Numbers: Exact and Approximate

The Wolfram Language keeps numbers exact unless you ask for a decimal approximation.

1/3 + 1/6
10/4
10/4.
N[10/4]
Sqrt[8]
2^100

1/2
5/2
2.5
2.5
2 Sqrt[2]
1267650600228229401496703205376

• Integers and fractions (rationals) stay exact, however large.
• A number with a decimal point, like 4., is approximate, and so is any result it touches.
• N[expr] gives a numerical approximation.
• Sqrt[8] simplifies to 2 Sqrt[2] — still exact.

Arbitrary precision

N[Pi]
N[Pi, 30]
N[Sqrt[2], 20]

3.14159
3.14159265358979323846264338328
1.4142135623730950488

N[expr, n] computes n significant digits — as many as you like. N[Pi] is a normal machine-precision number (about 16 digits); the notebook just displays six.

Integer operations

Quotient[17, 5]
Mod[17, 5]
20!
FactorInteger[360]
Prime[10]
PrimeQ[2^31 - 1]
GCD[12, 18]

3
2
2432902008176640000
{{2, 3}, {3, 2}, {5, 1}}
29
True
6

FactorInteger[360] means 360 = 2³ × 3² × 5.

Rounding

Round[2.5]
Round[3.5]
Floor[-2.5]
Ceiling[2.1]
Round[3.14159, 0.01]

2
4
-3
3
3.14

Round rounds halves to the nearest even integer, so 2.5 becomes 2.

Constants and functions

Built-in constants and functions use capital letters: Pi, E, I (the imaginary unit), Infinity, Degree.

Sin[Pi/6]
Sin[30 Degree]
Log[E]
Log[10, 1000]
Sqrt[-4]

1/2
1/2
1
3
2 I

Trigonometric functions use radians; multiply by Degree to use degrees. Log[b, x] is the logarithm to base b.

Common mistakes

• Writing sin(x) — it's Sin[x]: capital letter, square brackets. Parentheses only group.
• Expecting 1/3 to give 0.333333 — it stays 1/3. Use N[1/3] or 1/3..

Practice

Compute 2 to the power 64 exactly, and e (the constant E) to 20 digits.

2^64
N[E, 20]

18446744073709551616
2.7182818284590452354