1 / 21

Section 1.3

Section 1.3. Operations with Integers. Multiplication and Division. Multiplication. Multiplication was originally introduced as a shortcut for repetitive addition

libra
Télécharger la présentation

Section 1.3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 1.3 Operations with Integers. Multiplication and Division.

  2. Multiplication Multiplication was originally introduced as a shortcut for repetitive addition In operation of multiplication both multiplied numbers are called factors or multiplicands. The result of multiplication is called a product. Factor 1 Factor 2 Product Multiplication could be denoted by a x sign or a ∙ dot or parenthesis ( ):

  3. Properties of Multiplication • Multiplication of any number by 0 or 0 by any number always results in 0 • Multiplication of any number by 1 or 1 by any number always results in an original number • Commutative Property of Multiplication The product of any two numbers a and b does not depend on the order in which they are multiplied • Associative Property of Multiplication The product of any three numbers a,b and c does could be found by grouping the numbers in any order of multiplication

  4. Example:

  5. Multiplication in Column When multiplying large numbers Multiplication in column is used. Align the right-most digits of each number and multiply the first factor by every digit of the second.

  6. Multiplication with Negatives So far we multiplied only positive integers – natural numbers. The result of multiplication of two positive numbers is always positive. What would happen if one of the factors is negative?

  7. Example: Multiply the following integers we used the fact that multiplication is the same as repetitive addition. We were not able to use repetitive addition, since we didn’t know how to add a number to itself times So we used the Multiplicative Property of Multiplication and switched the order of the factors.

  8. Example: To see what happens when we multiply two negative integers we should consider the following pattern: When the second factor decreases by 1 the product increases by 3, so according to this pattern the second factor decreased by 1 should become -1 while the product will become 3

  9. The Product of Two Integer Numbers • If both integers have the same sign (either both are positive or both are negative) their product is positive. • If the integers have different signs (one is positive, while the other is negative) their product is negative.

  10. Division Operation of division originated as splitting of some amount into equal quantities. Example: If we want to evenly split 20 apples between 4 people, how many apples would each person get? So each person will receive 5 apples.

  11. Division as a Repetitive Subtraction. Think about the same problem of equally sharing 20 apples between 4 people. • Start with giving each person 1 apple. • We shared 4 apples, now we have 16 apples left to share. • Give another apple to each person. Each person has 2 apples now and we are left with 12 apples. • Give one more apple to each person. Each person has 3 apples now and we are left with 8 apples. • Give another apple to each person. Each person has 4 apples now and we are left with 4 apples. • Give another apple to each person. Each person has 5 apples now and we are left with 0 or no apples. We are finished now since there are no more apples left and each person has 5 apples.

  12. Division In operation of division the number that is divided is called dividend, the number by which the dividend is divided is called divisor. The result of division is called the quotient Dividend Divisor Quotient Algebraically the quotient , where is that unique number c for which

  13. Example: Bag 771 apples so there are 3 apples in one bag.  How many bags are needed?You can start by putting 3 apples to one bag, which leaves you 768 apples. Then for each bag you subtract 3 apples and keep counting the bags you use, until you hit zero apples. 771 -3 -3 -3 -3 … keep subtracting 1bag 1bag1bag1bag … keep counting It just takes quite a long time, doesn't it?  Instead you can take a 'shortcut' and initially subtract 300 apples (taking 100 bags) or some other big multiple of 3. 771 -300 -300 -30 -30 -30 … 100bags 100bags 10bags 10bags10bags … Let's figure it out and keep count of the bags as we subtract (put in bags) the apples. 

  14. Division (continued) So total needed is bags to bag all the apples.  And it all went even - no apples left over!  In other words,

  15. Example: Split 23 apples between 5 people evenly. • If we split 20 apples between 5 people, each person will get 4 apples. • 3 apples will remain, since we cannot divide 3 apples between 5 people without breaking them. • So if we try to divide 23 by 5 we will get a 4 as a quotient, and 3 will be called a remainder.

  16. Long Division Divide two integers: Check:

  17. Long Division Divide two integers: Check:

  18. Long Division Divide two integers: Check:

  19. Division of Integers If we divide two positive integers, the result is positive. What would be the result of division of two negative integers or negative and positive integers?

  20. Example: Divide integer numbers: • since • since • since • since • undefined, since there is no number such that multiplied by 0 it will be equal to 5

  21. Properties of Division

More Related