Real Numbers

Miss Johnson’s 8th Grade Math Class

Standard 8.NS.1

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RATIONAL NUMBERS IRRATIONAL NUMBERS

MULTIPLYING IRRATIONAL NUMBERS

TERMINATING VS. REPEATING

TERMINATING DECIMAL QUIZ

Rational Numbers

A rational number is any real number that can be expressed as the quotient or fraction of two integer

Example

1.5 is a rational number because 1.5= 3/2

It can be written as a fraction

More Examples

Number As a Fraction

Rational?

5 5/1 Yes1.75 7/4 Yes.001 1/1000 Yes-0.1 -1/10 Yes

0.111… 1/9 YesSquare

root of 2 ? NO!

The Square root of 2 cannot be written as a simple fraction. Since it cannot be written as a rational number is called irrational.

Irrational Numbers

An irrational number is a real number that cannot be written as a simple fraction.

Irrational means not Rational.

Symbol Definition

Pi is a famous irrational number. People have calculated Pi to over a quadrillion decimal places and still there is no pattern. The few digits look like this: 3.14159265358979323846433832795…

The number e (Euler’s Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this: 2.7182818284590452353602874713527….

The Golden Ratio is an irrational number. The first few digits look like this: 1.61803398874989484820…

Many square roots, cube roots, etc are also irrational numbersSquare root of 3Square root of 99Not all roots are irrational.

Examples

Note on Multiplying Irrational Numbers

π × π = π2 is irrational

But √2 × √2 = 2 is rational

Multiplying irrational numbers might result in a rational number

Terminating vs. Repeating Decimals

As mentioned earlier, decimal expansion is the form of a number that has a decimal point, either actual or implied. Examples of numbers with 'actual' decimal points are 10.2 and 0.0084. An example of a number with 'implied' decimal points is the whole number 17, which could actually be written as 17.000000000000. We often leave off the repeating zeros for ease of reading and figuring.

Continued…

Rational decimals that end with repeating zeroes are known as finite decimals. They're the opposite of infinite, or forever. Rational decimals that forever repeat are known as repeating decimals. This becomes important only when thinking about some of the patterns we see in the decimal forms of some numbers.

Examples of Terminating Decimal

Rational numbers with implied repeating zeros are finite, like 6/3 = 2 or 10/2.5 = 4. The other rational decimals that are finite are those which originate from a fraction with a denominator that is a product of 2's, 5's or both. For example, 3/8, 13/25 and 7/50 are all finite decimals because 8 = 2^3, 25 = 5^2 and 50= 5^2 x 2.

Repeating Decimals

Repeating decimals originate from fractions whose denominators are not entirely products of 2's and 5's only. For example, 3/14 and 7/22 are repeating decimals because 14 = 2 x 7 and 22 = 2 x 11. The number of digits that repeat are referred to as the period of the repeating decimal. For example, thirds have a period of 1 (1/3 = 0.3333333...), elevenths have a period of 2 (2/11 = 0.1818181818….) and sevenths have a period of 6 (2/7 = 0.285714285714285714…..).

Question 1

Which one of the following is not irrational?

A. √2

C. √4 D. √5

B. √3

Correct Answer is…. C

Question 2

How many numbers in the following list are irrational?

√3, π,e, 1/3, 0,5 √2,22/7, √36

A. 3

C. 5 D. 6

B. 4

The Correct Answer is B

Question 3

Is the number 0.09009000900009000009… rational or irrational?

Irrational!!!

Question 4

• Is the number 0.201 rational or irrational?

Rational!!!

Question 5

The square root of 25 is rational or irrational?

Rational!!!

Decimal Expansion

We use place notation to give each digit a value based on a negative power of ten.

Each “decimal place” consists of a digit 0-9 arranged such that each digit is multiplied by the power of 10.

The powers decrease from left to right, and with a decimal place indicating the 100 = 1s place.

Example

To the right is a table of the fractions and their decimal expansions, from 1/1 - 1/10. Keep in mind that a repeating decimal is indicated by an overlie, or a bar over particular digits. So 1/6 = 0.1666666666……

Natural Numbers

A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number. The set of natural numbers, denoted N, can be defined in either of two ways:

Integers

The number line goes on forever in both directions. This is indicated by the arrows.

Whole numbers greater than zero are called positive integers. These numbers are to the right of zero on the number line.

Whole numbers less than zero are called negative integers. These numbers are to the left of zero on the number line.

The integer zero is neutral. It is neither positive nor negative.

Continued…

The sign of an integer is either positive (+) or negative (-), except zero, which has no sign.

Two integers are opposites if they are each the same distance away from zero, but on opposite sides of the number line. One will have a positive sign, the other a negative sign. In the number line above, +3 and -3 are labeled as opposites.

Example

Name the opposite of each integer.

-12 +12

+21 -21

-17 +17

+9 -9

Great Job!!!This concludes our lesson on irrational numbers!!!!