MATHEMATICA TUTORIAL

# Types of Numbers

Four underlying types of numbers are built into Mathematica.

 Integer arbitrary-length exact integer Rational integer/integer in lowest terms Real approximate real number, with any specified precision Complex complex number of the form number+number I

Intrinsic types of numbers in Mathematica.

Rational numbers always consist of a ratio of two integers, reduced to lowest terms.
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Approximate real numbers are distinguished by the presence of an explicit decimal point.
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An approximate real number can have any number of digits.
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Complex numbers can have integer or rational components.
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They can also have approximate real number components.
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 123 an exact integer 123. an approximate real number 123.0000000000000 an approximate real number with a certain precision 123.+0.I a complex number with approximate real number components

Several versions of the number 123.

You can distinguish different types of numbers in Mathematica by looking at their heads. (Although numbers in Mathematica have heads like other expressions, they do not have explicit elements which you can extract.)

The object is taken to be an exact integer, with head Integer.
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The presence of an explicit decimal point makes Mathematica treat as an approximate real number, with head Real.
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 NumberQ[x] test whether x is any kind of number IntegerQ[x] test whether x is an integer EvenQ[x] test whether x is even OddQ[x] test whether x is odd PrimeQ[x] test whether x is a prime integer Head[x]===type test the type of a number

Tests for different types of numbers.

NumberQ[x] tests for any kind of number.
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is treated as a Real, so IntegerQ gives False.
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If you use complex numbers extensively, there is one subtlety you should be aware of. When you enter a number like , Mathematica treats it as an approximate real number, but assumes that its imaginary part is exactly zero. Sometimes you may want to enter approximate complex numbers with imaginary parts that are zero, but only to a certain precision.

When the imaginary part is the exact integer , Mathematica simplifies complex numbers to real ones.
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Here the imaginary part is only zero to a certain precision, so Mathematica retains the complex number form.
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The distinction between complex numbers whose imaginary parts are exactly zero, or are only zero to a certain precision, may seem like a pedantic one. However, for example, when the interpretation of powers and roots of complex numbers is discussed in "Functions That Do Not Have Unique Values", the distinction becomes significant.

One way to find out the type of a number in Mathematica is just to pick out its head using Head[expr]. For many purposes, however, it is better to use functions like IntegerQ that explicitly test for particular types. Functions like this are set up to return True if their argument is manifestly of the required type, and to return False otherwise. As a result, IntegerQ[x] will give False unless has an explicit integer value.

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