Natural Numbers and Integers - The Basics

You've been using natural numbers your whole life without realising it! These are simply the counting numbers: 1, 2, 3, 4, and so on. They're called "natural" because they feel natural to use when counting objects.

Integers take things a step further by including zero and all the negative numbers too. Think of them as the complete family: ...-3, -2, -1, 0, 1, 2, 3... The symbol for integers is Z, which comes from the German word "Zahlen" meaning numbers.

Quick Tip: Remember that natural numbers don't include zero, but integers do! This catches lots of students out in tests.

The number line is your best friend here. Numbers get bigger as you move right and smaller as you move left. So -1 is actually bigger than -5 because it's further to the right!

Adding and Subtracting Integers

Adding integers might seem tricky at first, but there are simple patterns to follow. When you're adding two positive numbers, it's just normal addition. When adding two negative numbers, add them up and keep the negative sign - think of it as getting "more negative".

The interesting bit happens when you mix positive and negative numbers. Picture it like a battle between the numbers! Find the difference between them (ignore the signs), then use the sign of the bigger number. So 7 + (-3) = 4 because 7 is bigger and positive.

For subtraction, use the "Keep, Change, Change" rule. Keep the first number, change the minus to a plus, and change the sign of the second number. So 8 - 5 becomes 8 + (-5) = 3.

Remember: Subtracting a negative is the same as adding a positive! The two negatives cancel out: 4 - (-3) = 4 + 3 = 7.

Multiplying and Dividing Integers

Good news - multiplying and dividing integers is much simpler than adding and subtracting! It's all about the signs, and there's just one rule to remember.

If the signs are the same (both positive or both negative), your answer is positive. If the signs are different (one positive, one negative), your answer is negative. So -5 × -2 = 10 $same signs = positive$, but 6 × -3 = -18 $different signs = negative$.

This works exactly the same for division. -12 ÷ -3 = 4 because both numbers are negative (same signs), while 20 ÷ -4 = -5 because the signs are different.

Memory Trick: Think "same = smiley face = positive" and "different = frowny face = negative"!

Working Through Examples

Let's tackle some real problems step by step. For -6 + 10 - (-4), work from left to right. First, -6 + 10 gives us 4 (different signs, so take the bigger number's sign). Then 4 - (-4) becomes 4 + 4 = 8.

Here's a practical example: A submarine starts 200 metres below sea level $that's -200m$ and dives another 50 metres. Since "below sea level" means negative, we calculate -200 + (-50) = -250m.

When you see brackets, always solve those first! For (-20 ÷ 5) × (-3), first work out -20 ÷ 5 = -4 $different signs = negative$. Then -4 × (-3) = 12 $same signs = positive$.

Pro Tip: Don't try to do everything in your head - write down each step. This prevents silly mistakes that cost marks!

Quick Test Revision

Here's everything you need for your test! Natural numbers are your basic counting numbers (1, 2, 3...), while integers include zero and all negative numbers too (...-2, -1, 0, 1, 2...).

For adding and subtracting, use the number line or remember that subtracting a negative equals adding a positive. The "Keep, Change, Change" rule is your friend for subtraction problems.

Multiplication and division follow one simple rule: same signs give positive answers, different signs give negative answers. Zero is an integer but it's neither positive nor negative.

Final Reminder: The biggest mistake students make is mixing up the rules for different operations. Adding two negatives stays negative, but multiplying two negatives becomes positive!