Switching a sign
One sign switch changes the value by an even number.
Plus or minus between four numbers
Take four consecutive numbers: 5, 6, 7, 8. Put a + or a − between them in every possible way. That gives 8 expressions, such as 5 + 6 + 7 + 8 and 5 − 6 − 7 + 8.
What will the 8 results look like?
What this lesson covers
The idea
Changing the sign before a term b changes the value by 2b, an even number, so a + b and a – b, and all the expressions a ± b ± c ± d, have the same parity.
Plus or minus between four numbers
Take four consecutive numbers: 5, 6, 7, 8. Put a + or a − between them in every possible way. That gives 8 expressions, such as 5 + 6 + 7 + 8 and 5 − 6 − 7 + 8.
What will the 8 results look like?
- All even
- All odd
- A mix of even and odd
Switch the signs
Tap a + or − to switch it. Watch how much the value changes each time, and the colour of the result.
2b is always even
Changing the sign before a term b changes the value by 2b, an even number, so a + b and a – b, and all the expressions a ± b ± c ± d, have the same parity.
(a + b − c − d) − (a − b − c − d) = a + b − c − d − a + b + c + d = 2b
The two values differ by an even number, so by the last lesson they have the same parity. Try the other numbers (2, 9, 4, 6): all 8 results are odd.
Notes
Changing the sign before a term b changes the value by 2b, an even number, so a + b and a – b, and all the expressions a ± b ± c ± d, have the same parity.
Check yourself
5 + 6 + 7 + 8 = 26. Now switch the sign before 7: 5 + 6 − 7 + 8. By how much does the value change?
Answer: 14
5 + 6 + 7 + 8 = 26 and 5 + 6 − 7 + 8 = 12. The change is 26 − 12 = 14 = 2 × 7.
For some four numbers, a − b + c − d is an odd number. What about a + b + c + d for the same numbers?
Take the numbers 2, 9, 4 and 6. Put + or − between them in all 8 ways. How many of the 8 results are even?
Answer: 0
2 + 9 + 4 + 6 = 21 is odd. Every other expression differs from it by an even number, so all 8 results are odd. The number of even results is 0.
Why can a sign switch never change the parity of the value?
- It is odd too — correct. Yes! Going from a − b + c − d to a + b + c + d switches two signs. The value changes by 2b + 2d, which is even, so the parity stays the same.
- It is even. Switching two signs changes the value by 2b + 2d, which is even. An even change cannot turn odd into even.
- It can be either. It cannot. The change is 2b + 2d, always even, so the parity never changes.
- The number b is always even. b can be odd or even. What is always even is the change 2b.
- +b and −b are the same number. +b and −b are different numbers. They differ by 2b.
- The value changes by 2b, which is even — correct. Yes! An even change keeps the parity: odd stays odd and even stays even.