- Take two negative integers:
( a < 0 ), ( b < 0 ).
- Write them as positives with a minus sign:
( a = -m ) where ( m > 0 ),
( b = -n ) where ( n > 0 ).
- Add them:
( a + b = (-m) + (-n) ).
- Combine the minus signs:
( (-m) + (-n) = -(m + n) ).
- Compare sizes:
( m + n > m ) and ( m + n > n ).
Putting a minus sign flips the inequality:
( -(m + n) < -m ) and ( -(m + n) < -n ).
- Back to ( a ) and ( b ):
( -(m + n) = a + b ),
( -m = a ), ( -n = b ).
So ( a + b < a ) and ( a + b < b ).
Number line idea:
- Adding a negative moves left.
- From a negative number, moving left gives a smaller number.
Example:
( -3 + (-5) = -8 );
( -8 < -5 ) and ( -8 < -3 ).
So the sum is smaller than both numbers.
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