NUMBERS

Mathematics is concerned with patterns and relationships. Mathematics is also a tool with which to solve problems as well as a language with which to communicate ideas about science, technology, music, and even art. And, of course, mathematics is concerned with numbers. The definitions that follow will help avoid errors and miscommunication.

Number: An abstract idea that represents quantity.

Numerals: Symbols used to represent numbers. For example, we use the numeral "6" to refer to six objects, say, whereas the Roman numeral for six is "VI."

Digits are the symbols we use in our counting system. There are 10 digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Counting or Natural Numbers: Children first learn to count using counting numbers. The counting numbers are

1, 2, 3, 4, 5. 6. 7. 8, 9, 10, 11, …

Whole Numbers: These are the counting numbers together with zero.

0, 1, 2, 3, …

Integers: The positive and negative whole numbers and zero.

… -3, -2, -1, 0, 1, 2, 3, …

Note that the set of whole numbers is included in the set of integers.

Rational Numbers: Numbers of the form where a and b are integers and b ¹ 0. (Remember that you can't divide by 0.)

Rational numbers are the fractions such as , …

The set of rational numbers includes all the integers, since, for example, .

Rational numbers can also be written as decimals. If a decimal is a rational number it will either terminate (end) or repeat.

Examples:

Real Numbers: The real numbers include the integers and the rational numbers as well as any number that can be represented on the real number line.

The decimal representations of and neither terminate nor do they have a repeating pattern. They are called irrational numbers.

For example, = 3.1415926... , = 1.4142135....

Complex Numbers: You won't see these but just to complete the discussion, the set of complex numbers includes all the real numbers as well as imaginary numbers like the square root of -1.

Summary: The following figure illustrates the relationships between counting numbers, whole numbers, integers, rational numbers, and real numbers.

In the examples that follow, two problem solving techniques are used: making a list and looking for a pattern.

Example: How many counting numbers are there between 10 and 17?

Answer: 6

Solution: Make a list. The numbers between 10 and 17 are 11, 12, 13, 14, 15, and 16.

Example: How many counting numbers have two digits?

Answer: 90

Solution: The counting numbers with two digits are those from 10 to 99. Now 99 - 10 = 89 but we need to add one more since we are also counting the endpoints of the set.

Example: How many numbers in the first 40 counting numbers contain the digit 8?

Answer: 4

Solution: Make a list: 8, 18, 28, and 38

Example: How many times does the digit "2" appear in the first 40 counting numbers?

Answer: 14

Solution: The numbers 2, 12, 20, 21, 23, 24, 25, 26, 27, 28, 29 and 32 each have one "2." The number 22 has two "2s."

Example: The fraction can be written 0.272727… = where the 27 repeats. Find the 15th digit to the right of the decimal point.

Answer: 7

Solution: Look for a pattern. Note that the first, third, and fifth digits are 2s and the second, fourth, and sixth digits are 7s. That is, starting at the decimal point and counting the digits to the right, the digits in the odd "slots" (the first, third and fifth digits) are 2s and the digits in the even "slots" are 7s. Since the 15th digit is in an odd "slot," that digit will be a 2. Note that the pattern consists of two numbers, (2s and 7s) and 15 divided by 2 has a remainder of 1.

Example: Find the last digit of .

A. 0

B. 2

C. 4

D. 6

E. 8

Answer: B

Solution: This number is too large to calculate so there must be a trick. Start by computing the last digits of the early powers of 2 and look for a pattern.

	Last digit
	2
	4
	8
	6
	2
	4
	8
	6
	2

Note that the pattern repeats after four numbers so that , , etc. all have the same last digit (which is 6), and , , , etc. all have the same last digit (which is 2), and so on. To generalize, if x represents a positive integer, all have the same last digit, all have the same last digit, etc. Since 65 = (4x16) + 1, the last digit will be 2. (Note that we can write as.)

As in the previous problem, note that the pattern consists of four numbers, (2s, 4s, 8s and 6s) and 65 divided by 4 has a remainder of 1.

Comment: Our solution does not employ rigorous mathematical reasoning and is not a formal proof. Recognizing patterns merely provides clues that can lead to formal proofs. A formal proof would require us to show that the pattern doesn't suddenly change when we get to , for example. However, many problems can be satisfactorily solved by examining patterns and trusting our intuition and logic.