chapter 1 – whole numbers and integers instructor: dr.gehan shanmuganathan

CHAPTER 1 – Whole Numbers and Integers

Instructor:

Dr.Gehan Shanmuganathan

Business Math, Ninth EditionCheryl Cleaves, Margie Hobbs & Jeffrey Nobel

© 2012 Pearson Education, Inc.Upper Saddle River, NJ 07458 All Rights Reserved

Learning Outcomes

Read whole numbers.

Write whole numbers.

Round whole numbers.

Read and round integers.

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Read whole numbers

Our system of numbers, the decimal number system uses 10 symbols called digits:

– 0,1, 2, 3, 4, 5, 6, 7, 8, and 9.

Place-value system: a number system that determines the value of a digit by its positionin a number.

Place Value and Our Number SystemSection 1-1

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Read whole numbers

Beginning with the ones place on the right, the digits are grouped with three digits in each group.

– For example: 286,418,917

Each group is called a period.


HOW TO:



Understanding place value

Each period has a name and a ones place, a tens place, and a hundreds place.

In a number, the first period from the left mayhave fewer than three digits.

In many cultures, the periods are separated by commas.


HOW TO:



Understanding place value

Identify the period name of the leftmost group.

Read the three digit number from left to right.

Name the period.

4,693,107 would read four million six hundred ninety-three thousand one hundred seven.


HOW TO:



Exceptions…

Do not read or name a period that is all zeros.

34,000,892 would read thirty-four million, eight hundred ninety-two.

Do not name the units period (892).

000, 892 34,




When reading whole numbers, remember…

The period name will be read at each comma.

Period names are read in the singular:

– (“thousand” not “thousands”).

Hundreds is not a period name.

Do not say the word “and” when readingwhole numbers.

Calculator displays ordinarily do not show commas; insert them when writing the number.


HOW TO:



Write whole numbers

Begin recording digits from left to right.

Insert a comma at each period name.

Every period after the first period must havethree digits.

Insert zeros as necessary.


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An Example…

Eight million, nine hundred three thousand, four hundred twenty-two…

…is written 8,903,422.

8, million

903, thousand 422 (units)




Round whole numbers

Rounding to a specific place:

– Identify the place.• “nearest hundred”, for example.

– Look at the digit immediately to the right.• Is it 5 or higher? Round up.

• Is it 4 or lower? Specified digit stays the same.

– All digits to the right of the specified placebecome zeros.


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Examples…

Round to the nearest hundred:

4,856

10,527

234,567

8,648,078

234,600

4,900

10,500

8,648,100




Key Terms…

Addends

– The numbers being added.

Sum or total

– The answer or result of addition.




Key Terms…

Commutative property of addition

– Two or more numbers can be added in either order without changing the sum.

• Example: 8 + 3 = 3 + 8 = 11

Associative property of addition

– When more than two numbers are being added, the addends can be grouped by two at a time in any way.

• Example: 5 + (2 + 1) = (5 + 2) + 1 = 8




Read and round integers

In the business world we sometimes want to express numbers that are smaller than 0.

– These are referred to as negative numbers.

When the set of whole numbers is expanded to include negative numbers, this set is called theset of integers.


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Read and round integers

When reading integers:

– The rules are the same as for reading whole numbers.• State the word negative or minus to read a number

less than zero.

When rounding integers:

– The rules are the same as for roundingwhole numbers.


HOW TO:



An Example…

Read the number for the U.S. national debt:

- $11,936,042,802,503

Negative eleven trillion, nine hundred thirty-six billion, forty-two million, eight hundred two

thousand, five hundred three dollars.




An Example…

Round the previous exampleto the nearest trillion:

The answer is -$12 trillion.


-$11,936,042,802,503 The trillions digit is 1.

-$11,936,042,802,503 The digit to the right is 9.

9 is more than 5, so increase the 1, by 1, to get 2.Replace all digits to the right of 2 with zeros.

-$12,000,000,000,000



Learning Outcomes

Add and subtract wholenumbers.

Add and subtract integers.

Multiply and divide integers.

Apply the standard order ofoperations. to a series of operations.

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Add and subtract whole numbers

To add whole numbers, write the numbers in a vertical column, aligning digits according to their place values.

– Beginning with the ones column, add the place digits.

– Add, if necessary, to the tens column.

Repeat the operation, adding to the hundreds column, if necessary, until you have reached the farthest column of digits to the left.

Operations With Whole Numbers and IntegersSection 1-2

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Add the ones column.– Place 8 at the bottom of

the ones column.

– Carry the 2 to the tens column.

Place the 4 in the tens column.– Carry the 2.

Finish the operation.

1 3 6 7 8

5 4 6 2

4 5 7 0 9

+ __ __ __ 9 9

= 8496 4

Operations with Whole Numbers and IntegersSection 1-2

An Example…

2211

Answer: 64,948

Add



Estimating

Estimate: to find a reasonable approximateanswer for a calculation.

– Use estimating as a quick tool when an exactnumber is not required.

Round whole numbers to the place desiredfor an estimate.


HOW TO:



What was the week’s total to the nearest hundred?

$200 + $900 + $500 + $600 + $1000 = $3,200

Sales for last week’sconcession stand.

Monday: $219

Tuesday: $877

Wednesday: $455

Thursday: $614

Friday: $980

An Example…Operations with Whole Numbers and IntegersSection 1-2



Subtract whole numbers

When subtracting whole numbers, the order ofthe numbers is important.

– Therefore, subtraction is not commutative.

9 – 4 ≠ 4 – 9

Grouping in subtraction is important.

– Subtraction is not associative.

(8 - 3) -1 = 5 – 1 = 4 but

8 - (3 -1) = 8 - 2 = 6

4 ≠ 6


HOW TO:



Minuend– The beginning amount or number that

a second number is being subtracted from. Subtrahend

– The number being subtracted. Difference

– The answer or result of subtracting. Borrow

– Regroup digits in the minuend by borrowing 1from the digit to the left of the specified place,and adding 10 to the specified place.

Key Terms…Operations with Whole Numbers and IntegersSection 1-2



18

Subtract Borrow 1 from the ten column,

add 10 to the ones column.

– Subtract 8 from 13.

Borrow 1 from the hundreds column, add 10 to the tens column.


Borrow 1 from the thousands column.


1 2 9 3

- 5 9 8

= 596

1


8181111 2 9 3

Answer: 695



Subtract 128 from 1,345 by rounding each number to the nearest hundred to estimatethe difference.

– 128 would become 100.

– 1,345 would become 1,300.

The estimated difference would be 1,200.

Using rounding in subtractionOperations with Whole Numbers and IntegersSection 1-2

HOW TO:



To add two negative integers, add the numbers without regard to the signs.

– Assign a negative to the sum.

Last year Murphy’s Used Car Co. lost $23,000. This year they lost another $16,000.

What is the total loss?

-$23,000 + (-$16,000) = -$39,000

The two-year loss is -$39,000.

Add and subtract integersOperations with Whole Numbers and IntegersSection 1-2

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To add a positive and a negative integer, subtract the numbers without regard to the signs.

Look at the numbers without the signs.

– Choose the larger of these numbers;• Assign the sum the sign of the larger number.

Add and subtract integersOperations with Whole Numbers and IntegersSection 1-2

HOW TO:



Jeremy has a bank balance of $47,then writes a check for $89.

What is the new balance,including a $30 overdraft fee?

$47 + (-$89) = -$42

-$42 + (-$30) = -$72

The final balance is -$72.




Numbers can be multiplied in any order without affecting the result

– Commutative property of multiplication.

8 x 12 = 12 x 8

96 = 96

Multiply whole numbersOperations with Whole Numbers and IntegersSection 1-2

HOW TO:



Multiplicand– The number being multiplied.

Multiplier– The number multiplied by.

Factor– Each number involved in multiplication.




Product– The answer or result of multiplication.

Partial product– The product of one digit of the multiplier and

the entire multiplicand.




Multiply

7 9

x 2 3

2 3 7

1 5 8 _

1 8 1 7

Multiplicand

Multiplier

Partial product

PRODUCT

Partial product


1



418 x 107 = ?

Answer: 44,726

88 x 120 = ?

Answer: 10,560

348 x 27 = ?

Answer: 9,396

Examples to try without a calculator…Operations with Whole Numbers and IntegersSection 1-2



To multiply a negative and a positive integer, multiply the two integers without regard tothe signs.

– Assign a negative sign to the product.

What is the total loss generated fromselling 87 frames each for $2 below cost?

87 x (-$2) = -$174

The total loss is -$174.

Multiply integersOperations with Whole Numbers and IntegersSection 1-2

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To multiply two negative or two positive integers, multiply the two integers without regard to the signs.

– The product is positive.

What is the product of (-16)(-3)?

16 x 3 = 48

The product is positive and is 48.

Multiply integersOperations with Whole Numbers and IntegersSection 1-2

HOW TO:



Division is used to find the number of equal parts into which a whole quantity can be separated.

A $40 tip is shared equally among 5 servers.

How much does each server receive?

$40 ÷ 5 servers = $8 each.

Divide integersOperations with Whole Numbers and IntegersSection 1-2

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Dividend– The number being divided or the total quantity.

Divisor– The number to divide by.

Quotient– The answer or result of the operation.

Whole-number part of the quotient– The quotient without regard to its remainder.




Remainder of quotient

– A number that is smaller than the divisor thatremains after division is complete.

Partial dividend

– The part of the dividend that is being consideredat a given step of the process.

Partial quotient

– The quotient of the partial dividend and the divisor.




There will be a remainder if an amount is toosmall to be further divided by the divisor.

– For example: 152 ÷ 3 = 50 R 2

That amount may be expressed as…

– A remainder (R 2).

– A fraction.

– A decimal.

RemaindersOperations with Whole Numbers and IntegersSection 1-2

HOW TO:



– Is it 1? No.– Is it 12? Yes.

– 5 goes into 12 two times.Place the 2 above the 2 in the dividend.

STEP 1

Beginning with its leftmost digit, identifythe first group of digits of the dividendthat is larger than or equal to the divisor.

1235 ÷ 5 = ?

25 1 235,

5 1 235,


HOW TO:

MORE



STEP 2

Multiply 2 by the divisor. Place 10 underthe 12 and subtract. The result is 2.

25 1 235,

102

STEP 3

Bring down the following digit whichis 3, and divide 5 into 23.

34

STEP 4

The result is 4. Place the 4 directlyabove the 3 in the dividend.


1235 ÷ 5 = ?

HOW TO:

MORE



STEP 5

Multiply 4 by the divisor. Place 20 underthe 23 and subtract. The result is 3.

25 1 235,

10220

353

STEP 6

Bring down the last digit, which is 5,and divide 5 into 35. The result is 7.

54

STEP 7

Place 7 directly above the 5. Youhave finished and the answer is 247.

3

7


1235 ÷ 5 = ?

HOW TO:



Adams-Duke Realty Company estimates thatits losses for the year will be $36,000,000.

What is the average loss per month?

Answer: -$3,000,000

Divide the following: 63,500,000 ÷ 1,000

(mentally eliminate the ending zeros from both numbers)

Answer: 63,500

Examples to try without a calculator…Operations with Whole Numbers and IntegersSection 1-2



STEP 1

Perform all operations that are inside grouping symbols, such as parentheses.

STEP 2

Perform all multiplications and divisions as they appear from left to right.

STEP 3

Perform all additions and subtractions as theyappear from left to right.

Apply the standard orderof operations to a series of operations

Operations with Whole Numbers and Integers

1-2-5Section 1-2



15 – (4 + 7) = ?

Answer: 4

(75 + 50 + 35 + 90) ÷ 5 = ?

Answer: 50

45 - 4 x 9 = ?

Answer: 9

Examples to try…Operations with Whole Numbers and IntegersSection 1-2



EXERCISES SET A



PRACTICE TEST

chapter 1 – whole numbers and integers instructor: dr.gehan shanmuganathan

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