Algebra I All-in-One For Dummies. Mary Jane Sterling

computations. Sometimes, changing the order in addition or multiplication situations can make the work much easier.

      

Q.

      A. You don’t really want to multiply fractions unless it’s necessary. Notice that the first and last factors are multiplicative inverses of one another:

      Just switch the order of the last two numbers:

      Q.

A. The second and last terms are reversed, and then the first two terms are grouped. .

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Relating Inverses and Identities

      In mathematics, inverses and identities are closely related. The definition of an inverse includes references to an identity. And when describing the identity of an operation, you call up the inverses.

      Investigating Inverses

      In mathematics, the inverse of a number is tied to a specific operation.

      The additive inverse of the number 5 is –5; the additive inverse of the number is . When you add a number and its additive inverse together, you always get 0, the additive identity. Every real number has an additive inverse, even the number 0. The number 0 is its own additive inverse. And all real numbers (except 0) and their inverses have opposite signs; the number 0 is neither positive nor negative, so there is no sign.

      The multiplicative inverse of the number 5 is ; the multiplicative inverse of the number is –3. When you multiply a number and its multiplicative inverse together, you always get 1, the multiplicative identity. Every real number except the number 0 has a multiplicative inverse. A number and its multiplicative inverse are always the same sign.

      

Q. Find the additive and multiplicative inverses of the number –14.

      A. The additive inverse is 14, because . The multiplicative inverse of –14 is , because .

      Q. Find the additive and multiplicative inverses of the number .

      A. The additive inverse is , because . The multiplicative inverse of is , because .

       Find the additive and multiplicative inverses of the given number.

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      Identifying Identities

      The term identity in mathematics is most frequently used in terms of a specific operation. When using addition, the additive identity is the number 0. You can think of it as allowing another number to keep its identity when 0 is added. If you add , the result is 7. The number 7 doesn’t change. When using multiplication, the multiplicative identity is the number 1. When you multiply , the result is 7. Again, the number 7 doesn’t change.

      When adding a number and its additive inverse together, you get the additive identity. So . And when multiplying a number and its multiplicative inverse together, you get the multiplicative identity. Multiplying, .

      

Q. Use an additive identity to change the expression to an expression with only the variable term.

      A. The additive inverse of 5 is –5. If you add –5 to the expression, you have . Use the associative property to group the 5 and –5 together: . The sum of a number and its additive inverse is 0, so the expression becomes . Because 0 is the additive identity, .

      Q. Use a multiplicative identity to change the expression into an expression without a fraction.

      A. Because the term