MATH COMPUTATION SKILLSSection I

DePaul Math Placement Test

There are different types of numbering systems that you must be familiar with in order to understand different properties of mathematical functions

• Whole Numbers: the set of counting numbers, including zero {0, 1, 2, 3, ..}.

• Natural Numbers: the set of all whole numbers except zero {1, 2, 3, 4, . . .}.

• Integers: the set of all positive and negative whole numbers, including zero. Fractions and decimals are not included {. . . , –3, –2, –1, 0, 1, 2, 3, . . .}.

• Rational Numbers: the set of all numbers that can be expressed as a quotient of integers. Any set of rational numbers includes all integers, and all fractions that contain integers in the numerator and denominator(m/n).

• Real Numbers: every number on the number line. The set of real numbers includes all rational and irrational numbers.

Different Types of Numbers

The following rules define how positive and negative numbers operate under various operations.

• Addition and Subtraction: Positive and negative numbers

• Multiplication:positive * positive = positive ( Example: 2*3=6)negative * negative = positive ( Example: -2*-3=6)positive * negative = negative ( Example: 4*-5=-20)

negative * positive = negative ( Example: -4*2=-8)

• Division: positive / positive = positive ( Example: 9/3=3)

negative / negative = positive ( Example: -12/-3=4)positive / negative = negative ( Example: 6/-3=-2)negative / positive = negative ( Example: -6/3=-2)

The rules for multiplication and division are exactly the same since any division operation can be written as a form of multiplication: a b = a/b = a 1/b.

Operations of Positive and Negative Numbers

The most basic form of mathematical expressions involving several mathematical operations can only be solved by using the order of PEMDAS. This catchy acronym stands for:

Parentheses: first, perform the operations in the innermost parentheses. A set of parentheses supercedes any other operation.

Exponents: before you do any other operation, raise all the required bases to the prescribed exponent. Exponents include roots, since root operations are the equivalent of raising a base to the 1 /n, where ‘n’ is any integer.

Multiplication and Division: perform multiplication and division.

Addition and Subtraction: perform addition and subtraction.

Order of Operations

Examples Order of Operations Here are two examples illustrating the usage of PEMDAS. Let’s work through a few examples to see how order of operations and PEMDAS work. 3 X 23 + 6 4. Since nothing is enclosed in parentheses, the first operation we carry out is exponentiation:3 X23 + 6÷4= 38+6÷4Next, we do all the necessary multiplication and division:

38+6÷4= 24÷1.5Lastly, we perform the required addition and subtraction. Our final answer is:

24÷1.5= 25.5 Here a few question to try yourself. Evaluate: 6 (232(5-3))Hint: Start solving from the innermost parenthesis first (Final Answer is 12). Here’s another question. Evaluate: 5-22 ⁄6+4.Hint: Solve the numerator and denominator separately.

Order of Operations

An exponent defines the number of times a number is to be multiplied by itself. For example, in ab, where a is the base and b the exponent, a is multiplied by itself b times. In a numerical example, 25 = 2 x 2 x 2 x 2 x 2. An exponent can also be referred to as a power.

The following are other terms related to exponents with which you should be familiar:Base. The base refers to the 3 in 35. It is the number that is being multiplied by itself however many times specified by the exponent.Exponent. The exponent is the 5 in 35. It indicates the number of times the base is to be multiplied with itself.Square. Saying that a number is squared means that the number has been raised to the second power, i.e., that it has an exponent of 2. In the expression 62, 6 has been squared. Cube. Saying that a number is cubed means that it has been raised to the third power, i.e., that it has an exponent of 3. In the expression 43, 4 has been cubed.

Exponents

In order to add or subtract numbers with exponents, you must first find the value of each power, then add the two numbers.

For example, to add 33 + 42, you must expand the exponents to get (3x 3 x 3) + (4 x 4), and then, 27 + 16 = 43.

However, algebraic expressions that have the same bases and exponents, such as 3x4 and 5x4, can be added and subtracted. For example, 3x4 + 5x4 = 8x4.

Adding and Subtracting Numbers with Exponents

To multiply exponential numbers raised to the same exponent, raise their product to that exponent: an x bn = (ax b)n = (ab)n

43x 53=(4x5)3= 203

To divide exponential numbers raised to the same exponent, raise their quotient to that exponent: an/bn = (a/b)n

125 / 35= (12/3)5 = 43

To multiply exponential numbers or tems that have the same base, add the exponents together: ambn = (ab)(m+n)

36 x32= 3(6+2)= 38

To divide two same-base exponential numbers or terms, just subtract the exponents: am/bn = (a/b)(m-n)

36/32 = 3(6-2) = 34

When an exponent is raised to another exponentIn cases, (32)4 and (x4)3. In such cases, multiply the exponents:

(am)n = a(mn)

(32)4 = 3(2x 4) = 38

Multiplying and Dividing Numbers with Exponents

As mentioned in the negative numbers section, when you multiply a negative number by a negative number, you get a positive number, and when you multiply a negative number by a positive number, you get a negative number. These rules affect how negative numbers function in reference to exponents.

•When you raise a negative number to an even number exponent, you get a positive number. For example, (–2)4 = 16.

Let’s break it down. (-2)4 means, –2 x –2 x –2 x –2. When you multiply the first two –2’s together, you get 4 because you are multiplying two negative numbers. Then, when you multiply the 4 by the next –2, you get –8, since you are multiplying a positive number by a negative number. Finally, you multiply the –8 by the last –2 and get 16, since you’re once again multiplying two negative numbers.

•When you raise a negative number to an odd power, you get a negative number. To see why, refer to the example above and stop the process at –8, which equals (–2)3.

Exponents and Negative Numbers

Special ExponentsThere are a few special properties of certain exponents that you also should know. ZeroAny base raised to the power of zero is equal to 1. If you see any exponent of the form x0, you should know that its value is 1. Note, however, that 00 is undefined. OneAny base raised to the power of one is equal to itself. For example, 21 = 2, (–67)1 = –67, and x1 = x. This can be helpful when you’re attempting an operation on exponential terms with the same base. For example:

3x6 x x4= 3x(6+4) = 3x7

Exponents- special cases