Identity and Equality Properties

© William James Calhoun To recognize and use the properties of identity and equality, and to determine the multiplicative inverse of a number. This section is all about mathematical properties. You do need to memorize these properties by name. You have been using these properties for years without knowing their names. Focus on understanding the meaning of the properties.

© William James Calhoun Question: What is 0 plus any number? Answer: 0 plus any number always gives you that number. additive identity - a number that can add to other numbers without changing their values Zero is the additive identity since it can add without changing values. ADDITIVE IDENTITY PROPERTY For any number a, a + 0 = 0 + a = a. Think about this… is the same as is the same as 5.

© William James Calhoun Question: What is 1 times any number? Answer: 1 times any number always gives you that number. multiplicative identity - a number that can multiply with other numbers without changing their values The number one is the multiplicative identity since it can multiply without changing values. MULTIPLICATIVE IDENTITY PROPERTY For any number a, a * 1 = 1 * a = a. Think about this… 1 * 8 is the same as 8 * 1 is the same as 8.

© William James Calhoun Question: What is 0 times any number? Answer: 0 times any number always gives you zero. This property of multiplying by zero is known as the: multiplicative property of zero - a property of zero that when it multiplies by any number, it yields zero MULTIPLICATIVE PROPERTY OF ZERO For any number a, a * 0 = 0 * a = 0. Think about this… 0 * 14 is the same as 14 * 0 is the same as 0.

© William James Calhoun Two more terms: multiplicative inverses - two numbers whose product is 1 reciprocals - two numbers whose product is 1 You need to remember these two terms – they will be used in many other sections! Since any number times 0 always gives you 0, 0 does NOT have a multiplicative inverse. You can not multiply it by anything to get to 1 - you would have to add.

© William James Calhoun MULTIPLICATIVE INVERSE PROPERTY For every nonzero number, where a and b  0, there is exactly one number such that. What this property says is that you can 1) Take any non-zero, rational number, 2) Write that number as a fraction, 3) Find its reciprocal (flip it over), 4) The two numbers multiplied will always yield one.

© William James Calhoun EXAMPLE 1α: Name the multiplicative inverse of each number or variable. Assume no variable equals zero. a. 5 b. x c. 2 / 3 Write 5 as a fraction 5/15/1 Flip it over to get the multiplicative inverse. Write x as a fraction x/1x/1 Flip it over to get the multiplicative inverse. Solution is 1 / 5.Solution is 1 / x. This is already written as a fraction so just flip it over to get the multiplicative inverse. Solution is 3 / 2. EXAMPLE 1β: Name the multiplicative inverse of each number or variable. Assume no variable equals zero. a. 3 b. c. xy

© William James Calhoun For Example 1, the instructions have “Assume no variable equals zero”? Why? Look at question 1α, part b. b. x The solution was 1/x and if x = 0, then the solution would be 1/0. Try putting 1/0 into your calculator. You get an error because YOU CAN NEVER DIVIDE BY ZERO. Dividing is breaking something up into groups. Divide by 2 means divide into 2 groups. You can not take something and divide it into nothing!

© William James Calhoun Question: Three students have 12-inch rulers. One student cuts a 3-inch piece off. The second cuts a 5-inch piece. The third keeps their piece whole. How much does each student have? Answer: Each student still has 12-inches of ruler. No matter how divided, the pieces of the ruler still add to inch + 9 inch = 9 inch + 3 inch = 12 inch reflexive property of equality - any quantity is equal to itself For any number a, a = a. REFLEXIVE PROPERTY OF EQUALITY Any number is equal to itself.

© William James Calhoun symmetric property of equality - if one quantity equals a second quantity, then the second quantity also equals the first. For any numbers a and b, if a = b, then b = a. SYMMETRIC PROPERTY OF EQUALITY If Mike is as old as Mary, then Mary is as old as Mike. transitive property of equality - if one quantity equals a second quantity and the second quantity equals a third quantity, then the third quantity equals the first quantity. For any numbers a, b, and c, if a = b and b = c, then a = c. TRANSITIVE PROPERTY OF EQUALITY If Mike is as old as Mary, and Mary is as old as Chuck, then Mike and Chuck are also the same age.

© William James Calhoun substitution property of equality - a quantity may be substituted for its equal in any expression If a = b, then a may be replaced by b in any expression. SUBSTITUTION PROPERTY OF EQUALITY If Mike and Mitch are tailbacks in football, Mike can substitute for Mitch and Mitch can substitute for Mike. Is that enough properties for you?!?

© William James Calhoun Please keep this in mind: I know these are a lot of properties to remember. You have been using most of these properties for years in your math classes. We are just giving them official names now. Do NOT let the names intimidate you! You only need to apply these properties - no need to memorize! The next slide is a summary of the properties in layman’s terms (where possible.)

© William James Calhoun