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Properties of Real Numbers

Module by: David Richards. E-mail the author

Summary: This module explores the properties of real numbers.

This module teaches the Properties of Real Numbers. These properties are the basis for how numbers are used in mathematics.

The Order of Operations

The order in which mathematical operations are performed makes a difference in the outcome of the operations. To ensure that all mathematicians, and students, get the same answers to mathematical equations, a particular order of operations has been agreed upon. Operations inside parenthesis are always performed from the inside out. This means, operations inside parenthesis are performed before any other operations. Exponentiation (raising numbers to powers) is performed next. Multipication and division are done next in order. Addition and subtraction are always done last.

A mnemonic is an abbreviation of a series of words or phrases that is itself a word or phrase that can be more easily remembered than the series of words or phrases. The mnemonic for the order of operations is a nonsense word, PEMDAS, representing the first letter of the operations in the correct order. A phrase that helps keep the operations in order, and is a better mnemonic, is Please Excuse My Dear Aunt Sally. Apparently Dear Aunt Sally needs excusing for some reason, possibly having to do with her mental state, but the first letters of the words in the phrase are the first letters of the operations in the correct order.

Now that we have a standard for the order of operations that we can all follow, let's look at some equations and see what properties these equations lead us to.

The Commutative Property

1. What is the sum of 3 and 4? What is the sum of 4 and 3? Is there a difference? Can you, using letters for numbers, generalize a principle using letters for numbers that would describe this property of real numbers? This property is called the commutative property of addition. Does this property hold if you take a difference of two numbers?

2. What is the product of 3 and 4? What is the product of 4 and 3? Is there a difference? Can you, using letters for numbers, generalize a principle using letters for numbers that would describe this property of real numbers? This property is called the commutative property of multiplication. Does this property hold for the division of two numbers?

The Associative Property

3. What is the sum of 2, 3, and 4? We can only add two numbers at a time so we must pick two numbers of the three to add first before adding the third. This requires us to group two of the numbers together. Does it matter which two numbers we group together first? Can you, using letters for numbers, generalize a principle using letters for numbers that would describe this property of real numbers? This property is called the associative property of addition. Does this property work if one or more of the operations is a difference?

4. What is the product of 2, 3, and 4? We can only multiply two numbers at a time so we must pick two numbers of the three to multiply first before multiplying the third. This requires us to group two of the numbers together. Does it matter which two numbers we group together first? Can you, using letters for numbers, generalize a principle using letters for numbers that would describe this property of real numbers? This property is called the associative property of multiplication. Does this property work if one or more of the operations is a division?

The Distributive Property

5. Using the correct order of operations, calculate the result of the following equation: 2×(3+4)2×(3+4). Calculate the result of the following equation: 6+86+8. How are these two equations related? What happened to the multiplication and the addition between the first equation and the second? Can you, using letters for numbers, generalize a principle using letters for numbers that would describe this property of real numbers? This property of real numbers is called the distributive property and is the agreed upon method for using multiplication and addition together. Can you undo the number 27 so that you have the product of a number and a sum like the first equation? This is the process of undoing a distribution and is called factoring. We will make extensive use of this property to combine a product over a sum and to undo a product over a sum.

6. Can you use the distributive property to properly distribute (2+3)×(4+5)(2+3)×(4+5)? Hint: You have to distribute twice, once for each term in the first equation.

The Identities

7. Can you think of a number that when added to any other number doesn't change the value of that second number? This number is called the additive identity because adding this number doesn't change a number's identity. Can you write a definition of the additive identity by using letters and numbers?

8. Can you think of a number that when multiplied to any other number doesn't change the value of the second number? This number is called the multiplicative identity because multiplying by this number doesn't change a number's identity. Can you write a definition of the multiplicative identity using letters and numbers?.

We will use this property to change the appearance of rational numbers without changing the rational numbers' values. Any rational number whose numerator and denominator are the same are equivalent to the multiplicative identity. Multiplying a rational number by a rational representation of the multiplicative identity changes the appearance of a rational number but doesn't change its value.

The Inverses

9. Can you think of an integer that when added to 3 will give you the additive identity? This integer is 3's additive inverse. Can you think of an integer that when added to -4-4 will give you the additive identity? This number is -4-4's additive inverse. Can you, using letters for numbers, generalize a principle that will allow you to find any real number's additive inverse?

Adding a number and its additive inverse to a number or equation adds the additive identity to the number or equation. This is called adding a well chosen zero and we will find many situations when this will become advantageous.

10. Can you think of a rational number that when multiplied to 3434 will give you the multiplicative identity? This rational number is 3434's multiplicative inverse. Can you think of a rational number that when multiplied to 5454 gives you the multiplicative identity? This rational number is 5454's multiplicative inverse. Can you,using letters for numbers, generalize a principle that would allow you to find the multiplicative inverse of any real number? The multiplicative inverse is also called the reciprocal.

Laws of Exponents

Exponentiation, or raising a number to a power, is multiple multiplications. In the number 2323, 2 is the base and 3 is the exponent. The exponent tells you how many times to multiply the base to itself. A number raised to the 2nd power is said to be squared. A number raised to the 3rd power is said to be cubed. Any number without an exponent is understood to have an exponent of 1. Any number raised to the 0 power equals the multiplicative identity.

Let's look at some consequences of this description of exponentiation.

11. Using the multiple multiplications of exponentiation, can you find a way to rewrite the product of 2222 and 2323 as a single exponent? Can you, using letters for numbers, generalize a principle that describes what happens to the exponents when you multiply two exponentiated numbers? Does it work for the product of 2323 and 3232? Can you make your principle more specific?

12. Using the multiple multiplications of exponentiation, can you find a way to rewrite (22)3(22)3 using a single exponent? Can you, using letters for numbers, generalize a principle that describes what happens to the exponents when you exponentiate an exponent? Why do you get the same answer for 2323 squared? Hint: Look at your previous properties.

13. Using the multiple multiplications of exponentiation, can you find a way to rewrite (2×3)3(2×3)3 as a product of two numbers without a parenthesis? Can you, using letters for numbers, generalize a principle that would allow you to describe what happens when you exponentiate a product of two numbers? What happens with (2+3)2(2+3)2. Do you get the right answer using the principle for exponentiating a product? (2+3)2(2+3)2 is actually the product of two sums. We've already seen how you would solve this equation. What was that method and how would you solve this equation?

14. Can you modify the principle in number #11 to simplify 23222322. Does your principle work for 23322332? What happens when you apply your principle to 22232223? What do you notice about negative exponents?

15. Can you modify the principle established in #12 and #13 to simplify 23332333? How could you simplify 23362336?

Laws of Roots

273=3273=3 if and only if 33=2733=27. In 234234 4 is the index, 2 is the base and 3 is the exponent. The 2nd root is called the square root. Any radical without an index is understood to be the square root. The 3rd root is called the cube root.

To reduce a radical, factor the base into its prime number factors, then, using the index, remove groups of factors by the index until you can't remove any more. Each time you remove a group, you count the base once on the outside of the radical. When you can remove no more groups, the remaining factors are multiplied back together again and the bases on the outside are multiplied together. Any radical with no base left inside it is a whole integer number. Any radical with a base left inside it is an irrational number. Most radicals are irrational numbers.

Using this description of radicals as a basis, let's look at some consequences.

16. Simplify 233233. Simplify 263263. What would you do with 283283? Can you, using letters for numbers, generalize a principle for the case where the index and the exponent on the base are the same?

17. Simplify 233×333233×333. Using the definition of a root, rewrite these two numbers a product under a single radical. Can you, using letters for numbers, generalize a principle that would describe this radical. Do the bases have to be the same? Do the indexes have to be the same? Can you do something similar with the number 233333233333? Using this principle and the previous principle, simplify 234364234364.

A radical can be written as a rational exponent. The exponent of the number under the radical is the numerator, and the index of the radical is the denominator. Express 265265 as a rational exponent. Using rational exponents for 17, do the indexes still have to be the same? Using rational exponents, can you simplify 23252325?

Here are the properties you should have discovered in this module. Your letters may be different, but the equations should be similar.

Commutative Property

a + b = b + a a + b = b + a

a × b = b × a a × b = b × a

Associative Property

a + ( b + c ) = ( a + b ) + c a + ( b + c ) = ( a + b ) + c

a × ( b × c ) = ( a × b ) × c a × ( b × c ) = ( a × b ) × c

Distributive Property

a × ( b + c ) = a × b + a × c a × ( b + c ) = a × b + a × c

Identities

a + 0 = a a + 0 = a

a × 1 = a a × 1 = a

Inverses

a + ( - a ) = 0 a + ( - a ) = 0

a × 1 a = 1 a × 1 a = 1

a b × b a = 1 a b × b a = 1

Laws of Exponents

a m × a n = a m + n a m × a n = a m + n

( a m ) n = a m × n ( a m ) n = a m × n

a m a n = a m - n a m a n = a m - n

a - m = 1 a m a - m = 1 a m

a m b n p = a m × p b n × p a m b n p = a m × p b n × p

Laws of Roots

a n n = a a n n = a

a m + n n = a × a m n a m + n n = a × a m n

a n × b n = a × b n a n × b n = a × b n

a n b n = a b n a n b n = a b n

a m n = a m / n a m n = a m / n

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