ATI TEAS -- MATHEMATICS: (Part 2) Numbers and Operations

Examples of adding, subtracting, multiplying, and dividing real numbers

1. -18 + 12 = -6 Since |-18| > |12|, the answer is negative. |-18| - |12| = 6, so the answer is -6. 2. -3.64 + (-2.18) = -5.82 Adding two negative numbers results in a negative number. Add the values. 3. 9.37 - 4.25 = 5.12 4. 86 - (-20) = 106 Change the subtraction to addition, change the sign of the second number, then add. 5. (10/3) x (-9/5) = -6 Multpily the numerators, multiply the denominators, then simplify: -90/15 = -6 6. -64/-10 = 6.4 A negative divided by a negative is a positive number. 7. (2.2)(2.3) = 7.26 The parentheses indicate multiplication. 8. -52/13 = -4

The rules for rounding are as follows:

1. Underline the place value being rounded to. 2. Locate the digit one place value to the right of the underlined value. If this value is less than 5, keep the underlined value and replace all the digits to the right of the underlined value with zero. If the value to the right of the underlined digit is more than 5, increase the underlined digit by one and replace all digits to the right of it with zero.

Prime Number

A prime number is a natural number greater than 1 which can only be divided evenly by 1 and itself. For example, 7 is a prime number because it can only be divided by the numbers 1 and 7.

Rational Number

A rational number can be represented as a fraction. Any decimal part must terminate, or resolve into a repeating pattern. Examples include: -12, -4/5, 0.36, 7.7, 26 1/2, etc.

Real number

A real number is a number that can be represented by a point on a number line. Real numbers include all the rational and irrational numbers.

Composite Number

A composite number is a natural number greater than 1 which can be evenly divided by at least one other number besides 1 and itself. For example, 6 is a composite number because it can be divided by 1,2,3, and 6.

Decimals

A decimal is a number that contains a decimal point. The place value for a decimal includes: - tenths (one place after the decimal) - hundredths (two places after the decimal) - thousandths (three places after the decimal) ***Decimals can be added, subtracted, multiplied, and divided***

Fractions

A fraction is a number that can be written in the form a/b where b is not equal to zero. The a part of the fraction is the numerator (top number) and the b part of the fraction is the denominator (bottom number).

Converting between fractions and decimals

A fraction is converted to a decimal by using long division until there is no remainder or a pattern of repeating numbers occur. To covert a decimal to a fraction, place the numbers to the right of the decimal over the appropriate base-10 power and simplify the fraction.

Mixed Number

A mixed number has a whole number part and a proper fraction part. Improper fractions can be converted to mixed numbers by dividing the numerator by the denominator, which gives the whole number part, and the remainder becomes the numerator of the proper fraction part.

Natural number

A natural number is greater than zero and has no decimal or fraction attached. These are also sometimes called counting numbers. {1,2,3,4,...}

SMURF

An acronym for dividing fractions. Same Multiply Upside down Rename Fraction

Exponent

An exponent shows how many times a number should be multiplied by itself.

Irrational Number

An irrational number cannot be represented as a fraction. An irrational decimal number never ends and never resolves into a repeating pattern. Examples include: pie and 0.32456789135....

Subtracting Real Numbers

Change the subtraction to addition, change the sign of the second number, and use addition rules Negative - Positive = Negative -7 - 8 = -7 + (-8) = -15 Positive - Negative = Positive 7 - (-8) = 7 + 8 = 15 Negative - Negative = Keep the sign of the number with the larger absolute value and subtract the absolute values of the numbers - 7 - (-8) = -7 + 8 = 1 - 8 - (-7) = - 8 + 7 = -1

Adding and Subtracting Fractions

Fractions with the same denominator (bottom number) can be added, or subtracted, by simply just adding or subtracting the numerators (top numbers) and the denominator will stay the same. If they do not have a common denominator, the least common multiple of the denominators must be found.

Positive Numbers

Greater than zero - can be shown on a number line

Proper Fraction

If the denominator of a fraction is greater than the numerator, the value of the fraction is less than 1 and it is called a proper fraction.

Factor tree

In a factor tree, the number 54 can be written as: 2 x 3 x 3 x 3 OR 2 x 3^3

Improper Fraction

In an improper fraction, the denominator is less than the numerator and the value of the fraction is greater than one.

Integer

Integers include positive and negative natural numbers and zero. {..., -4,-3,-2,-1,0,1,2,3,4,...}

Order of operations

Must be used to get the correct answer. PEMDAS: 1. Parenthesis: Calculate expressions inside parenthesis, brackets, braces, etc... 2. Exponents: Calculate exponents and square roots 3/4. Multiplication/Division: Calculate any remaining multiplication and division in order from left to right 5/6. Addition/Subtraction: Calculate any remaining addition and subtraction in order from left to right ***the steps: "Multiple-Divide AND Addition-Subtraction" go in order from left to right***

Negative Numbers

Less than zero - can be shown on a number line

Types of Numbers

Natural number Whole number Integers Rational number Irrational number Real number Prime Number Composite Number

Adding real numbers

Positive + Positive = Positive 7 + 8 = 15 Negative + Negative = Negative -7 + (-8) = -15 Negative + Positive OR Positive + Negative = Keep the sign of the number with the larger absolute value and subtract the absolute values of the numbers -7 + 8 = 1 7 + (-8) = -1

Multiplying Real Numbers

Positive x Positive = Positive 8 x 4 = 32 Negative x Negative = Positive -8 x (-4) = 32 Positive x Negative OR Negative x Positive = Negative 8 x (-4) = -32 -8 x 4 = -32

Dividing Real Numbers

Positive/Positive=Positive 8/4 = 2 Negative/Negative=Positive -8/-4 = 2 Positive/Negative OR Negative/Positive = Negative 8/-4 = -2 -8/4 = -2

Dividing Fractions

To divide two fractions, invert the second fraction (swap the numerator and denominator) then multiply normally. If there is a mixed number/fraction, make sure to first change it into an improper fraction.

Multiplying Decimals

To multiply decimals, first multiply the numbers without decimal points. Then, add the number of decimal places to the right of the decimal point in the original numbers ad place the decimal point in the answer so that there are that many places to the right of the decimal. **If you are unsure which way to move the decimal after multiplying, remember that changing the decimal should always make the final answer smaller.

Dividing Decimals

When dividing decimals, move the decimal point to the right in order to make the divisor a whole number and move the decimal the same number places in the dividend. Divide the numbers without regard to the decimal. Then, place the decimal point of the quotient directly above the decimal point of the dividend.

Multiplying Fractions

When multiplying fractions, simply multiply each numerator together and each denominator together.

Whole number

Whole numbers are natural numbers and the number zero. {0,1,2,3,4,...}

Scientific Notation

a method of representing very large and small numbers in the form: a x 10^n --where a is a value between 1 and 10, and n is a nonzero integer For example, the number 927,000,000 is written in scientific notation as 9.27 x 10^8

Ratio

is a comparison of two numbers. The two numbers represent a constant relationship, not a specific value: for every a number of items in the first group, there will be b number of items in the second.

Rounding

is a way of simplifying a complicated number. The result of rounding will be a less precise value that is easier to write or perform operations on.

Proportion

is an equation which states that two ratios are equal. A proportion is solved using cross-multiplication to crease an equation with no fractional components.

Estimation

is when numbers are rounded and then an operation is performed. This process can be used when working with large numbers to find a close, but not exact, answer.

Percentages

means per hundred and is expressed with a percent symbol (%). For example, 54% means 54 out of every 100. A percent can be converted to a decimal by removing the % symbol and moving the decimal two places to the left, while a decimal can be converted into a percent by moving the decimal point two places to the right and adding the % sign. A percent can be converted to a fraction by writing the percent as a fraction with 100 as the denominator and reducing. A fraction can be converted to a percent by performing the indicated division, multiplying the result by 100 and attaching the % sign.