U Can: Algebra I For Dummies. Sterling Mary Jane

.

       Examples

      Q. –14 + (14 + 23) =

      A. Re-associate the terms and then add the first two together: –14 + (14 + 23) = (–14 + 14) + 23 = 0 + 23 = 23.

      Q. 4(5 · 6) =

      A. You can either multiply the way the problem is written, 4(5 · 6) = 4(30) = 120, or you can re-associate and multiply the first two factors first: (4 · 5) 6 = (20)6 = 120.

       Practice Questions

      1. 16 + (–16 + 47) =

      2. (5 – 13) + 13 =

      3.

      4.

       Practice Answers

      1. 47.

      2. 5.

      3. 70.

      4. 110.

Computing by Commuting

      Before discussing the commutative property, take a look at the word commute. You probably commute to work or school and know that whether you’re traveling from home to work or from work to home, the distance is the same: The distance doesn’t change because you change directions (although getting home during rush hour may make that distance seem longer).

      The same principle is true of some algebraic operations: It doesn’t matter whether you add 1 + 2 or 2 + 1, the answer is still 3. Likewise, multiplying 2 · 3 or 3 · 2 yields 6.

       Tip: The commutative property means that you can change the order of the numbers in an operation without affecting the result. Addition and multiplication are commutative. Subtraction and division are not. So,

      In general, subtraction and division are not commutative. The special cases occur when you choose the numbers carefully. For example, if a and b are the same number, then the subtraction appears to be commutative because switching the order doesn’t change the answer. In the case of division, if a and b are opposites, then you get –1 no matter which order you divide them in. By the way, this is why, in mathematics, big deals are made about proofs. A few special cases of something may work, but a real rule or theorem has to work all the time.

      Take a look at how the commutative property works:

       ✓ 4 + 5 = 9 and 5 + 4 = 9, so 4 + 5 = 5 + 4

       ✓ 3 · (–7) = –21 and (–7) · 3 = –21, so 3 · (–7) = (–7) · 3

       ✓ 6.3 + 5.7 = 12 and 5.7 + 6.3 = 12, so 6.3 + 5.7 = 5.7 + 6.3

      ✓

      You can use this rule to your advantage when doing math computations. In the following two examples, the associative rule finishes off the problems after changing the order.

       Examples

      Q.

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

. The second and last factors were reversed.

      Q. –3 + 16 + 303 =

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

      – 3 + 16 + 303 = –3 + 303 + 16 = (–3 + 303) + 16 = 300 + 16 = 316.

       Practice Questions

      1. 8 + 5 + (–8) =

      2. 5 · 47 · 2 =

      3.

      4. –23 + 47 + 23 – 47 + 8 =

       Practice Answers

      1. 5.

      2. 470.

      3. 78.

      4. 8.

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.

       Examples

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

      A. The additive inverse is 14, because –14 + 14 = 0. 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

.

       Practice Questions

      Find the additive and multiplicative inverses of the number given.

      1. 11

      2.

      3.