What Type of Number Will You Get?

This Web page is about the types of numbers you get as solutions/correct answers to math problems depending on the arithmetic operation used. (Except division; out of all 4 aritmetic operations, division is the most unpredictable & tends to gives you random types of quotients!) For more information about number types, see the page entitled "Types of Numbers".

You can select the 3 Different Pairs of Directly Opposite Number Types here:

  1. Even Numbers & Odd Numbers
  2. Positive Numbers & Negative Numbers
  3. Real Numbers & Imaginary Numbers

First, Even Numbers & Odd Numbers

Even + Even = Even

Odd + Odd = Even

Even + Odd = Odd

(Addition is Commutative!)

Even - Even = Even

Odd - Odd = Even

Even - Odd = Odd

Odd - Even = Odd


Even × Even = Even

Odd × Odd = Odd

Even × Odd = Even

(Multiplication is Commutative!)

Positive Numbers & Negative Numbers

Positve + Positive = Positive

Negative + Negative = Negative

Positive + Negative = Sign That's Greater In Absolute Value or Zero If They're Additive Inverses

(Addition is Commutative!)

Positive - Positive = Positive or Zero If Both Numbers Are Equal or Negative If 2nd Number Is Greater In Absolute Value

Negative - Negative = Negative or Zero If Both Numbers Are Equal or Positive If 2nd Number Is Greater In Absolute Value

Positive - Negative = Positive

Negative - Positive = Negative


Positive × Positive = Positive

Negative × Negative = Positive

Positive × Negative = Negative

(Multiplication is Commutative!)

Finally, Real Numbers & Imaginary Numbers

Real + Real = Real

Imaginary + Imaginary = Imaginary

Real + Imaginary = Complex

(Addition is Commutative!)

Real - Real = Real

Imaginary - Imaginary = Imaginary

Real - Imaginary = Complex

Imaginary - Real = Complex


Real × Real = Real

Imaginary × Imaginary = Real

Real × Imaginary = Imaginary

(Multiplication is Commutative!)

In conclusion, these are the only 3 different pairs of directly opposite number types because with indirectly opposite types, the number type of the solution is random! Multiplying an irrational number by an irrational number, for example, could give you either a rational product or an irrational product. The square root of 2 multiplied by itself gives you 2, which is rational; however, the square root of 2 multiplied by the square root of 3 = the square root of 6, which is irrational! Also, the square root of 2 + the square root of 3the square root of (2 + 3), which is 5; but yes, the sum of those 2 irrational numbers is also irrational; it's approximately equal to π + .0046717164... But adding 2 irrational numbers might give a rational number! (The square root of 2 + (1 - the square root of 2) = 1!)

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© Derek Cumberbatch