ACT math

Rational Numbers

Any number that repeats or terminates. Examples of these types of numbers are: 3, -3, 1/2, .3333333, and -.66666.

Irrational Numbers

Any number that does not repeat or terminate. Examples of these types of numbers are: 1.32453345245 or 3.141592635897 (Mathematical Pie)

Real Numbers

These are all numbers that include any rational or irrational number.

Imaginary Numbers

This is any number that is not rational or irrational. The number i = -1 i^2 = -1

Factors

A factor of a number is a number that is equally divisible into another number. For example: The number 20 has six factors 1,2,4,5,10, and 20.

Greatest Common Factor

When you have two numbers such as the number 12 and 20 and you are list out its factors. 12 - 1,2,3,4,6, and 12. 20 - 1,2,4,5,10, and 20. The least common factor between 12 and 20 is the number 4.

Multiples

A number that can be divided by another number without a remainder. The multiples of 5 are: 5, 10, 15, 20, 25 and so on.

Remainder

The amount leftover when a number is not equally divisible by another number. For example: 23 7 has a remainder of 2.

Adding and Subtracting Fractions

When adding and subtracting fractions, you must have a common denominator. The denominator is the number found at the bottom of a fraction 3/4. To add or subtract fractions you add the numerator - which is the number found at the top of the fraction 3/4 - only if the denominators are the same. 3/5 + 1/5 = 4/5 or 7/8 - 2/8 = 5/8

Multiplying Fractions

When multiplying fractions, you multiply the numerator and the denominator of both fractions. 3/5 X 4/5 = 12/25

Dividing Fractions

When dividing fractions, you multiply the first fraction to the reciprocal of the second fraction. You find the reciprocal of a fraction by flipping the fraction upside down. 2/5 / 4/5 = 2/5 X 5/4 which then equals 10/20 simplified to 1/2.

Exponents

A quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression. An example is: 2^3 = 2 x 2 x 2 which is equivalent to 8.

Multiplying Exponents

The exponent “product rule” tells us that when multiplying two powers that have the same base, you can add the exponents. 2^4 X 2^5 = 2^9 or 512.

Dividing Exponents

To divide exponents with the same base you simply subtract the exponents or the “powers”. 2^7 / 2^3 = 2^4 which is equivalent to 16.

Raising an Exponent to an Exponent

When an exponent is raised to another exponent, you simply multiply the exponents to achieve the result. (2^4)^3 = 2^12 which is equivalent to 4096.

Raising an Exponent to a Fractional Exponent

When you raise an exponent to a fractional exponent, the numerator of the fraction is the power you raise the base to and the denominator is the root that you place the base number underneath. 2^(3/4) = The fourth root of the number 2^3.

Absolute Value

The absolute value of a number is defined as the distance away from zero on a number line.

Percent Increase/Decrease

To determine the percentage increase or decrease of an original number, you need to take the the difference of the two numbers and divide by the original number. Then, take this value and multiply it by 100. For example, John had 120 books in his library originally. He sold 80 of his books. What is the percent decrease in the amount of books. 120-80 = 40 40/120 = .33333 .3333 X 100 = 33.3%

Mean (Average)

To find the mean or average of a set of numbers, you take the total or sum of all the numbers and then divide by the amount of numbers are in the set. The most important aspect of any mean or average problem is the total or sum of the set of numbers. For example: 2+3+4+5+6+7+8 = 35. 35/7 = 5.

Median

The median is the middle number of any set of numbers. You first must place the numbers in the set in numerical order. If the set has an odd amount of numbers inside it, it is the middle number. If the set has an even amount of numbers inside it, it is the average of the two middle numbers of the set. For example: 4, 9, 2, 11, 1, 6, 31…. 1, 2, 4, 6, 9, 11, 31. Thus, the median is 6. 4, 9, 2, 11, 1, 6, 31, 15…. 1, 2, 4, 6, 9, 11, 15, 31. You must take the average of 6 and 9 which is 7.5.

Mode

The mode is the most frequently found number in a set. A set may have no mode, one more or numerous modes. For Example: 1,2,6,5,3,11… has no mode. 1,2,2,3,4,5… has a mode of 2. 1,2,2,3,3 has a mode of 2 and 3.

Ratio

A ratio is a relationship of a part to a part. For example: 1:3 is 1 part flour per 3 part sugar in a recipe. Also, there is 4 total parts of flour and sugar.