Foundations of Mathematics
Arithmetic Review
Core Operations Revisited
Arithmetic is built on four fundamental operations. Addition and subtraction are inverses of each other; one undoes the other. If you know that $7 + 5 = 12$, you also know that $12 - 5 = 7$.
Similarly, multiplication and division are inverse operations. Multiplication can be seen as repeated addition, while division is essentially repeated subtraction. For instance, is the same as adding 4 to itself 3 times (). Conversely, asks how many times you can subtract 4 from 12 before reaching zero.
Properties of Numbers
Beyond operations, numbers themselves have inherent properties that help us classify them and understand their relationships. The most basic classification is between even and odd numbers.
Even Number
noun
An integer that is divisible by 2. Even numbers always end in 0, 2, 4, 6, or 8.
Odd Number
noun
An integer that is not divisible by 2. Odd numbers always end in 1, 3, 5, 7, or 9.
The rules for operating on even and odd numbers are consistent and predictable. Understanding these patterns can help you quickly check the reasonableness of an answer.
| Operation | Example | Result |
|---|---|---|
| even + even | 4 + 6 = 10 | Even |
| odd + odd | 3 + 5 = 8 | Even |
| even + odd | 4 + 3 = 7 | Odd |
| even × even | 2 × 8 = 16 | Even |
| odd × odd | 3 × 7 = 21 | Odd |
| even × odd | 2 × 5 = 10 | Even |
Another crucial classification distinguishes between prime and composite numbers. This property relates to a number's factors, which are the integers that can be multiplied to get that number.
Prime Number
noun
A natural number greater than 1 that has no positive divisors other than 1 and itself.
The first few prime numbers are 2, 3, 5, 7, 11, 13, and 17. Notice that 2 is the only even prime number. All other even numbers are divisible by 2, so they have more than two factors.
The number 1 is not considered prime because it only has one distinct factor: itself.
Composite Number
noun
A natural number that has at least one divisor other than 1 and itself.
Divisibility Shortcuts
Divisibility rules are efficient shortcuts to determine whether one number can be evenly divided by another without performing the full division. They are especially handy for simplifying fractions or finding factors of large numbers.
| Divisible by... | The Rule |
|---|---|
| 2 | The last digit is even (0, 2, 4, 6, 8). |
| 3 | The sum of the digits is divisible by 3. |
| 4 | The last two digits form a number divisible by 4. |
| 5 | The last digit is 0 or 5. |
| 6 | The number is divisible by both 2 and 3. |
| 9 | The sum of the digits is divisible by 9. |
| 10 | The number ends in 0. |
Let's test the number 5,265.
- It ends in 5, so it's divisible by 5.
- The sum of its digits is $5 + 2 + 6 + 5 = 18$. Since 18 is divisible by 3 and 9, the number 5,265 is also divisible by 3 and 9.
- It's not divisible by 2 (it's odd), so it cannot be divisible by 6 or 10.
Ready to test your knowledge?
If , which of the following statements is also true based on the concept of inverse operations?
The number 2 is the only even prime number.
These fundamentals form the bedrock for more complex mathematical reasoning.