Negative Numbers (OCR GCSE Maths): Revision Notes

Example 1: $2−7$

1. Start at $2$ on the number line.
2. Move left $7$ spaces because you are subtracting $7$.
3. Final position is at $-5$.

Example 2: $−4+6$ 4. Start at $-4$ on the number line. 5. Move right $6$ spaces because you are adding $6$. 6. Final position is at $2$.

Example 3: $−1−4$ 7. Start at $-1$ on the number line. 8. Move left $4$ spaces because you are subtracting $4$. 9. Final position is at $-5$.

Example 4: $56−89$ 10. Visualize Starting at $56$:

• Imagine your finger is at $56$ on the number line.

11. Move Left to Zero:

• To reach zero, you must move $56$ spaces to the left.

12. Move Further Left:

• You still need to subtract more, so continue $33$ more spaces left to complete the subtraction (since $89−56=33$).

13. Final Position:

• Your final position on the number line is at $−33$.

Money Analogy:

• Imagine you have $£56$ and someone takes away $£89$. You lose all $£56$, and you're now $£33$ in debt.

Example 5: $−102+217$ 14. Visualize Starting at $−102$:

• Imagine your finger is at $−102$ on the number line.

15. Move Right to Zero:

• To reach zero, you must move $102$ spaces to the right.

16. Move Further Right:

• You still need to add more, so continue $115$ more spaces right to complete the addition (since $217−102=115$).

17. Final Position:

• Your final position on the number line is at $115$.

Money Analogy:

• Imagine you are $£102$ in debt, and someone gives you $£217$. You first pay off your debt and are left with $£115$.

Example 1: $−4+(−8)$ Identify the Touching Signs:

• Here, $+$ and $-$ are touching. Apply the Rule:

     •  Replace :highlight[$+ -$ with $-$].
  

Simplify the Expression:

• The sum now becomes $−4−8$. Calculate the Result:

• Using a number line or mentally, move $8$ spaces left from $-4$.

Final Answer: $−4+(−8)=−12$

Example 2: $5−(−6)$ Identify the Touching Signs:

• Here, $-$ and $-$ are touching. Apply the Rule:

• Replace $-$ $-$ with $+$. Simplify the Expression:

• The sum now becomes $5+6$. Calculate the Result:

• Add $6$ to $5$.

Final Answer: $5−(−6)=11$

Example 3: $−22−(−9)$ Identify the Touching Signs:

• Here, $-$ and $-$ are touching. Apply the Rule:

• Replace $-$ $-$ with $+$. Simplify the Expression:

• The sum now becomes $−22+9$. Calculate the Result:

• Add $9$ to $-22$ by moving $9$ spaces right from $-22$.

Final Answer: $−22−(−9)=−13$

Example 4: $−6−10$

• Here, the signs are not touching, so we don't apply the rule. Proceed with Calculation:

• The sum remains $−6−10$. Calculate the Result:

• Move $10$ spaces left from $-6$.

Final Answer: $−6−10=−16$

Key Rule:

1. Perform the multiplication or division as you normally would, ignoring the plus and minus signs.
2. Count the number of minus signs in the problem:

• If there is one minus sign, the result is negative.
• If there are two minus signs, the result is positive.
• If there are three minus signs, the result is negative.
• If there are four minus signs, the result is positive.
• And so on...

Example 1: $−20÷4$ 18. Perform the Division:

19. Count the Minus Signs:

• There is one minus sign in the problem.

20. Determine the Sign:

• One minus makes the answer negative. Final Answer: $−20÷4=−5$

Example 2: $−6×−9$ 21. Perform the Multiplication:

22. Count the Minus Signs:

• There are two minus signs in the problem.

23. Determine the Sign:

• Two minuses make the answer positive. Final Answer: $−6×−9=54$

Example 3: $−3×−2×−5$ 24. Perform the Multiplication:

25. Count the Minus Signs:

• There are three minus signs in the problem.

26. Determine the Sign:

• Three minuses make the answer negative. Final Answer: $−3×−2×−5=−30$

Example 4: $\frac{−88}{−4}$ 27. Perform the Division:

28. Count the Minus Signs:

• There are two minus signs in the problem.

29. Determine the Sign:

• Two minuses make the answer positive. Final Answer: $\frac{−88}{−4}=22$

📑Example 1: $-2 \times (3 + 5)$ 3. Apply BIDMAS/BODMAS:

• Brackets first: Solve the expression inside the brackets.

• Now, the expression becomes $−2×8.$

4. Multiply:

• Perform the multiplication:

Final Answer: $-2 \times (3 + 5) = -16$

📑Example 2: $-3 + \frac{-8 \div 4}{-2}$ 5. Apply BIDMAS/BODMAS:

• Division first:

• Substituting back into the expression gives:

6. Simplify the Fraction:

• The fraction $\frac{-2}{-2} = 1$ (because a negative divided by a negative is positive).

7. Combine with the Original Expression:

Final Answer: $-3 + \frac{-8 \div 4}{-2} = -2$

📑Example 3: $\frac{-4 \times -3}{-3 - (-9)}$ 8. Apply BIDMAS/BODMAS:

• Multiplication first on the numerator:

• Simplify the denominator by handling the negative signs:

• Now, the expression becomes:

9. Simplify the Fraction:

Final Answer: $\frac{-4 \times -3}{-3 - (-9)} = 2$

📝Question: Simplify the expression $−5×(2−4)+3$