Rational and Irrational Real Numbers Rational and Irrational

Let’s look at the relationships between number sets Let’s look at the relationships between number sets. Notice rational and irrational numbers make up the larger number set known as Real Numbers

A number represents the value or quantity of something… Like how much money you have.. Or how many marbles you have… Or how tall you are. As you may remember from earlier grades there are different types of numbers.

A number line - is an infinitely long line whose points match up with the real number system.

Here are the rational numbers represented on a number line.

Integers The coldest temperature on record in the U.S. is -80° F, recorded in 1971 in Alaska

Integers are used to represent real-world quantities such as temperatures, miles per hour, making withdrawals from your bank account, and other quantities. When you know how to perform operations with integers, you can solve equations and problems involving integers.

By using integers, you can express elevations above, below, and at sea level. Sea level has an elevation of 0 feet. Badwater Basin in Utah is -282 below sea level, and Clingman’s Dome in the Great Smokey Mountains is +6,643 above sea level.

If you remember, the whole numbers are the counting numbers and zero: 0, 1, 2, 3,… Integers - the set of all whole numbers and their opposites. This means all the positive integers and all the negative integers together.

Opposites – two numbers that are equal distance from zero on a number line; also called additive inverse. The additive inverse property states that if you add two opposites together their sum is 0 -3 + 3 = 0

Integers increase in value as you move to the right along a number line. They decrease in value as you move to the left. Remember to order numbers we use the symbol < means “less than,” and the symbol > means “is greater than.”

A number’s absolute value - is it’s distance from 0 on a number line A number’s absolute value - is it’s distance from 0 on a number line. Since distance can never be negative, absolute values are always positive. The symbol || represents the absolute value of a number. This symbol is read as “the absolute value of.” For example |-3| = 3.

Finding absolute value using a number line is very simple Finding absolute value using a number line is very simple. You just need to know the distance the number is from zero. |5| = 5, |-6| = 6

Lesson Quiz Compare, Use <, >, or =. 1) -32 □ 32 2) 26 □ |-26| 3) -8 □ -12 4) Graph the numbers -2, 3, -4, 5. and -1 on a number line. Then list the numbers in order from least to greatest. 5) The coldest temperature ever recorded east of the Mississippi is fifty-four degrees below zero in Danbury, Wisconsin, on January 24, 1922. Write the temperature as an integer.

Rules for Integer Operations

Adding Integers When we add numbers with the same signs, 1) add the absolute values, and 2) write the sum (the answer) with the sign of the numbers. When you add numbers with different signs, 1) subtract the absolute values, and 2) write the difference (the answer) with the sign of the number having the larger absolute value.

Try the following problems 1) -9 + (-7) = -16 2) -20 + 15 = -5 3) (+3) + (+5) = +8 4) -9 + 6 = -3 5) (-21) + 21 = 0 6) (-23) + (-7) = -30

Subtracting Integers You subtract integers by adding its opposite

Try the following problems 1) -5 – 4 = -5 + (-4) = -9 2) 3 – (+5) = 3 + (-5) = -2 3) -25 – (+25) = -25 + (-25) = -50 4) 9 – 3 = 9 + (-3) = +6 5) -10 – (-15) = -10 + (+15) = +5

Multiplying and Dividing Integers If the signs are the same, the answer is positive. If the signs are different, the answer is negative.

Try the following problems Think of multiplication as repeated addition. 3 · 2 = 2 + 2 + 2 = 6 and 3 · (-2) = (-2) + (-2) + (-2) = -6 1) 3 · (-3) = Remember multiplication is fast adding = 3 · (-3) = (-3) + (-3) + (-3) = -9 2) -4 · 2 = Remember multiplication is fast adding = -4 · 2 = (-4) + (-4) = -8

same sign positive different signs negative Dividing Integers Multiplication and division are inverse operations. They “undo” each other. You can use this fact to discover the rules for division of integers. 4 · (-2) = -8 -4 · (-2) = 8 -8 ÷ (-2) = 4 8 ÷ (-2) = -4 same sign positive different signs negative The rule for division is like the rule for multiplication.

Try the following problems 1) 72 ÷ (-9) 72 ÷ (-9) Think: 72 ÷ 9 = 8 -8 The signs are different, so the quotient is negative. 2) -144 ÷ 12 -144 ÷ 12 Think: 144 12 = 12 -12 The signs are different, so the quotient is negative. 3) -100 ÷ (-5) Think: 100 ÷ 5 = 20 -100 ÷ (-5) The signs are the same, so the quotient is positive.

Evaluate x + y for x = -2 and y = -15 3) 3 – 9 = 4) -3 – (-5) = Lesson Quiz Find the sum or difference 1) -7 + (-6) = 2) -15 + 24 + (-9) = Evaluate x + y for x = -2 and y = -15 3) 3 – 9 = 4) -3 – (-5) = Evaluate x – y + z for x = -4, y = 5, and z = -10 Find the product or quotient 1) -8 · 12 = 2) -3 · 5 · (-2) = 3) -75 ÷ 5 = 4) -110 ÷ (-2) = 5) The temperature in Bar Harbor, Maine, was -3 F. During the night, it dropped to be four times as cold. What was the temperature then?

Fractions and Decimals Rational Numbers Fractions and Decimals

Rational numbers – numbers that can be written in the form a/b (fractions), with integers for numerators and denominators. Integers and certain decimals are rational numbers because they can be written as fractions.

Remember you can simplify a fraction into a decimal by dividing the denominator into the numerator, or you can reduce a decimal by placing the decimal equivalent over the appropriate place value. O.625 = 625/1000 = 5/8

Hint: When given a rational number in decimal form (such as 2 Hint: When given a rational number in decimal form (such as 2.3456) and asked to write it as a fraction, it is often helpful to “say” the decimal out loud using the place values to help form the fraction.

Write each rational number as a fraction:

Hint: When checking to see which fraction is larger, change the fractions to decimals by dividing and comparing their decimal values.

Examples of rational numbers are: 6 or 6/1 can also be written as 6.0 -2 or -2/1 can also be written as -2.0 ½ can also be written as 0.5 -5/4 can also be written as -1.25 2/3 can also be written as .66 2/3 can also be written as 0.666666… 21/55 can also be written as 0.38181818… 53/83 can also be written as 0.62855421687… the decimals will repeat after 41 digits

Examples: Write each rational number as a fraction: 0.3 0.007 -5.9 0.45

Since Real Numbers are both rational and irrational ordering them on a number line can be difficult if you don’t pay attention to the details. As you can see from the example at the left, there are rational and irrational numbers placed at the appropriate location on the number line. This is called ordering real numbers.

√2 = 1.414213562… no perfect squares here Irrational numbers √2 = 1.414213562… no perfect squares here

An irrational number cannot be expressed as a fraction. Irrational number – a number that cannot be expressed as a ratio of two integers (fraction) or as a repeating or terminating decimal. An irrational number cannot be expressed as a fraction. Irrational numbers cannot be represented as terminating or repeating decimals. Irrational numbers are non-terminating, non-repeating decimals.

Below are three irrational numbers Below are three irrational numbers. Decimal representations of each of these are nonrepeating and nonterminating

Examples of irrational numbers are: = 3.141592654… √2 = 1.414213562… 0.12122122212 √7, √5, √3, √11, 343√ Non-perfect squares are irrational numbers Note: The √ of perfect squares are rational numbers. √25 = 5 √16 = 4 √81 = 9 Remember: Rational numbers when divided will produce terminating or repeating decimals.

NOTE: Many students think that is a terminating decimal, 3.14, but it is not. Yes, certain math problems ask you to use as 3.14, but that problem is rounding the value of to make your calculations easier. It is actually an infinite decimal and is an irrational number.

There are many numbers on a real number line that are not rational There are many numbers on a real number line that are not rational. The number is not a rational number, and it can be located on a real number line by using geometry. The number is not equal to 22/7, which is only an approximation of the value. The number is exactly equal to the ratio of the circumference of a circle to its diameter.

Enjoy your Pi