Absolute value: A number's absolute value is its numerical value without taking the sign into consideration, and it is represented by the sign \(|a|\); where a is any number.
\(e.g.\)\(|a|=a\) and \(|-a|=a\)
For any two integers \(a\) and \(b\)
1. \(a+b=\) sum (addition) of both integers
\(e.g.\) \(3+5=8\)
2. \((i)\) \(a-b=+\)(difference (subtraction) of both integers) if \(|a|>|b|\)
\(e.g.\) \(7-5=+2\) because \(|7|>|5|\)
\((ii)\) \(a-b = -\)(difference (subtraction) of both integers) if \(|b|>|a|\)
\(e.g.\) \(5-7=-2\) because \(|7|>|5|\)
3. \((i)\) \(a+(-b)=a-b\)
\(=+\)(difference (subtraction) of both integers) if \(|a|>|b|\)
\(e.g.\) \(7+(-5)=7-5\)
\(=+2\) because \(|7|>|5|\)
\((ii)\) \(a+(-b)=a-b\)
\(= -\)(difference (subtraction) of both integers) if \(|b|>|a|\)
\(e.g.\) \(5+(-7)=5-7\)
\(=-2\) because \(|7|>|5|\)
4. \((i)\) \(-a+b=+\)(difference (subtraction) of both integers) if \(|b|>|a|\)
\(e.g.\) \(-5+7=-2\) because \(|7|>|5|\)
\((ii)\) \(-a+b=-\)(difference (subtraction) of both integers) if \(|a|>|b| \)
\(e.g.\) \(-7+5=-2\) because \(|7|>|5|\)
5. \(-a+(-b)=-a-b\)
\(=-\)(sum (addition) of both integers)
\(e.g.\) \(-7+(-5)=-7-5\)
\(=-12\)
6. \((i)\) \(-a-(-b)=-a+b\)
\(=+\)(difference (subtraction) of both integers) if \(|b|>|a|\)
\(e.g.\) \(-5—7=-5+7\)
\(=-2\) because \(|7|>|5|\)
\((ii)\) \(-a-(-b)=-a+b\)
\(=-\)(difference (subtraction) of both integers) if \(|a|>|b|\)
\(e.g.\) \(-7+5=-7(-5)\)
\(=-2\) because \(|7|>|5|\)
7. \(a-(-b)=a+b \)
\(=\)sum (addition) of both integers
\(e.g.\) \(3-(-5)=3+5=8\)
Note: 1) If we have opposite signs in and before the parenthesis, it will turn negative.
\(e.g.\) \(-(+a)=-a\), also \(+(-a)=-a\)
2) If we have same signs in and before parenthesis, it will turn into negative.
\(e.g.\) \(-(-a)=+a\), also \(+(+a)=+a\)