Welcome to Order of Operations and Negative Numbers!
Have you ever wondered why a calculator gives you a different answer than your friend for the same math problem? Usually, it's because of the order of operations. In this chapter, we are going to learn the "rules of the road" for mathematics. We will also explore the world of negative numbers—those numbers below zero that help us describe things like debt, freezing temperatures, or diving deep into the ocean.
By the end of these notes, you will be able to handle complex calculations with confidence, making sure you get the right answer every single time!
1. Understanding Negative Numbers
Negative numbers are numbers less than zero. We write them with a minus sign in front, like \(-5\). You can think of them using a number line. Positive numbers live to the right of zero, and negative numbers live to the left.
Adding and Subtracting with Negative Numbers
Don't worry if this seems tricky at first! A great way to visualize this is to imagine yourself standing on a number line at the first number:
- Adding a positive number: Move to the right (getting bigger).
- Subtracting a positive number: Move to the left (getting smaller).
But what happens when we have two signs next to each other? Here is a simple trick:
The Double Sign Rule:
If the signs are the same, they become a plus: \(+ (+)\) or \(- (-)\) becomes \(+\)
If the signs are different, they become a minus: \(+ (-)\) or \(- (+)\) becomes \(-\)
Example 1: Calculate \(5 - (-3)\)
The two minuses next to each other become a plus: \(5 + 3 = 8\).
Example 2: Calculate \(-2 + (-4)\)
The plus and minus are different, so they become a minus: \(-2 - 4 = -6\). (Imagine starting at \(-2\) on the number line and moving \(4\) steps further left).
Multiplying and Dividing with Negative Numbers
The rules for multiplication and division are very similar to the "Double Sign Rule" above:
- Positive \(\times\) Positive \(=\) Positive: \(3 \times 4 = 12\)
- Negative \(\times\) Negative \(=\) Positive: \(-3 \times -4 = 12\)
- Positive \(\times\) Negative \(=\) Negative: \(3 \times -4 = -12\)
- Negative \(\times\) Positive \(=\) Negative: \(-3 \times 4 = -12\)
Quick Review: If the signs are the same, the answer is positive. If the signs are different, the answer is negative! This works exactly the same for division (\(/\) or \(\div\)).
Key Takeaway:
Always check your signs before you start calculating. A single missing minus sign can change your whole answer!
2. The Order of Operations (BIDMAS / BODMAS)
When a math problem has many parts (like addition, multiplication, and powers all at once), we must follow a specific order. Without these rules, people would get different answers for the same sum!
We use the word BIDMAS (or sometimes BODMAS) to help us remember the order:
B - Brackets \(( \dots )\)
I - Indices (This means Powers like \(x^2\), Roots like \(\sqrt{x}\), and Reciprocals)
D - Division \((\div)\)
M - Multiplication \((\times)\)
A - Addition \((+)\)
S - Subtraction \((-)\)
Important Note on Priority
Division and Multiplication have equal priority. Similarly, Addition and Subtraction have equal priority. If you have a sum with both, just work from left to right.
Step-by-Step Example:
Let's solve: \(10 + (2 \times 3^2) - 5\)
- Brackets first: Look inside \((2 \times 3^2)\). Inside the bracket, we have a power (\(3^2\)).
- Indices: \(3^2 = 9\). So the bracket is now \((2 \times 9)\).
- Solve the Bracket: \(2 \times 9 = 18\). Our sum is now \(10 + 18 - 5\).
- Addition and Subtraction: Move left to right. \(10 + 18 = 28\). Then \(28 - 5 = 23\).
The final answer is \(23\).
Common Mistake to Avoid:
Many students see \(2 + 3 \times 5\) and want to do the addition first because it comes first in the sentence. Don't do it! Multiplication always comes before addition. Correct: \(2 + 15 = 17\). Incorrect: \(5 \times 5 = 25\).
3. Powers and Roots with Negative Numbers
In Key Stage 3, we also look at Integer Powers (whole number powers) and Roots.
Powers
Remember that a power tells you how many times to multiply a number by itself. For example, \(a^2 = a \times a\).
- \((-3)^2 = -3 \times -3 = 9\) (A negative squared becomes positive).
- \((-2)^3 = -2 \times -2 \times -2 = -8\) (A negative cubed stays negative).
Roots
A root is the inverse (opposite) of a power. For example, the square root of \(16\) is \(4\) because \(4^2 = 16\).
Did you know? You can find the square root (\(\sqrt{\dots}\)) or the cube root (\(\sqrt[3]{\dots}\)) of numbers. While we often use calculators for these, it's important to recognize common powers of \(2, 3, 4,\) and \(5\) to help your mental math!
4. Working with Different Types of Numbers
The rules of negative numbers and order of operations don't just apply to whole numbers (integers). They also apply to decimals and fractions (both proper and improper).
Example: \(-\frac{1}{2} \times \frac{2}{3} = -\frac{2}{6} = -\frac{1}{3}\)
The same multiplication sign rules apply! (Negative \(\times\) Positive \(=\) Negative).
Note: For more details on how to calculate with fractions, check out the chapter on "Operations with Fractions".
Summary and Key Takeaways
- Negative Number Addition/Subtraction: Use a number line or the "Double Sign Rule." Same signs make a plus; different signs make a minus.
- Negative Number Multiplication/Division: Same signs give a positive result; different signs give a negative result.
- BIDMAS: Always follow the order: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
- Left to Right: If operations have equal priority (like \(+\) and \(-\)), work from left to right.
- Powers: A negative number squared is always positive.
Practice these rules often, and they will become second nature! When in doubt, write out your calculation one step at a time, just like we did in the BIDMAS example.