Natural numbers are the most basic types of numbers we use in everyday life—the ones we use to count things. If you are counting apples in a basket, you start with one.
Definition: Counting numbers starting from 1, 2, 3, 4, and so on.
Notation: The set of natural numbers is denoted by the letter N.
The Set: $N = \{1, 2, 3, 4, …\}$
Whole numbers are simply natural numbers, but they include the concept of “nothingness” or zero.
Definition: All natural numbers along with ‘0’ are called whole numbers.
Notation: The set of whole numbers is denoted by the letter W.
The Set: $W = \{0, 1, 2, 3, 4, …\}$
Key Tip: Every natural number is a whole number, but not every whole number is a natural number (because of zero!).
Even numbers are numbers that can be divided perfectly into two equal groups.
Definition: Numbers which give zero as a remainder when divided by 2.
Examples: $0, 2, 4, 6, 8, …$
Odd numbers are the “opposites” of even numbers. They cannot be divided into two equal whole groups.
Definition: Numbers which give 1 as a remainder when divided by 2.
Examples: $1, 3, 5, 7, 9, …$
Multiples are what you get when you multiply a specific number by integers. Think of them as the results found in a multiplication table.
Definition: The products obtained when a number is multiplied by 1, 2, 3, 4, etc.
Example (Multiples of 3): $3, 6, 9, 12, 15, …$
Example (Multiples of 5): $5, 10, 15, 20, 25, …$
| Number Type | Starts With | Includes Zero? | Example |
| Natural | 1 | No | 1, 5, 100 |
| Whole | 0 | Yes | 0, 5, 100 |
| Even | 0 | Yes | 2, 4, 12 |
| Odd | 1 | No | 1, 3, 15 |
Factors are the building blocks of a number.
Definition: The numbers which divide a given number without leaving a remainder are called its factors.
Alternative Name: A factor is also commonly referred to as a divisor.
How to find them: You can find factors by looking for pairs of numbers that multiply together to reach your target.
Example: Factors of 6
To find the factors of 6, we see that:
$1 \times 6 = 6$
$2 \times 3 = 6$
Therefore, the factors of 6 are 1, 2, 3, and 6.
Example: Factors of 12
To find the factors of 12, we check multiplication pairs:
$1 \times 12 = 12$
$2 \times 6 = 12$
$3 \times 4 = 12$
The factors of 12 are 1, 2, 3, 4, 6, and 12.
In algebra, we often use symbols to represent numbers that can change.
Definition: A symbol which can take on various numerical values is called a variable or a literal.
Examples: Common variables include letters like $x, y, z, a, b, c$, etc.
Unlike variables, constants never change.
Definition: A symbol which has a fixed, unchanging value is called a constant.
Examples: Specific numbers like $5, 2, 3, 7$, etc.
When writing math involving variables, we often simplify the way we show multiplication.
General Rule: $2 \times a$ can be written simply as $2a$.
Example: $3 \times x$ is written as $3x$.
Example: $5 \times x$ is written as $5x$.
Caution: This shorthand only works with variables! You cannot do this with two constants. For example, writing $2 \times 3$ as “$23$” is incorrect. $2 \times 3$ is always 6.1. Repeated Addition (Multiplication by a Number)
When you add the same variable multiple times, you are performing repeated addition. In algebra, we simplify this by using a coefficient (the number in front of the letter).
The Concept: Adding $a$ to itself twice is $2a$. Adding it three times is $3a$.
The General Rule: If you add $a$ a total of $n$ times, the result is $na$.
Think of the variable $a$ as an object (like an apple). If you have 3 apples and add 2 more, you have 5 apples.
Addition: $3a + 2a = 5a$
Subtraction: $7a – 3a = 4a$
Key Takeaway: When adding or subtracting, you only change the number in front (the coefficient). The variable stays exactly the same.
When you multiply a variable by itself, you aren’t just increasing the count; you are increasing the power (or exponent).
Square: $a \times a = a^2$
Read this as “a squared” or “a to the power of 2.”
Cube: $a \times a \times a = a^3$
Read this as “a cubed” or “a to the power of 3.”
It is a common mistake to mix these two up. Here is a quick reference table to keep them straight:
| Operation | Expression | Result | Name |
| Addition | $a + a$ | $2a$ | Coefficients |
| Multiplication | $a \times a$ | $a^2$ | Exponents |
Building on our previous look at algebraic variables, let’s dive into Exponential Form—the mathematical shorthand for repeated multiplication.
If you’ve ever felt overwhelmed by long strings of numbers, exponents are about to become your favorite tool.
Exponential form is a way to write the multiplication of the same number multiple times in a concise way.
The Definition: If a number $a$ is multiplied by itself $n$ times, we write it as $a^n$.
How to Read It: You would say this as “a to the power n”.
The Formula: $N = a \times a \times a \times \dots n \text{ times} = a^n$.
Squaring a number ($a^2$) is one of the most common operations in math. Here is a quick reference guide based on the chart:
$1^2 = 1$
$2^2 = 4$
$3^2 = 9$
$4^2 = 16$
$5^2 = 25$
$6^2 = 36$
$7^2 = 49$
$8^2 = 64$
$9^2 = 81$
$10^2 = 100$
11–20: Ranges from $11^2 = 121$ up to $20^2 = 400$.
21–30: Ranges from $21^2 = 441$ up to $30^2 = 900$.
Notice the pattern for numbers ending in 5? They all end in 25:
$15^2 = 225$
$25^2 = 625$
$35^2 = 1225$
$95^2 = 9025$
When you multiply a number by itself three times, it’s called a “cube” ($a^3$).
$1^3 = 1$
$2^3 = 8$
$3^3 = 27$
$4^3 = 64$
$5^3 = 125$
$10^3 = 1000$
Squaring a number means multiplying it by itself once ($a \times a$).
| Number | Square | Number | Square | Number | Square |
| 1 | $1$ | 11 | $121$ | 21 | $441$ |
| 2 | $4$ | 12 | $144$ | 25 | $625$ |
| 5 | $25$ | 15 | $225$ | 30 | $900$ |
| 10 | $100$ | 20 | $400$ | 95 | $9025$ |
Cubing a number means multiplying it by itself twice ($a \times a \times a$).
$2^3 = 8$
$3^3 = 27$
$4^3 = 64$
$5^3 = 125$
$10^3 = 1000$
The number 2 is unique because its powers grow very quickly and are used constantly in computer science and advanced algebra.
$2^4 = 16$
$2^5 = 32$
$2^7 = 128$
$2^{10} = 1024$
If you have a large number and need to find its exponential form, you can use successive division (also known as prime factorization).
Example: Express 1024 in exponential form with base 2.
Divide 1024 by 2 repeatedly.
Count how many times you divided.