TX PACT: Essential Academic Skills 700 (Subtest III: Mathematics - 703) (2024/2025) (Passed with 100%)

TX PACT: Essential Academic Skills 700 (Subtest III: Mathematics - 703) (2024/2025) (Passed with 100%) Real numbers - Answer ️️ -are numbers that have a specific value. Real numbers can be rational or irrational. Within rational numbers, there are integer, whole, and natural numbers. Rational numbers - Answer ️️ -are numbers that can be expressed as a ratio (fraction or comparison) of two integers, a/b where b ≠ 0 (because zero can't be the denominator of a fraction). All rational numbers can be written exactly in both fraction and decimal forms, and can be graphed on a number line. Note that exact is a formal term in mathematics meaning a value that has not been rounded or approximated. rational numbers - Answer ️️ -ex. mixed numbers - like 4½ which have both a whole number part, like the "4," and a fractional part, like the "½" ex. decimal numbers that end, such as 0.25 ex. decimal numbers that repeat the same pattern forever, such as 0.727272... (also written 0.overline{72}0.72), which is 8/11 when written in fraction form Natural Numbers - Answer ️️ -real numbers that can be pointed to in nature - one tree, two people, three dogs. This set is also known as the counting numbers; it starts at 1 and goes up by 1s in the same way that children first learn to count, yielding the infinitely large set of values 1, 2, 3, 4, 5,... Whole Numbers - Answer ️️ -the set of natural numbers and zero; it starts at 0 and goes up by 1s to create the infinite set 0, 1, 2, 3, 4,... Integers - Answer ️️ -positive and negative counting numbers and zero; the infinite set ...-3, -2, -1, 0, 1, 2, 3,... Irrational Numbers - Answer ️️ -All include a symbol such as square root or pi ex. must be written a symbol or radical. real numbers that CANNOT be represented exactly as a ratio of two integers. In order to be expressed exactly (without rounding), irrational numbers are often written with symbols (like π, e, or sqrt{2}2). When irrational numbers are written using only ordinary numerals, they must be rounded. Irrational numbers, by definition, cannot be written in fraction form, and when expressed as decimals, they have infinitely many, non-repeating digits. Irrational numbers can be approximated (π approx 3.14≈3.14 approx≈ 22/7; e approx 2.72≈2.72, and sqrt{2} approx 1.412≈1.41) and then positioned on a number line. Real numbers - Answer ️️ -can be represented in a variety of ways, with the most common being fractions, percents, and decimals. Fractions - Answer ️️ -usually represent partial numbers but can represent whole numbers. ½ means one half of one unit 4/1 means 4 of one unit The top part of a fraction is referred to as the fraction's numerator. The bottom of a fraction is its denominator. You can remember this with the mnemonic "denominator is down." The fraction bar itself should be understood to be a division symbol. Percentages - Answer ️️ -usually represent part of a whole but can represent larger values. 40% means 4 out of 10 parts of 1 unit or 40 out of 100 units. 200% means twice as much as 1 unit Decimals - Answer ️️ -are numbers less than 1 displayed using place values and the powers of ten. They are usually first introduced as decimal fractions, which are fractions with denominators that are powers of 10. For example, 1/10 is shown to be equivalent to 0.1. When turning a fraction into a decimal, be sure to divide in the correct order: - Answer ️️ - numerator divided by denominator. For example, 2/3 becomes 2 ÷ 3 and equals the repeating decimal 0.666.... If accidentally done in reverse order, a calculator would show the terminating decimal to be 1.5. It is important to realize that "2/3" represents having two parts of three total possible parts so 2/3 should be less than one. To convert a decimal into a percent - Answer ️️ -move the decimal place twice to the right (multiply by 100) and put the % symbol behind the number. Example conversions: What is 0.25 expressed as a percentage? Multiply by 100 by moving the decimal two places to the right and add a percent sign. 0.25 rightarrow 25%0.25→25% What is ⅔ expressed as a percentage? - Answer ️️ -Convert first to decimal form by dividing. 2/3 is 0.66666... in a calculator. Multiply by 100 by moving the decimal point twice to the right and round as needed. ⅔ approx 66.67%⅔≈66.67% To convert a percent to a fraction or decimal - Answer ️️ -recall that "percent" means "for every one hundred." Rewrite a quantity