Real Numbers and Their Properties Section 1.1 – The Real Numbers Week 1 Real Numbers and Their Properties Section 1.1 – The Real Numbers

Week 1 Objectives This week students will: Utilize proper math terminology in written explanations. Compute basic operations with signed numbers. Exemplify properties of real numbers. Execute simplification techniques on expressions and equations.

Counting and Whole Numbers Numbers are everywhere, around us. When we know that 6 people are coming to our house-party and each of them is going to consume three cans of sodas and thus we have to buy 18 cans of sodas, we are not only dealing with numbers but also with numerical operations like addition and/or multiplication. The numbers like {1, 2, 3, 4, 5, ….} are what are known as counting numbers or natural numbers. When you add the zero to the set of counting numbers, you have whole numbers (what are also known as natural numbers) = {0, 1, 2, 3, 4, 5, ….}. {0, 1, 2, 3, 4, 5,…} are also known as positive numbers. Think of positive numbers as if you have gained $2 in a gambling game, you have a positive 2 in your pocket. However, if you have lost $2 in the gambling game, you have a negative 2 in your pocket and such numbers {…., -4, -3, -2, -1} are known as negative numbers. All the positive and negative numbers together are known as integers = {….., -4, -3, -2, -1, 0, 1, 2, 3, 4, …..}

Rational Numbers Rational numbers are of the form a/b (also written as ) where a and b are integers. However b, the number in the bottom, can never be equal to zero. You might recognize rational numbers as fractions and that’s absolutely fine. Examples of rational numbers are Note: you can have zero in the top but never in the bottom; the numbers can be either positive or negative numbers; the three dots at the end indicates that many such examples are out there.

Irrational Numbers Numbers that are not rational are known as irrational numbers. Duh! This is a no-brainer definition. So which numbers are not rational numbers? Any numbers that cannot be expressed as a fraction. Examples are Each of the above numbers are irrational numbers because you can never express them as fractions. The dash on top of 3 in the first example indicates that 3 is repeating continuously as 0.33333333333333…….; the second number is what is known as pi in mathematics and tells us about the ratio of circumference of a circle to its diameter; the last number is what is known as e, one of the most famous mathematical number as it appears everywhere in natural world, from share-market, finance to growth of human population.

Real Numbers All numbers – whole, positive integers, negative integers, rational numbers, irrational numbers – together make up the set of real numbers. Examples: 2 is a positive integer as well as a real number -10 is a negative integer as well as real number 7/10 is a rational number as well as real number is a irrational number as well as real number. If you are bit confused, think of how all the countries make up this world; but each country has its own identity. Thus, integers, whole, natural, rational, irrational numbers are like countries each with their own identities and properties; but all these different numbers make up the real numbers (the world.

Number Line Number line helps us to visualize the whole numbers. Draw a horizontal line. Put a zero in the middle of the line. To the right of the zero, mark off equally spaced points along the line and label the points with positive numbers. To the left of the zero, mark off equally spaced points along the line and label the points with negative numbers. The arrows at the end of the lines indicate the process continues to infinity in both left and right direction. You can put the rational and irrational numbers in between the positive and negative numbers. Thus, ½ corresponds to a point half-way between 0 and 1. You can think of the points as measuring distance from zero. If the distances are measure to the right of zero, they are all positive; if the distances are measured to the left of zero, they are all negative. 1 2 3 -1 -2 -3

Absolute value The absolute value of a number is the number’s distance from zero. It is denoted as |a| where a can be any number. Example: |5| = |-5| = 5 because the distance between 0 and 5 is 5; the distance between 0 and -5 is also 5 as shown in the picture below: More advanced definition: The above definition is basically telling us that the number can be either positive or negative (like -5 or 5 in our example). However, absolute value of the number, whether positive or negative, is always going to be same. -5 5

Intervals An interval is a range of numbers, that is, all the numbers between two given numbers. The two given numbers are known as endpoints. Example: [2, 4] is an interval and it means all the numbers between 2 and 4 (keep in mind that 3 is not the only number between 2 and 4; you have all the fractions, decimals, integers, everything, thus 2, 2.1, 2.000005, 2.00234,….all these sorts of numbers). An interval is always enclosed in either open brackets (), closed brackets[] or mixed-brackets (] or [). The open brackets simply mean that the endpoints are not included; closed brackets mean that the endpoints are included. Example: (2,4) means you only consider all the numbers in between but not 2 and 4; [2, 4] means you consider all the numbers in between including 2 and 4; [2, 4) means you consider all the number in between including 2 but not 4 because there is an open bracket after 4; similarly, (2, 4] means you consider all the numbers in between including 4 but not 2 because now the open bracket is with 2.

Graphical Representation of Intervals Look at this website to learn about graphical representation of intervals: http://www.regentsprep.org/Regents/math/ALGEBRA/AP1/IntervalNot.htm The picture is from the above website: