Calculations with integers

Welcome to the World of Integers!

Hello! In this chapter, we are going to dive into Calculations with Integers. This is a foundational part of your OCR GCSE (9-1) Mathematics course. Think of integers as the "building blocks" of math. Once you master how to add, subtract, multiply, and divide them, everything else—from algebra to finance—becomes much easier!

An integer is simply a whole number. This includes positive numbers, negative numbers, and zero. We see them everywhere: in bank balances, temperatures, and even sports scores. Let’s get started!

1. Adding and Subtracting Integers

Adding and subtracting integers can feel a bit like moving back and forth on a ladder or a thermometer. Positive numbers move you up (or right), and negative numbers move you down (or left).

The Number Line Method

When you add a positive number, you move to the right.
When you subtract a positive number, you move to the left.

Dealing with "Double Signs"

Sometimes you will see two signs right next to each other, like \( 5 - (-3) \). Don't worry if this looks weird at first! There is a simple trick to simplify them:

• Two signs the same become a plus: \( + (+) \) or \( - (-) \) becomes \( + \)
• Two signs that are different become a minus: \( + (-) \) or \( - (+) \) becomes \( - \)

Real-World Analogy: The Bank Account
Think of positive numbers as money you have and negative numbers as debt.
Example 1: \( 5 + (-2) \) is like having \( \$5 \) but adding \( \$2 \) of debt. You now have \( \$3 \).
Example 2: \( 5 - (-2) \) is like having \( \$5 \) and someone taking away \( \$2 \) of your debt. Taking away debt makes you richer! So, \( 5 + 2 = 7 \).

Quick Review Box:
\( 10 + (-4) = 10 - 4 = 6 \)
\( 10 - (-4) = 10 + 4 = 14 \)

Key Takeaway: If signs are the same, replace them with a plus. If they are different, replace them with a minus.

2. Multiplying and Dividing Integers

Multiplying and dividing integers is actually often easier than adding them because the rules are very consistent. You just do the calculation like normal, and then decide if the answer is positive or negative.

The Rule of Signs

• Same signs give a Positive answer. (\( + \times + = + \) or \( - \times - = + \))
• Different signs give a Negative answer. (\( + \times - = - \) or \( - \times + = - \))

Step-by-Step Process:
1. Ignore the signs and multiply or divide the numbers.
2. Look at the original signs.
3. Apply the rule above to find the final sign.

Example: Calculate \( -6 \times 4 \)
1. \( 6 \times 4 = 24 \)
2. One is negative (\( - \)), one is positive (\( + \)).
3. The signs are different, so the answer is \( -24 \).

Memory Aid: The "Good/Bad" Rule
- A good thing (\( + \)) happening to a good person (\( + \)) is good (\( + \))
- A bad thing (\( - \)) happening to a bad person (\( - \)) is good (\( + \))
- A bad thing (\( - \)) happening to a good person (\( + \)) is bad (\( - \))

Key Takeaway: Same signs = Positive result. Different signs = Negative result.

3. Priority of Operations (BIDMAS)

When a calculation has many parts, we must follow a specific order. If we didn't, everyone would get different answers! We use the acronym BIDMAS (or sometimes BODMAS) to remember the order.

• Brackets: Do anything inside \( ( ) \) first.
• Indices: Powers like \( 2^2 \) or square roots.
• Division and Multiplication: Do these from left to right.
• Addition and Subtraction: Do these last, from left to right.

Example: Calculate \( 5 + 2 \times (10 - 7) \)
1. Brackets first: \( 10 - 7 = 3 \). Now we have \( 5 + 2 \times 3 \).
2. Multiplication next: \( 2 \times 3 = 6 \). Now we have \( 5 + 6 \).
3. Addition last: \( 5 + 6 = 11 \).
Common Mistake: Many students just go from left to right and do \( 5 + 2 \) first. Don't do this! Always check for Multiplication before Addition.

Did you know? Multiplication and Division have the same priority. If you see both in a row, just work from left to right like you're reading a book!

Key Takeaway: Always follow BIDMAS to ensure your calculations are accurate.

4. Inverse Operations

Inverse is just a fancy math word for opposite. Knowing the opposite of an operation helps you "undo" a calculation or check your work.

• The inverse of Addition is Subtraction.
• The inverse of Multiplication is Division.
• The inverse of Squaring is Square Root.

Using Inverse to Simplify Calculations:
You can sometimes make a tricky non-calculator question easier by adjusting both numbers.
Example: \( 25 \times 12 \)
Double the first number: \( 50 \)
Halve the second number: \( 6 \)
Now calculate \( 50 \times 6 = 300 \). Much easier!

"I'm thinking of a number" Problems:
If a problem says: "I think of a number, multiply it by 3 and get 15," you use the inverse to solve it. Work backwards! The opposite of multiplying by 3 is dividing by 3. \( 15 \div 3 = 5 \).

Key Takeaway: Use inverse operations to "work backwards" and check if your answers make sense.

5. Common Pitfalls to Avoid

• The "Minus Minus" Trap: Remember that \( -5 - 3 \) is NOT positive. You are at \( -5 \) and going further down by \( 3 \), so the answer is \( -8 \). Only two signs touching each other (like \( - - \)) become a plus.
• Dividing by Zero: You cannot divide a number by zero. If you see this in a problem, the answer is "undefined" or impossible!
• Forgetting BIDMAS: Always pause before you start a long string of numbers and ask: "Which part do I do first?"

Final Encouragement: Calculations with negative numbers can be confusing at first—even for professional mathematicians! If you find yourself stuck, draw a quick number line or think about money. You've got this!