QR - Arithmetic

Question 1

Properties of integers

Answer 1

• Integer can be any number - positive and negative
• If y = xn and n is an integer, y is said to be divisible by x where x is a divisor (factor) of y
• If x and y are positive integers, there exist unique integer q and r, called quotient and remainder respectively
  Y = xq +r and 0=< r

Question 2

Fractions
A. Addition and Subtraction
B. Multiplication and division
C. Mixed numbers

Answer 2

2 fractions are said to be equivalent if they represent the same number

A. Need the same denominator before doing anything
Otherwise, fine the new denominator by choosing
The least common multiple (LCM)

B. To divide by a fraction, just multiply by its reciprocal

C. A number that consists of a whole number and a fraction, for example 7 2/3 is a mixed number and means 7 +2/3
To change a mixed number into a fraction, multiply the whole number by the denominator of the fraction and add this number to the numerator of the fraction; then put the result over the denominator of the fraction.
For example, 7 2/3 = [(3*7) +2 ]/3 = 23/3

Question 3

Decimals

Answer 3

• The decimal point is moved to the right if the exponent is positive
• The decimal point is moved to the left if the exponent is negative
• Multiplication of decimals: multiply the numbers as if they were whole numbers and then insert the decimal point in the product so that the number of digits to the right of the decimal pint is equal to the sum of the numbers of digits to the right of the decimal points in the number being multiplied

Ex: 2.09 (2 digits to the right) * 1.3 (1 digit to the right) = 2.717 (2+1 = 3 digits to the right)

• Division: to divide a number (the dividend) by a decimal (the divisor), move the decimal point of the divisor to the right until the divisor is a whole number. Then move the decimal point of the dividend the same number of places to the right , and divide as you would by a whole number

Question 4

Real numbers

Answer 4

• The distance between a number and zero on the number line is called the absolute value of the number

Question 5

Ratio

Answer 5

• The ratio of a to b is a:b or a/b
• A proportion is a statement that 2 ratios are equal
  Ex: 2/3 = 8/12 is a proportion

Question 6

Powers and square roots

Answer 6

• Squaring a number that is greater than 1, or raising it to a higher power, results in a larger number
  Ex: 3^2 = 9 and 9>3
• Squaring a number between 0 and 1 results in a smaller number
  Ex: (1/3)^2 = 1/9 and 1/9< 1/3 or (0.1)^2 = 0.01

Question 7

Descriptive statistics

• Mean or Average
• Median
• Mode
• Range
• Standard deviation
• Frequency distribution

Answer 7

• Average or the mean
  It locates a type of “center” for he data
  The average of n numbers is defined as the sum of n numbers divided by n
• Median
  Is another type of center for a list of numbers
  To calculate median of n numbers
  1. Order the numbers from least to greatest
  If n is odd, the median is defined as the middle number
  If n is even, the median is defined as the average of the two middle numbers
  NOTE: The median of a set of data can be less, equal to or greater than the mean
• Mode
  1. The mode of a list of numbers is the number that occurs most frequently in the list
  2. A list of numbers may have more than 1 mode
• Range
  Defined as the greatest value in the numerical data minus the least value
• Standard deviation
  Generally speaking, the more the data are spread away from the mean, the greater the standard deviation
  It can be calculated as follows:
  1. Find the arithmetic mean
  2. Find the differences between the mean and each of the n numbers
  3. Square each of the differences
  4. Find the average of the squared differences
  5. Take the nonnegative square root of this average
• Frequency distribution
  Useful for data that have values occurring with varying frequencies. If you end up re-arranging the numbers in a table, do so in ascending order!
  1. Mean of the frequency distribution =
  (Xf + X2f2 + X3f3 etc)/n

2. Median

Question 8

Sets

Answer 8

• If s is a set having finite number of elements, then the number of elements is denoted by |s|= x
• Rule |S U T| = |S|+ |T|- |S n T|
• Two sets that have no elements in common are said to be disjoint or mutually exclusive and therefore |S n T| = 0

Question 9

Factorial

Answer 9

• If n is an integer greater than 1, then n factorial, denoted by “n!” And is defined as all the integers from 1 to n
  Ex:
  2! = (1) (2) = 2
  3! = (1) (2) (3) = 6
• Rule: 0! = 1! = 1
• Factorial is useful for counting the number of ways that a set of objects can be ordered