# Number Theory: Types Of Numbers

Number theory is very important on the SAT math section as it forms the foundation for understanding higher algebra. A number is a mathematical object used to count, label, and measure. There are different sets of numbers such as, natural numbers, rational and irrational numbers, odd numbers, even numbers, whole numbers, prime numbers, composite numbers, real numbers etc.

The ** set of natural numbers** is another name given to the

**and it is represented by the**

*set of counting numbers***symbol**

*N.**The set of natural numbers, N = {1, 2, 3…..}.*

The ** set of whole numbers** is the set of natural numbers or

**and**

*counting numbers***zero**. It is represented by the

**. N.B. it is obvious from the statement above that zero is not a natural number.**

*symbol W**The set of whole numbers, W = {0, 1, 2, 3……}.*

The ** set of integers** can be accepted as the set of

**and**

*positive*

*negative***or**

*natural***and**

*whole numbers***Note however, that zero is neither positive nor negative. The set of integers can be represented by**

*zero.***.**

*the symbol Z**The set of integers, Z = {…, -3, -2, -1, 0, 1, 2, 3….}. *

The set of rational numbers is really the set of numbers that can be written as a fraction. It is the set of positive and negative fractions. e.g. -2/3, -1/2, 3/5 and 8/9.

** A rational number** can always be written as a decimal, whether terminating or recurring. E.g. 0.8, 0.65, 0.3 and 0.6.

*The set of rational numbers is represented by the symbol Q.*

*The set of rational numbers, Q = {n/d, d **≠ 0, n, d element of Z and n and d have no common factor}. *

Where n = numerator, d = denominator and n/d is a fraction in simplest form.

The ** set of rational numbers** is the

**that cannot be written as fractions.**

*set of numbers*e.g. -√3, √7, -√5/3 and √2/7.

Further, when ** irrational numbers** are written as decimal they do not terminate or recur.

E.g. π = 3.142 592 7……. (Correct to 7 d.p)

√3 = 1.732 050 8…….. (Correct to 7 d.p)

*The set of irrational numbers is represented by the symbol Q’ or I*

*The set of irrational numbers, Q’ or I ≠ {n/d, ≠ 0, n, d element of Z and, d and n have no common factor}.*

** Even numbers** are numbers that cannot only be divided by 2 and it self.

e.g. {2, 4, 6, 8….}

** Prime numbers** are numbers that can only be divided by itself and 1

e.g. {3, 5, 7, 11….}

** Odd numbers** are numbers that when divided by 2 it leaves a remainder

e.g. {3, 5, 7, 9, 11…..}

** Composite numbers** are numbers that have factors other than itself and 1

e.g. {4, 6, 8, 9, 10, 12….}

The set of real numbers is the union of the ** set of rational and irrational numbers**, and it is represented by the

*symbol R.*Imaginary numbers is not exactly classified as a set per se as the other types of numbers. Imaginary numbers are numbers that when squared it produce a negative answer. That is, a^{2} = -b.

There are no known number that one can multiply by itself to produce a negative number. We can try by playing around with some numbers:

-2 × 3 = -6

-1 × -1 = 1

– 3 × 7 = -21

We can try every possible number we know, we will never get it. Therefore, the concept of *imaginary* came into play. So let us *imagine* that such a number exist, we could call it ‘*i*’ for imaginary.

Thus, by taking the square of both side *i **× i* = i^{2 }= -1

√*i*^{2} = √-1

*i* = √-1

Therefore the value of imaginary number is √-1. Let us just use one example of how it applies

Example:

Find the value of √-16.

What we do is to find some way we can rewrite the value that it makes sense.

√-16 = √ (16 × -1) which is the same as √16 × √-1 = 4√-1. Since the √-1 is the value for ‘*i*‘ we can just substitute it.

√-16 = 4*i*

Do you really need to know all of these???? YES! The SAT expects you to know the difference between all these types of numbers. The SAT also covers imaginary numbers, so make sure you know it!!!