Week 1 Real Numbers and Their Properties (Section 1.4: Multiplication and Division of Real Numbers)

Week 1 Objectives This week students will: Utilize proper math terminology in written explanations. Compute basic operations with signed numbers. Exemplify properties of real numbers. Execute simplification techniques on expressions and equations.

Meaning of multiplication To begin with, think of multiplication as adding a number certain times, that is repeated addition. Notation for multiplication can be *, Thus, 4 * 8 = 8 + 8 + 8 + 8, that is 8 repeatedly added four times. You can also look at it as 4 + 4 + 4 + 4+ 4 + 4+ 4 + 4, that is repeatedly adding 4 eight times. The answer is 32 in both the cases. A good video to watch is at http://www.youtube.com/watch?v=056TtC6DSyg To do multiplication, you need to know your times-table of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. One such table can be found at http://www.mathsisfun.com/tables.html and another interactive at http://illuminations.nctm.org/ActivityDetail.aspx?ID=155

Multiplication process Suppose you have to multiply 131 * 7. To do this multiplication, you can follow the following steps. These steps will also help you when you have to multiply more complex multiplications like 131 * 72 or 131*721 Step 1: Here 131 is known as multiplicand. 7 is the multiplier. Write down the multiplicand numbers vertically so that the unit, tens, hundreds places are all lined up. Step 2: Multiply the multiplicand and multiplier numbers in the unit’s place. If you get a single digit number, write it down in the unit’s place. If you get a two digit number, write down the unit place number in the unit place. Carry over the ten’s place number. Step 3: Multiply the numbers in the ten’s place. Whatever result you get, add to it the number that was carried over to the ten’s place. Step 4: Continue the process. A nice page explaining, step-by-step, the multiplication process is http://www.aaamath.com/mul48ax2.htm

Example 1 Step 1: 131 7 Step 2: Multiply the unit place numbers: 1 * 7 = 7. Write it down in the unit place: 131

Example 1-- continues Step 3: Multiply the 7 to the number in the tens place: 3* 7 = 21. Write down 1 below the ten’s place and carry 2 to the hundred’s place 2 131 7 1 7 Step 4: Multiply 7 to 1, the number in the hundred’s place: 1 * 7 = 7. Add 2 to the result that was carried over to get 9 9 1 7 Answer

Short quiz Complete the multiplication and check your answer with the given answer. 47 * 9 = 423 129 * 6 = 774

Multiplication of two whole numbers Multiply 72 * 37 To do this multiplication, break it up into two parts: First part: multiply 72 * 7 Second part: multiply 72 * 3 Then, add the results. While doing the addition, keep in mind that 3 was in hundred’s place. Thus, we will apply a trick to ensure the fact that 3 was in hundred’s place.

First part -- 72*7 1 72 7 4 (multiply 2* 7 = 14. Write down 4 and carry over 1) 50 4 (multiply 7*7 = 49. Add 1, that was carried over, to 49 to get 50).

Second part -- 72*3 72 3 6 (multiply 2* 3 = 6. Write down 6 in the unit place) 21 6 (multiply 7*3 = 21).

The Final Answer So, we got 72*7 = 504 and 72*3 = 216. Recall that 3 was in the hundred place. So, whatever result that you got by multiplying 72 with 3, add a zero to the end of the result. Thus, it becomes 72*3 = 2160 (the 0 is added). Now, add 504 + 2160 = 2664 and that is the final answer.

Quiz and multiplying three digit numbers You have to multiply 216 * 612. So, as taught in the previous slides, break up the multiplication. This is your quiz: do the multiplications like taught in the previous slides and check your answer with the given answers. First part: 216 *2= 432 Second part: 216*1 = 216 Third part: 216 *6 = 1296 Now, you have to get the final answer. So, again realize that 1 was in tens place. So, put a zero at the end of 216 to get 2160. 6 was in the hundreds place. So, put two zeros at the end of 1296 to get 129600. Add the numbers to get the final answer: 432 + 2160 + 129600 = 132192

Factors Factors of a product are the numbers that, when multiplied together, give us the product back. Thus, the factors of 10 are 2, 5, 10, 1 because 2*5 = 10 and 1*10 = 10 The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 because 1*24 = 24, 2*12 = 24, 4*6 = 24, 3*8 = 24, 2*3*4 = 24

Quiz Find all the factors of 48 Factors are 1, 2, 3, 4, 6, 8, 12, 24, 48

Properties of multiplication Distributive Property: the sum of two numbers times a third number is equal to the sum of each number multiplied to the third number: 2(3 + 4) = 2*3 + 2* 4 = 6 + 8 = 14 Multiplication Property of zero: any number multiplied to zero gives us a zero: 2 * 0 = 0, 7 * 0 = 0, 10 * 0 = 0 Multiplication property of 1: any number multiplied to one gives us back the number: 2 * 1 = 2, 4 * 1= 4 Commutative property: the order of the numbers in the product does not affect the result. Thus, 2*3 = 3*2 = 6 Associative property: Grouping does not affect the result: (2*3)*1 = 2(3*1) = 6

Vocabulary and various notations 3 times 5 = 3*5 = 15 3 multiplied to 5 = 3*5 = 15 Product of 3 and 5 = 3*5 = 15 3(5) = 15 5(3) = 15 3∙5 = 15 3*5 = 15 3 5 = 15

Meaning of division Division is the opposite of multiplication. When you slice up a pizza and distribute it among 4 hungry-eaters, you are dividing up the pizza into 4 equal parts. So, you can think of division as distributing a quantity among certain parts, separating a quantity into certain parts.

notations 4/2 = 2

vocabulary While doing division, we will across words like divisor, dividend, quotient and reminder. 4 1 Thus, 2 (divisor) goes into 5 (dividend) two times (quotient) and the remainder is 1 because 2* 2 + 1 = 5 Quotient Divisor Dividend Reminder

Division by one digit number The process is explained through the example: divide 84 by 7. Step 1: Write down the divisor and dividend using the symbol: Step 2: 7 goes into 8 only once. Thus, write 1 in the quotient place and 7 underneath 8 because 7*1 = 7: 7

Division by one digit number – Continues – Part 1 Step 3: Subtract 7 from 8, write down the answer and bring down 4 beside it: 7 1 4

Division by one digit number – Continues – Part 2 Step 4: decide how many times 7 goes into 14. The answer is 2. Thus write 2 in the quotient place and 7*2 = 14 underneath 7 1 4 14 Answer: 12 is the quotient, 0 is the remainder. Thus, 84/7 = 12

Division by two digit numbers Divide 1152/12 Follow the same process. Step 1: Since 12 is a two digit number, we are going to consider the first two digits of the dividend. The first two digits are 11. 11 is smaller than 12. So, we will consider the first three digits, 115. We will ask the question how many times 12 goes into 115? The answer is 9

Division by two digit number continues Step 2: 12*9 = 108. So, write 108 under 115 and subtract. Bring the 2, from the dividend, down. 108 72

Division by two digit number continues Step 3: how many times 12 goes into 72? Answer is 6 because 12*6 = 72. So, write 6 in the quotient, 72 down and subtract. 108 72 Answer is 1152/12 = 96

Division with remainder Divide 1153/12. Follow the same steps (note I am not explaining them here as you can consider it as a quiz) 108 73 72 1 Answer : 1153/12 = 96 with a remainder 1. You can check whether you have done the division correctly or not. Multiply 96 with 12 and add 1, you should get 1153, i. e, 1153 = 12*96 +

Division with zero In mathematics, dividing any number with zero is undefined, that is unacceptable. Thus, you can NEVER DO anything like 4/0 or 10/0.

Multiplication with negative numbers As we have discussed in week 1, multiplication is nothing but repeated addition. So, when we try to compute 5 * (-3), you can think of multiplication now as of repeated subtractions (or addition of negative numbers) as -3 + (-3) + (-3) + (-3) + (-3) = -15 Another example: (-4) * 2 = -4 + (-4) = -8

Rule (-a)*(-b) = a*b, that is negative of a multiplied to negative of b gives you a multiplied to b Example 1: -2 * -4 = 2 * 4 = 8 Example 2: -6 * -7 = 6 * 7 = 42 Multiply -2 * - 4 * -3 To do this multiplication, break it up into two parts: -2 *-4 = 8 Then, multiply 8 * -3 = -24 Example: -8 * -1 * -2 = 8 * -2 = -16

Quiz Multiply the following: -2 * -10 . Answer is 20

Important table to remember (while doing multiplication) Original numbers have The answer is Example Same signs Positive 2*3 = 6 Different signs Negative -2*3 = -6 2 * -3 = -6 -2 * -3 = 6

Division involving negative numbers Division involving negative numbers follow the following rules: Divisor and Dividend have The answer is Example Same signs Positive 6/2 = 3 Different signs Negative -6/2 = -3 6/-2 = -3 -6/-2 = 3 Example 1: 8/-2 = -4 Example 2: -16/-8 = 2 Example 3: -48/12 = -4

Quiz Do the following divisions: -30/15. The answer is -2