The sum of two opposites
The sum of two opposites (additive inverses) is always zero: for any number a, a + (-a) = 0. So (+92) + (-92) = 0, and (+3) + (-3) = 0. This Grade 7 math skill from Saxon Math, Course 2 is the Additive Inverse Property and is fundamental to all algebraic equation solving — every time you add a number to both sides of an equation to eliminate a term, you are using this property. It also explains why debts cancel out gains and temperatures above and below zero cancel.
Key Concepts
Property The sum of two opposites is always zero.
Examples Taking 8 steps to the right and 8 steps to the left gets you back to the start: $(+8) + ( 8) = 0$. If you borrow 20 dollars ($ 20$) and then earn 20 dollars ($+20$) to repay the debt, your balance is zero: $( 20) + (+20) = 0$. On a number line, an arrow from $0$ to $+15$ and an arrow from $+15$ back to $0$ show that $(+15) + ( 15) = 0$.
Explanation Adding two opposite numbers is like taking five steps forward and then taking five steps backward—you end up exactly where you started: at zero! They perfectly cancel each other out, like a superhero and their arch nemesis in a final battle where both disappear in a flash. Poof! Nothing is left but a perfect, balanced zero.
Common Questions
What is the sum of two opposites?
The sum of any two opposite numbers (additive inverses) is always zero. For example, (+7) + (-7) = 0, and (-15) + (+15) = 0.
What are additive inverses?
Additive inverses are pairs of numbers that sum to zero. The additive inverse of 5 is -5, and the additive inverse of -8 is +8.
How is the sum of two opposites used in solving equations?
When solving x + 7 = 12, you add -7 to both sides: the +7 and -7 cancel (sum to 0), leaving x = 5. This is the Additive Inverse Property in action.
What is a real-world example of the sum of two opposites?
A gain of $450 and a loss of $450 results in $0 net change. Walking 10 steps forward and then 10 steps backward brings you back to the start.
When do students learn the Additive Inverse Property?
This property is introduced in Grade 6-7. Saxon Math, Course 2 covers it in Chapter 5 as part of integer operations and equation solving.
Is zero its own opposite?
Yes. The opposite of zero is zero: 0 + 0 = 0. Zero is the only number that is its own additive inverse.
How does the sum of two opposites differ from the Multiplicative Inverse?
The additive inverse sums to 0 (a + (-a) = 0). The multiplicative inverse (reciprocal) multiplies to 1 (a times 1/a = 1). Both are essential properties in algebra.