Number SystemCommon Core 8th Grade

8.NS.A.1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Rational Numbers A Rational Number is a real number that can

be written as a simple fraction (i.e. as a ratio).Example: 1.5 is rational, because it can be written as the ratio 3/2 7 is rational, because it can be written as the ratio 7/1 0.333... (3 repeating) is also rational, because it can

be written as the ratio 1/3

The following number sets consist of rational numbers:

Natural (Counting); Whole; Integers; Fractions; Repeating Decimals; Terminating Decimals

More formally we would say:A rational number is a number that can be in the

form p/q where p and q are integers and q is not equal to zero.

Fun Fact:The ancient greek mathematician Pythagoras believed that all

numbers were rational (could be written as a fraction), but one of his students Hippasus proved (using geometry, it is thought) that you could not represent the square root of 2 as a fraction, and so it was irrational.

However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned!

8.NS.A.1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Irrational NumbersAn Irrational Number is a real number that cannot

be written as a simple fraction.Irrational means not RationalExample: π Pi is a famous irrational number. π = 3.1415926535897932384626433832795 (and

more...)You cannot write down a simple fraction that equals Pi. The popular approximation of 22/7 = 3.1428571428571... is close but

not accurate.

Another clue is that the decimal goes on forever without repeating.

Square roots of non-perfect squares are also irrational.

Name ALL the set(s) of numbers to which each number belongs.-5/6 35.9904 1/8

-8012/3

-3.243

5

100

4

8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers.

Estimating square roots:In math it is sometimes important to estimate square

roots. This can be done following a few simple steps.a. Find the closest perfect square below and above the

number that you are trying to find the square root of. Example: 53 is between the perfect squares 49 and 64

b. Find the square root of the two perfect squares. so is between 7 and 8.Since 49 is closer to 53 than 64 try a value a little more

than 7 such as 7.2. If you square 7.2, the value is 51.84 so you need to try a larger value. Repeat the process until you are close to 53.

53

749 864 53

Now you try:11 200 613