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MAT082 :: Lecture Note :: Week 02
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Base-10 Place Values
[due: class 2 of week 2]
[due: class 2 of week 2]
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The place values for whole numbers in the decimal (base-10) number system are defined using positive integer powers of
10.100 is 1 ... ones 101 is 10 ... tens 102 is 100 ... hundreds 103 is 1,000 ... thousands 104 is 10,000 ... ten thousands 105 is 100,000 ... hundred thousands 106 is 1,000,000 ... millions ... 10^9 is billions; 10^12 is trillions; 10^15 is quadrillions 10^100 is googol; 10^googol is googolplex caret ^ is calculator notation for exponents (raising a number to a power) n^m = nm (e.g. 5^3 = 53)Example: The integer number
4,096has four units of one thousand, zero units of one hundred, nine units of one ten, and six units of one. The digit '4' is in the "thousands place," the digit '0' is in the "hundreds place," the digit '9' is in the "tens place" and the digit '6' is in the "ones place."4 x 1000 = 4000 0 x 100 = 0 9 x 10 = 90 6 x 1 = 6 4000 + 0 + 90 + 6 = 4096The place values for decimal digits in the base-10 number system are defined using negative integer powers of ten.
10-1 equals 1/10 equals 0.1 10-2 equals 1/100 equals 0.01 10-3 equals 1/1000 equals 0.001 10-4 equals 1/10,000 equals 0.0001 ... 10-9 equals 1/1,000,000,000 equals 0.000000001To the left of the decimal point are the ones, tens, hundreds, thousands, and so on. On the fractional side of decimal number are the tenths, hundreths, thousandths, and so on. There are no oneths.
The number
0.5214is less than one and greater than zero. The digit '5' is in the tenths place, the digit '2' is in the hundreths place, the digit '1' is in the thousandths place and the digit '4' is in the ten thousandths place.5 x 0.1 = 0.5 2 x 0.01 = 0.02 1 x 0.001 = 0.001 4 x 0.0001 = 0.0004 0.5 + 0.02 + 0.001 + 0.0004 = 0.5214PurpleMath.com:: Number Bases
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Addition Terminology
Addition is the combination of two or more numbers. The result of an addition is called a sum.
general equation: a + b = c'a' is called the augend, 'b' is called the addend, while 'c' is called the sum.
5 + 3 = 8 5 is the augend 3 is the addend 8 is 5 added with 3 + is the arithmetic addition operator = is the equals operatorSubtraction Terminology
Subtraction is the arithmetic operation in which the difference between two numbers is calculated.
general equation: a - b = c'a' is called the minuend, 'b' is called the subtraend, while 'c' is called the difference.
5 - 3 = 2 5 is the minuend 3 is the subtrend 2 is the difference 5 and 3 - is the arithmetic subtraction operator = is the equals operatorSubtraction is related to addition as follows. If
a + b = c, thenc - b = aandc - a = b.If you subtract a larger number from a smaller number, then the difference is a negative number (i.e. it is less than zero). Negative numbers are prefixed with a dash
-character.3 - 5 = -2 3 minus 5 equals a negative 2 (i.e. -2) 5 subtracted from 3 is a difference of -2 the difference between 3 and 5 is -2Multiplication Terminology
Multiplication is a technique for adding identical numbers.
4 * 5 is equal to 4 + 4 + 4 + 4 + 4 9 * 2 is equal to 9 + 9 3 * 8 is equal to 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3general equation: a * b = c'a' is called the multiplicand, 'b' is called the multiplicator, and 'c' is called the product.
For example, it is
11miles between my home in Tempe and the SCC campus. The type (i.e. unit of measurement) ismilesand11is a value. Round-trip my commute is22miles (2times11). Two times eleven can be written in the following ways.2 × 11 2(11) 2 ⋅ 11 2 * 11It is a good idea to memorize the 12-by-12 multiplication table.
Division Terminology
Division is the reverse operation of multiplication.
general equation: a / b = c 'a' and 'b' and 'c' are "variables" variables are assigned "values"'a' is called the dividend, 'b' is called the divisor, and 'c' is called the quotient.
Exercise: If fourteen (14) Artie Artichoke dolls are to be split evenly between two (2) people, how many dolls will each person receive?
14 / 2 14 ÷ 2 _______ 2)14 14 divided by 2 equals 7 Given the general equation: a / b = c In this specific problem: 'a' has the value 14, 'b' has the value 2, and 'c' has the value 7.[side-bar]
14is an even number because when divided by2there is zero remainder. In other words,14is evenly divisable by2; therefore,14is an even number.7 / 2 = 3 with a reminder of 1Division is the reverse operation of multiplication. If
a * b = candbis not zero, then the equation is equal toa = c / b.a * b = c --------- let 'a' equal 4 and 'b' equal 2 4 * 2 = 8 a = c / b --------- 4 = 8 / 2Division by zero is undefined (i.e not allowed). {MathForum.org:: Ask Dr. Math: Dividing by Zero}
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Wikipedia says..."
0is a number and a numerical digit."Zero represents nothing (or null or nil or void or absence of value).
let 'a' represent any whole number a + 0 = a 55 + 0 = 55 a - 0 = a 18 - 0 = 18 a * 0 = 0 33 * 0 = 0 a / 0 = not defined cannot divide by zero 0 / a = 0 0 / 13 = 0 0 is neither positive nor negative 0 is not a prime numberMost definitions indicate that 0 is an even number; however, some people believe 0 is neither even nor odd. Even numbers are integers that are evenly divisable by 2; thus, according to this definition, 0 is an even number.
GDT::BAB:: Google's Calculator Has Stopped Dividing-By-Zero [28 July 2005]
Wikipedia.org:: 0 (number)
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The number one often represents a unit (i.e. "a single undivided whole).
Let 'a' represent any whole number. a + 1 = (a incremented by 1) 7 + 1 = 8 a - 1 = (a decremented by 1) 9 - 1 = 8 a * 1 = a 55 * 1 = 55 a / 1 = a 33 / 1 = 33 1 is an odd number 1 is not a prime number 1 is the first whole number?A fraction with one as its numerator is called a unit fraction.
Wikipedia.org:: 1 (number)
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Tables are used to display data in an informational way.
Tables generally have one or more rows with each row having one or more columns. The intersections of rows and columns form cells that are used to display data. Typcially, one row and one column are allocated for displaying headings. Headings play a major role in turning data into information. Some tables have borders around the cells. A table often has a caption that describes the table (i.e. provide information about the table as a whole).
The first row and column are often used to display headings that help turn data into information.
Spreadsheats are a form of table. Computer spreadsheats are a form of interactive table.
Finance.Yahoo.com displays historical stock prices using a table. {Finance.Yahoo.com:: DSCM: Historical Prices for Drugstore.com}
The following is a table of Zelmo Zeroman's DSCM trading activity.
------- Zelmo Zeroman's DSCM Trades ------- number per share action date shares price ($) (MM/DD/YYYY) =========================================== 100 2.92 Buy 01/23/2005 50 4.00 Sell 07/28/2005 200 2.62 Buy 12/26/2005 100 2.71 Buy 05/31/2006From the table, we see that on
01/23/2005Zelmo purchased100 sharesat a price of$2.92 per share. He sold50 sharesat$4.00 per shareon07/28/2005.
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Precedence means "priority of importance."
Given an expression such as
3 + 5 x 8we need to concern ourselves about the order of evaluation. For example, if we add 3 and 5 and multiply the sum by 8, then we get 64 for an answer; however, if we multiply 5 by 8 and add 3 to the product, then we get an answer of 43.order of operations =================== groupings () [] ---- exponents multiply, divide [equal precedence; left-to-right] addition, subtraction [equal precedence; left-to-right] 3 + 5 * 8 Multiply has a higher precedence than addition; therefore, do it first followed by the addition. 5 * 8 is 40, add 3 gives 43 (3 + 5) * 8 Grouping has the highest precedence; therefore, do it first and then mulitply. 3 + 5 is 8, times 8 gives 64 8 + 3 - 2 Addition and subraction have the same precedence; therefore, evaluate left-to-right. 8 + 3 is 11, subtract 2 gives 9 (4 + 1)2 Grouping has highest precedence; therefore, do it first. 4 + 1 is 5, 5 squared is 25 4 + 12 Exponent has higher precedence than addition. 1 squared is 1, plus 4 gives 5 3 + 2 ----- 2 * 5 Separately evaluate the numerator (top) and denominator (bottom) and then divide the denominator into the numerator. 3 + 2 = 5 ... 2 * 5 = 10 ... 5 divided-by 10 = 0.5 Note: (3 + 2) / (2 * 5) does not equal 3 + 2 / 2 * 5What's PEMDAS?
The following was from a Fall 2004 student.
Please excuse my dear aunt sally. P - parenthesis E - exponent M - multiply D - divide A - add S - subtractThere are some special cases when using PEMDAS.
- brackets
[]go ahead of parenthesis()- multiply and divide are at the same level of precedence and they are evaluated left-to-right
- add and subtract are at the same level of precedence and they are evaluated left-to-right
B - brackets E - exponents D - divide M - multiply A - add S - subtractExternal Hyperlinks
- Wikipedia.org:: Order of operations
- PurpleMath.com:: Order of Operations: PEMDAS
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