0.1 Number Theory
- 1. 0.1 Number Theory Unit 0 Review of Basic Concepts
- 2. Concepts and Objectives ⚫ Number Theory ⚫ Identify subsets of numbers ⚫ Simplify expressions using order of operations ⚫ Identify real number axioms ⚫ Rational number operations
- 3. Number Systems ⚫ What we currently know as the set of real numbers was only formulated around 1879. We usually present this as sets of numbers.
- 4. Number Systems ⚫ The set of natural numbers () and the set of integers () have been around since ancient times, probably prompted by the need to maintain trade accounts. Ancient civilizations, such as the Babylonians, also used ratios to compare quantities. ⚫ One of the greatest mathematical advances was the introduction of the number 0.
- 5. Set Notation ⚫ The basic definition of a set is a collection of objects, and the objects that belong to that set are called the elements or members of that set. ⚫ We use braces, { }, to list the elements in a set, for example, {1, 2, 3, 4}. To show that 4 is a member of the set, we use the symbol, , and write 4 {1, 2, 3, 4}. ⚫ We use ellipsis (…) to show that a set continues according to an established pattern. Ex.: {1, 2, 3, …, 10} ⚫ {x | x and 2 < x < 7} would be read “the set of all elements x such that x is a natural number between 2 and 7.”
- 6. Set Notation (cont.) ⚫ If a set has no elements, it is referred to the null set or empty set, and it is denoted as either { } or . For example, the set of all odd numbers ending in 2 would be { }. ⚫ If every member of set A is a member of set B, then set A is a subset of set B, which is written A B. ⚫ Given two sets A and B, the set of all elements belonging to both set A and set B is called the intersection of the two sets, written A B. The set of all elements belong to either set A or set B is called the union of the two sets, written A B.
- 7. Order of Operations ⚫ Parentheses (or other grouping symbols, such as square brackets or fraction bars) – start with the innermost set, following the sequence below, and work outward. ⚫ Exponents ⚫ Multiplication ⚫ Division ⚫ Addition ⚫ Subtraction working from left to right working from left to right
- 8. Order of Operations ⚫ Use order of operations to explain why ( )− − 2 2 3 3
- 9. Order of Operations ⚫ Use order of operations to explain why ⚫ We can think of –3 as being –1 3. Therefore we have It should be easier now to see that on the left side we multiply first and then apply the exponent, and on the right side, we apply the exponent and then multiply. ( )− − 2 2 3 3 ( )− − 2 2 1 3 1 3
- 10. Order of Operations Work the following examples without using your calculator. 1. 2. 3. − + 2 5 12 3 ( ) ( )( )− − + − 3 4 9 8 7 2 ( )( ) ( ) 8 4 6 12 4 3 − + − − − −
- 11. Order of Operations Work the following examples without using your calculator. 1. 2. 3. − + 2 5 12 3 ( ) ( )( )− − + − 3 4 9 8 7 2 ( )( ) ( ) − + − − − − 8 4 6 12 4 3 1. –6 2. –60 − 6 3. 7
- 12. Rational Numbers ⚫ A number is a rational number () if and only if it can be expressed as the ratio (or Quotient) of two integers. ⚫ Rational numbers include decimals as well as fractions. The definition does not require that a rational number must be written as a quotient of two integers, only that it can be.
- 13. Examples ⚫ Example: Prove that the following numbers are rational numbers by expressing them as ratios of integers. 1. 2-4 4. 2. 64-½ 5. 3. 6. –5.4322986 4 20.3 0.9 6.3
- 14. Examples ⚫ Example: Prove that the following numbers are rational numbers by expressing them as ratios of integers. 1. 2-4 4. 2. 64-½ 5. 3. 6. –5.4322986 4 20.3 0.9 6.3 1 16 1 8 4 1 7 = 1 61 20 3 3 − 54322986 10000000
- 15. Real Numbers ⚫ The number line is a geometric model of the system of real numbers. Rational numbers are thus fairly easy to represent: ⚫ What about irrational numbers? Consider the following: • (1,1) 2 This comes from the Pyth. Theorem. 12 + 12 = c2 , so 2c =
- 16. Real Numbers ⚫ In this way, if an irrational number can be identified with a length, we can find a point on the number line corresponding to it. ⚫ What this emphasizes is that the number line is continuous—there are no gaps.
- 17. Properties of Real Numbers ⚫ Closure Property ⚫ a + b ⚫ ab ⚫ Commutative Property ⚫ a + b = b + a ⚫ ab = ba ⚫ Associative Property ⚫ (a + b) + c = a + (b + c) ⚫ (ab)c = a(bc) ⚫ Identity Property ⚫ a + 0 = a ⚫ a 1 = a ⚫ Inverse Property ⚫ a + (–a) = 0 ⚫ ⚫ Distributive Property ⚫ a(b ± c) = ab ± ac For all real numbers a, b, and c: 1 =1a a
- 18. Properties of Real Numbers ⚫ The properties are also called axioms. ⚫ 0 is called the additive identity and 1 is called the multiplicative identity. ⚫ Notice the relationships between the identities and the inverses (called the additive inverse and the multiplicative inverse). ⚫ Saying that a set is “closed” under an operation (such as multiplication) means that performing that operation on numbers in the set will always produce an answer that is also in the set – there are no answers outside the set.
- 19. Properties of Real Numbers ⚫ Examples ⚫ The set of natural numbers () is not closed under the operation of subtraction. Why? ⚫ –20 5 = –4. Does this show that the set of integers is closed under division?
- 20. Properties of Real Numbers ⚫ Examples ⚫ The set of natural numbers () is not closed under the operation of subtraction. Why? ⚫ You can end up with a result that is not a natural number. For example, 5 – 7 = –2, which is not in . ⚫ –20 5 = –4. Does this show that the set of integers is closed under division? ⚫ No. Any division that has a remainder is not in .
- 21. Finite and Repeating Decimals ⚫ If the decimal representation of a rational number has a finite number of digits after the decimal, then it is said to terminate. ⚫ If the decimal representation of a rational number does not terminate, then the decimal is periodic (or repeating). The repeating string of numbers is called the period of the decimal. ⚫ It turns out that for a rational number where b > 0, the period is at most b – 1. a b
- 22. Finite and Repeating Decimals ⚫ Example: Use long division to find the decimal representation of and find its period. What is the period of this decimal? 462 13 462 35.538461 13 = 6
- 23. Finite and Repeating Decimals ⚫ The repeating portion of a decimal does not necessarily start right after the decimal point. A decimal which starts repeating after the decimal point is called a simple-periodic decimal; one which starts later is called a delayed-periodic decimal. 1 2 30. ... td d d d ( 0)td 0.3, 0.142857, 0.09, 0.076923 1 2 30. ... pd d d d 0.16, 0.083, 0.0714285, 0.06 Type of Decimal Examples General Form terminating 0.5, 0.25, 0.2, 0.125, 0.0625 simple-periodic delayed-periodic + + + +1 2 3 1 2 30. ... ...t t t t t pd d d d d d d d
- 24. Decimal Representation ⚫ If we know the fraction, it’s fairly straightforward (although sometimes tedious) to find its decimal representation. What about going the other direction? How do we find the fraction from the decimal, especially if it repeats? ⚫ We can state that the terminating decimal 0.d1d2d3…dt can be written as . Example: 1 2 3... 10 t t d d d d = = 845 169 0.845 1000 200
- 25. Decimal Representation ⚫ For simple-periodic decimals, the “trick” is to turn them into fractions with the same number of 9s in the denominator as there are repeating digits and simplify: To put it more generally, to convert , we can write it as . 3 1 0.3 9 3 = = = = 6 2 0.06 99 33 153846 2 0.153846 999999 13 = = 1 2 30. ... pd d d d 1 2 3... 999...9 pd d d d
- 26. Decimal Representation ⚫ For delayed-periodic decimals, the process is a little more complicated. Consider the following: What is the decimal representation of ? is a product of what two fractions? Notice that the decimal representation has characteristics of each factor (2 terminating digits and 1 repeating digit). 1 12 0.083 1 12 1 1 4 3
- 27. Decimal Representation ⚫ It turns out you can break a delayed-periodic decimal into a product of terminating and simple-periodic decimals, so the general form is also a product of the general forms: The decimal can be written as the fraction , where N is the integer d1d2…dtdt+1dt+2…dt+p – d1d2…dt . (whew!) + + +21 2 10. ... ... pt t t td d d d d d ( )−10 10 1pt N
- 28. Decimal Representation ⚫ Example: Convert the decimal to a fraction. Comparing this to the formula, we can see that t = 1 (the terminating part) and p = 3 (the repeating part). Thus: Another way to look at the denominator is that it is 3 nines (repeating) followed by 1 zero (terminating). 4.9331 ( )1 3 5 49 4 49331 49282 24641 331 9990 49910 10 .9 1 − = = = −
- 29. Decimal Representation ⚫ Example: Convert the decimal to a fraction. 0.467988654
- 30. Decimal Representation ⚫ Example: Convert the decimal to a fraction. It’s possible this might reduce, but we can see that there are no obvious common factors (2, 3, 4, 5, 6, 8, 9, or 10), so it’s okay to leave it like this. 0.467988654 − = = 988654 99 0 467 467 00 0 467 467988187 . 9 9 88654 999909 9 99009
- 31. Classwork ⚫ Classwork: 0.1 WS ⚫ If you aren’t able to finish before you leave, it will be due at the beginning of our next class.