can divide within 20 minutes. If the conditions stay optimal, how many bacteria are there after 8 hours?