unit 1 whole numbers
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Unit 1 Whole Numbers - PresentationTRANSCRIPT
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