Unit 1

Whole Numbers

PLACE VALUE

• The value of any digit depends on its place value

• Place value is based on multiples of 10 as follows:

UNITSTENSHUNDREDSTHOUSANDS TENTHOUSANDS

HUNDREDTHOUSANDSMILLIONS

2 , 6 7 8 , 9 3 2

382 can be written in expanded form as:

3 hundreds + 8 tens + 2 ones

EXPANDED FORM• Place value held by each digit can be

emphasized by writing the number in expanded form

( 3 100) ( 8 10) ( 2 1) or

ESTIMATING• Used when an exact mathematical answer

is not required

• A rough calculation is called estimating or approximating

• Mistakes can often be avoided when estimating is done before the actual calculation

• When estimating, exact values are rounded

ROUNDING

• Used to make estimates• Rounding Rules:

– Determine place value to which the number is to be rounded

– Look at the digit immediately to its right.• If digit to right is less than 5, replace that digit

and all following digits with zeros• If digit to right is 5 or more, add 1 to the digit in

the place to which you are rounding. Replace all following digits with zeros

ROUNDING EXAMPLES

• Round 612 to the nearest hundredSince 1 is less than 5, 6 remains unchanged

–Ans: 600

• Round 175,890 to the nearest ten thousand7 is in the ten thousands place value, so look

at 5Since 5 is greater than or equal to 5, change

7 to 8 and replace 5, 8, and 9 with zeros–Ans: 180,000

ROUNDING TO THE EVEN

• Many technical trades use a process of rounding to even

• Reduces bias when several numbers are added

ROUNDING TO THE EVEN• Rounding Rules:

– Determine place value to which the number is to be rounded

– This is the same as the previous method – The only difference is if the digit to the right

is 5 followed by all zeros, • Increase the digits at the place value by 1 if it is

an odd number (1, 3, 5, 7, or 9)• Do not change the digits place if it is an even

number (0, 2, 4, 6, 8)

• Round 4,250 to the nearest hundred2 is in the hundreds place so look at 5

5 is followed by zeros and 2 is an even number so drop the 5 and leave the 2– Ans: 4,200

• Round 673,500 to the nearest thousand3 is in the thousands place so look at 5

5 is followed by zeros and 3 is odd so change the 3 to a 4– Ans: 674,000

ROUNDING TO EVENS EXAMPLES

• The result of adding numbers is called the sum

• The plus sign (+) indicates addition

• Numbers can be added in any order

ADDITION OF WHOLE NUMBERS

• Commutative property of addition:– Numbers can be added in any order– Example: 2 + 4 + 3 = 3 + 4 + 2

• Associative property of addition:– Numbers can be grouped in any way and

the sum is the same– Example: (2 + 4) + 3 = 2 + (4 + 3)

PROPERTIES OF ADDITION

PROCEDURE FOR ADDING WHOLE NUMBERS

• Example: Add 763 + 619– Align numbers to be added as shown;

line up digits that hold the same place value

– Add digits holding the same place value, starting on the right, 9 + 3 = 12

– Write 2 in the units place value and carry the one

PROCEDURE FOR ADDING WHOLE NUMBERS

– Continue adding from right to left

– Therefore,

763 + 619 = 1,382

• Subtraction is the operation which determines the difference between two quantities

• It is the inverse or opposite of addition

• The minus sign (–) indicates subtraction

SUBTRACTION OF WHOLE NUMBERS

• The quantity subtracted is called the subtrahend

• The quantity from which the subtrahend is subtracted is called the minuend

• The result is the difference

SUBTRACTION OF WHOLE NUMBERS

PROCEDURE FOR SUBTRACTING WHOLE NUMBERS

• Example: Subtract 917 – 523

– Align digits that hold the same place value

– Start at the right and work left: 7 – 3 = 4

PROCEDURE FOR SUBTRACTING WHOLE NUMBERS

– Since 2 cannot be subtracted from 1, you need to borrow from 9 (making it 8) and add 10 to 1 (making it 11)

• Now, 11 – 2 = 9; 8 – 5 = 3; Therefore, 917 – 523 = 394

MULTIPLICATION OF WHOLE NUMBERS

• Multiplication is a short method of adding equal amounts

• There are many occupational uses of multiplication

• The times sign (×) is used to indicate multiplication

MULTIPLICATION OF WHOLE NUMBERS

• The number to be multiplied is called the multiplicand

• The number by which the multiplicand is multiplied is called the multiplier

• Factors are the numbers used in multiplying• The result of multiplying is called the product

PROPERTIES OF MULTIPLICATION

• Commutative property of multiplication:– Numbers can be multiplied in any order– Example: 2 x 4 x 3 = 3 x 4 x 2

• Associative property of multiplication:– Numbers can be grouped in any way and

the product is the same– Example: (2 x 4) x 3 = 2 x (4 x 3)

PROCEDURE FOR MULTIPLICATION

• Example: Multiply 386 × 7– Align the digits on the right

– First, multiply 7 by the units of the multiplicand; 7 ×6 = 42

– Write 2 in the units position of the answer

PROCEDURE FOR MULTIPLICATION

– Multiply the 7 by the tens of the multiplicand; 7 × 8 = 56

– Add the 4 tens from the product of the units; 56 + 4 = 60

– Write the 0 in the tens position of the answer

PROCEDURE FOR MULTIPLICATION

– Multiply the 7 by the hundreds of the multiplicand; 7 × 3 = 21

– Add the 6 hundreds from the product of the tens; 21 + 6 = 27

– Write the 7 in the hundreds position and the 2 in the thousands position

– Therefore, 386 × 7 = 2,702

DIVISION OF WHOLE NUMBERS

• In division, the number to be divided is called the dividend

• The number by which the dividend is divided is called the divisor

• The result is the quotient

• A difference left over is called the remainder

DIVISION OF WHOLE NUMBERS

• Division is the inverse, or opposite, of multiplication

• Division is the short method of subtraction

• The symbol for division is ÷

• Division can also be expressed in fractional form such as

• The long division symbol is

DIVISION WITH ZERO

• Zero divided by a number equals zero– For example: 0 ÷ 5 = 0

• Dividing by zero is impossible; it is undefined– For example: 5 ÷ 0 is not possible

PROCEDURE FOR DIVISION

• Example: Divide 4,505 ÷ 6‒ Write division problem with divisor

outside long division symbol and dividend within symbol

‒ Since, 6 does not go into 4, divide 6 into 45. 45 6 = 7; write 7 above the 5 in number 4505 as shown

‒ Multiply: 7 × 6 = 42; write this under 45‒ Subtract: 45 – 42 = 3

PROCEDURE FOR DIVISION‒ Bring down the 0

‒ Divide 30 6 = 5; write the 5 above the 0

‒ Multiply: 5 × 6 = 30; write this under 30

‒ Subtract: 30 – 30 = 0‒ Since 6 can not divide into 5,

write 0 in the answer above the 5. Subtract 0 from 5 and 5 is the remainder

‒ Therefore 4,505 6 = 750 r5

ORDER OF OPERATIONS• All arithmetic expressions must be

simplified using the following order of operations:1. Parentheses

2. Raise to a power or find a root

3. Multiplication and division from left to right

4. Addition and subtraction from left to right

ORDER OF OPERATIONS

• Example: Evaluate (15 + 6) ×3 – 28 ÷ 7

21 ×3 – 28 ÷ 7

63 – 4

63 – 4 = 59– Therefore: (15 + 6) ×3 – 28 ÷ 7 = 59

Do the operation in parentheses first (15 + 6 = 21)

Multiply and divide next (21 ×3 = 63) and (28 ÷ 7 = 4)

Subtract last

PRACTICAL PROBLEMS• A 5-floor apartment building has 8 electrical

circuits per apartment. There are 6 apartments per floor. How many electrical circuits are there in the building?

PRACTICAL PROBLEMS• Multiply the number of apartments per

floor times the number of electrical outlets

• Multiply the number of floors times the number of outlets per floor obtained in the previous step

• There are 240 outlets in the building