GE141 - Course Introduction
1. Whole Numbers
Instructor Emeritus
8/15/22
1 Chapter 1: Whole Numbers
1.1 Numerical Representation
\[\small\begin{matrix} \phantom{000,000,00}9 & 9\times 10^0 & nine \\ \phantom{000,000,0}80 & 8\times 10^1 & eighty \\ \phantom{000,000,}700 & 7\times 10^2 & seven\ hundred \\ \hline \phantom{000,00}6,000& 6\times 10^3&six\ thousand\\ \phantom{000,0}50,000& 5\times 10^4& fifty\ thousand\\ \phantom{000,}400,000& 4\times 10^5& four\ hundred\ thousand\\ \hline \phantom{00}3,000,000& 3\times 10^6& three\ million\\ \phantom{0}20,000,000& 2\times 10^7& twenty\ million\\ 100,000,000& 1\times 10^8& one\ hundred\ million\\ \hline 123,456,789&&\\ \end{matrix}\]
1.2 Word form
123,456,789 \(\longrightarrow\)
one hundred twenty-three million,
four hundred fifty-six thousand,
seven hundred eighty-four1.3 Orders of magnitude
\[\small\begin{matrix} 10^3 & K & Kilo & Thousand \\ 10^6 & M & Mega & Million \\ 10^9 & G & Giga & Billion \\ 10^{12} & T & Tera & Trillion \\ 10^{15} & P & Petra & Quadrillion \\ 10^{18} & E & Exa & Pentillion \\ 10^{21} & Z & Zetta & Sextillion \\ 10^{24} & Y & Yotta & Septillion \\ \end{matrix}\]
2 Addition
2.1 Addition Table
\[\tiny\begin{matrix} &|& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ \hline 0&|& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 1&|& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\ 2&|& 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11\\ 3&|& 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10& 11 & 12\\ 4&|& 4 & 5 & 6 & 7 & 8 & 9 & 10& 11 & 12 & 13\\ 5&|& 5 & 6 & 7 & 8 & 9 & 10& 11 & 12 & 13 & 14\\ 6&|& 6 & 7 & 8 & 9 & 10& 11 & 12 & 13 & 14 & 15\\ 7&|& 7 & 8 & 9 & 10& 11 & 12 & 13 & 14 & 15 & 16\\ 8&|& 8 & 9 & 10& 11 & 12 & 13 & 14 & 15 & 16 & 17\\ 9&|& 9 & 10& 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18\\ \end{matrix}\]
2.2 Addition examples
\[\begin{matrix} 123&=&1\times 10^2 + 2\times 10^1 + 3\times 10^0\\ 234&=&2\times 10^2 + 3\times 10^1 + 4\times 10^0\\ \hline 357&=&3\times 10^2 + 5\times 10^1 + 7\times 10^0\\ \end{matrix}\]
2.3 Addition with carrys
\[\begin{matrix} 789 &=& 7\times 10^2 + 8\times 10^1 + 9\times 10^0 \\ 678 &=& 6\times 10^2 + 7\times 10^1 + 8\times 10^0 \\ \hline & & 13\times 10^2 + 15\times 10^1 + 17\times 10^0 \\ \hline \left({1110\atop\ 357 }\right) &=& {1\times 10^3 + (3+1)\times 10^2 +\atop (5+1)\times 10^1 + 7\times 10^0} \\ \hline 1467 &=& 1\times 10^3 + 4\times 10^2 + 6\times 10^1 + 7\times 10^0 \\ \end{matrix}\]
3 Subtraction
\[\large\begin{eqnarray} a - b &\ne& b - a\\ 2-3 &\ne& 3 - 2\\ \square\square - \blacksquare\blacksquare\blacksquare &\ne& \square\square\square - \blacksquare\blacksquare\\ \blacksquare&\ne&\square\\ \end{eqnarray}\]
3.1 Subtraction Table
\[\tiny\begin{matrix} &|&0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ \hline 0&|&0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 1&|&-1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 2&|&-2 &-1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 3&|&-3 & -2& -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6\\ 4&|&-4 & -3 & -2& -1 & 0 & 1 & 2 & 3 & 4 & 5\\ 5&|&-5 &-4 & -3 & -2& -1 & 0 & 1 & 2 & 3 & 4\\ 6&|&-6 &-5 &-4 & -3 & -2& -1 & 0 & 1 & 2 & 3\\ 7&|&-7 &-6 &-5 &-4 & -3 & -2& -1 & 0 & 1 & 2\\ 8&|&-8 &-7 &-6 &-5 &-4 & -3 & -2& -1 & 0 & 1\\ 9&|&-9 &-8 &-7 &-6 &-5 &-4 & -3 & -2& -1 & 0\\ \end{matrix}\]
3.2 Subtraction example
\[\small\begin{matrix} 789 &=& 7\times 10^2 + 8\times 10^1 + 9\times 10^0\\ 123 &=& 1\times 10^2 + 2\times 10^1 + 3\times 10^0\\ \hline 666 &=& 6\times 10^2 + 6\times 10^1 + 6\times 10^0\\ \end{matrix}\]
3.3 Substraction with carry
\[\begin{matrix} 621 &=& 6 \times 10^2 + 2\times 10^1 + 1\times 10^0\\ 489 &=& 4\times 10^2 + 8\times 10^1 + 9\times 10^0\\ \hline & & 2 \times 10^2 -6\times 10^1 -8 \times 10^0\\ \hline & & {(2-1) \times 10^2 + (10-6-1)\times 10^1 +\atop (10-8) \times 10^0}\\ \hline 132 &=& 1 \times 10^2 + 3\times 10^1 + 2 \times 10^0\\ \end{matrix}\]
4 Multiplication
\[\large\begin{eqnarray} a \times b &=& b \times a\\ 2 \times 3 &=& 3 \times 2\\ \left(\begin{matrix}\square&\square\\ \square&\square\\ \square&\square\\ \end{matrix}\right) &=& \left(\begin{matrix}\square&\square&\square\\ \square&\square&\square\\ \end{matrix}\right)\\ \end{eqnarray}\]
4.1 Multiplication Table
\[\tiny\begin{matrix} &|& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0&|& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1&|& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 2&|& 0 & 2 & 4& 6 & 8 & 10 & 12 & 14 & 16 & 18 \\ 3&|& 0 & 3 & 6 & 9 & 12 & 15 & 18 & 21 & 24 & 27 \\ 4&|& 0 & 4 & 8 & 12 & 16 & 20 & 24 & 28 & 32 & 36 \\ 5&|& 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 \\ 6&|& 0 & 6 & 12 & 18 & 24 & 30 & 36 & 42 & 48 & 54 \\ 7&|& 0 & 7 & 14 & 21 & 28 & 35 & 42 & 49 & 56 & 63 \\ 8&|& 0 & 8 & 16 & 24 & 32 & 40 & 48 & 56 & 64 & 72 \\ 9&|& 0 & 9 & 18 & 27 & 36 & 45 & 54 & 63 & 72 & 81 \\ \end{matrix}\]
5 Division
\[\begin{eqnarray} Quotient &=& {Dividend \over Divisor}\\ 3\quad &=& \quad\frac{6}{2}\\ \left[\blacksquare\square\blacksquare\square\blacksquare\square\right]&\longrightarrow&\left[\begin{matrix}\left[\blacksquare\square\right]\\ \left[\blacksquare\square\right]\\ \left[\blacksquare\square\right]\\ \end{matrix}\right] \end{eqnarray}\]
5.1 Division methods
Long division
\[\begin{matrix} &\phantom{)\ }16\\ 27&\overline{)\ 459}\\ &\phantom{\ }\underline{27}\\ &\phantom{\ }189\\ &\phantom{\ }\underline{189}\\ & \phantom{\ 00}0\\ \end{matrix}\]
Factor analysis
\[\frac{459}{27} = \frac{9\times 51}{9\times 3} = \frac{17\times 3}{3} = 17\]
5.2 Divisibility
6 Exponent
\[\begin{matrix} 1^2 & 2^2 & 3^2 & 4^2 & 5^2\\ 1\times 1 & 2\times 2 & 3\times 3 & 4\times 4 & 5\times 5\\ \left[\square\right] & \left[\begin{matrix}\square\square\\ \square\square\\\end{matrix}\right] & \left[\begin{matrix}\square\square\square\\ \square\square\square\\ \square\square\square\\ \end{matrix}\right] & \left[\begin{matrix}\square\square\square\square\\ \square\square\square\square\\ \square\square\square\square\\ \square\square\square\square\\ \end{matrix}\right] & \left[\begin{matrix}\square\square\square\square\square\\ \square\square\square\square\square\\ \square\square\square\square\square\\ \square\square\square\square\square\\ \square\square\square\square\square\\ \end{matrix}\right]\\ \\ \end{matrix}\]
6.1 Exponent
\[\tiny\begin{matrix} 2^0 &=& 1 &=& 1 & | & 10^0 &=& 1&=& 1 \\ 2^1 &=& 2 &=& 2 & | & 10^1 &=& 10 &=& 10 \\ 2^2 &=& 2\cdot 2&=& 4 & | & 10^2 &=& 10\cdot 10&=& 100 \\ 2^3 &=& 2\cdot 2\cdot 2 &=& 8 & | & 10^3 &=& 10\cdot 10\cdot 10&=& 1,000 \\ 2^4 &=& 2\cdot 2\cdot 2\cdot 2&=& 16 & | & 10^4 &=& 10\cdot 10\cdot 10\cdot 10&=& 10,000 \\ 2^5 &=& 2\cdot 2\cdot 2\cdot 2\cdot 2&=& 32 & | & 10^5 &=& 10\cdot 10\cdot 10\cdot 10\cdot 10 &=& 100,000 \\ 2^6 &=& 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2 &=& 64 & | & 10^6 &=& 10\cdot 10\cdot 10\cdot 10\cdot 10 \cdot 10 &=& 1,000,000 \\ 2^7 &=& 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2&=& 128 & | & 10^7&=& 10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10 &=& 10,000,000 \\ 2^8 &=& 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2&=& 256 &| & 10^8 &=& 10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10 &=& 100,000,000 \\ \end{matrix}\]
7 Average
\[\bar x = \frac{1}{n}\sum_{i=1}^n x_i\]
\[\bar x = \frac{20+24+25+29+35}{5} = \frac{133}{5} = 26.6\]
7.1 Standard deviation
\[\small std\ dev = \sqrt{\frac{1}{n-1}\displaystyle\sum_{x=1}^5 (\bar x - x_i)^2}\]
\[\tiny\begin{matrix} Sample& Values& \bar x - x_i & (\bar x - x_i)^2\\ \hline 1 & 20& 6.6 & 43.56\\ 2 & 24& 2.6 & 6.76\\ 3 & 25& 1.6 & 2.56\\ 4 & 29& -2.4& 5.76\\ 5 & 35& -8.4& 70.56\\ \hline \sum & 133 & & 129.2\\ \bar x & 26.6& \sum/4 & 32.3\\ & & std\ dev& 5.68\\ \end{matrix} \]
8 Word problems
8.1 Solving Word Problems
- Read the problem carefully.
- Choose a strategy.
- Decide which basic operation(s) are relevant and then translate the words into mathematical symbols.
- Perform the operations.
- Check the solution to see if the answer is reasonable.
8.2 Basic math operations
\[\begin{matrix} + & Combining\\ - & Taking\ away\\ \times & Adding\ repeatedly\\ \div & Splitting\ up\\ \end{matrix}\]
8.3 Clue words
| \(+\) | \(-\) | \(\times\) | \(\div\) |
|---|---|---|---|
| add | subtract | multiply | divide |
| sum | difference | product | quotient |
| total | take away | times | over |
| plus | minus | double | split up |
| more | less | twice | fit into |
| increase | decrease | triple | per |
| gain | loss | of | goes into |
8.4 Example of clue words
| Phrase | Math symbol |
|---|---|
| The patient’s weight increased by 5 kgs. | \(+5\) |
| The number of unemployed people tripled. \(\times 3\) | |
| The area of the bedroom is 8 square feet less than of the kitchen. | \(- 8\) |
| The company’s earnings were split among the four partners. | \(\div 4\) |
8.5 Sample Problem 1
In retailing, the difference between the gross sales and customer returns and allowances is called the net sales. If a store’s gross sales were $2,538 and customer returns and allowances amounted to $388, what was the store’s net sales?
8.6 Sample Problem 2
The population of the United States in 1800 was 5,308,483. Ten years later, the population had grown to 7,239,881. During this period of time, did the country’s population double?
8.7 Sample Problem 3.
A delivery van travels 27 miles west, 31 miles east, 45 miles west, and 14 miles east. How far is the van from its starting point?
8.8 Sample Problem 4.
Recycling one aluminum can saves enough energy to run a television for three hours. The average American watches 3,048 hours of television a year. For a year, how many aluminum cans would it take to power a television for the average American?
8.9 Sample Problem 5.
A blue whale weighs about 300,000 pounds, and a great white shark weighs about 4,000 pounds. How many times the weight of a great white shark is the weight of a blue whale?
8.10 Sample Problem 6.
A sales representative flew from Los Angeles to Miami (2,339 miles), then to New York (1,092 miles), and finally back to LA (2,451 miles). How many total miles did he fly?
8.11 Sample Problem 7.
A movie fan installed shelves for his collection of 400 DVDs. If 36 DVDs fit on each shelf, how many shelves did he need to house his entire collection?
8.12 Sample Problem 8.
Two major naval disasters of the twentieth century involved the sinking of British ships—the Titanic and the Lusitania. The Titanic, which weighed about 93,000,000 pounds, was the most luxurious liner of its time; it struck an iceberg on its maiden voyage in 1912. The Lusitania, which weighed about 63,000,000 pounds, was sunk by a German submarine in 1915. How much heavier was the Titanic than the Lusitania?
PYU GE141 2022/1: GE 141 Mathematics for daily life [1]