← Grade 8: Rational and Irrational Numbers
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Grade 8: Rational and Irrational Numbers
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Grade 8: Rational and Irrational Numbers — Flashcards
| Front | Back |
|---|---|
| What is the origin on a number line? | A chosen point on a line, marked 0, from which distances to other points are measured. |
| How is a rational number p/q represented on the number line? | By a length equal to p copies of one qth of the unit interval, placed to the right of the origin if p/q is positive, or to the left if negative. |
| What is the integral part of a positive number a? | The integer N such that N ≤ a < N + 1. |
| How is the decimal expansion of a number a constructed geometrically? | By repeatedly dividing intervals into ten equal parts and counting how many parts fit between the previous approximation and a, generating digits d1, d2, d3, … |
| What did the "tilted squares" construction reveal (Pythagorean discovery)? | That some lengths, such as the diagonal of a unit square, do not correspond to any rational number — an observation made by the Pythagorean society about 2500 years ago. |
| What does the symbol √A represent? | A number a whose square equals A (a² = A); it only makes sense when A is not negative. |
| What is a perfect square? | A positive integer whose square root is also a positive integer. |
| What does the symbol ³√V represent? | The side length of a cube whose volume is V (the cube root of V). |
| What is the Pythagorean theorem? | For a right triangle with legs a and b and hypotenuse c, a² + b² = c². |
| What is an irrational number? | A number that cannot be expressed as a quotient (fraction) of two integers. |
| What rule determines whether a rational number has a terminating decimal? | A rational number has a terminating decimal only if its denominator (in lowest terms) is a product of twos and fives. |
| What kind of decimal represents a rational number that is not terminating? | A repeating decimal. |
| What can be said about a decimal expansion that is neither terminating nor repeating? | It represents an irrational number. |
| What is true about √N for a whole number N? | Either N is a perfect square (so √N is an integer), or √N is irrational (not a quotient of integers). |
| What is Newton's method used for in this chapter? | Approximating square roots of a number N to any required degree of accuracy using repeated recursion. |
| What is the recursion formula in Newton's method for approximating √N? | a_new = ½(a_old + N / a_old), starting from a reasonable initial estimate a_old. |
| Why must care be taken when doing arithmetic with irrational numbers? | To achieve a specified accuracy in the final result, the original numbers may need to be known to much greater accuracy. |
| How can a point (a, b) be located in the coordinate plane? | By moving a horizontal distance a along the x-axis, then a directed vertical distance b, using the same unit interval as the number line. |
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