Transcript PowerPoint Presentation 3: Signed Numbers

PRESENTATION 3
Signed Numbers
SIGNED NUMBERS
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In algebra, plus and minus signs are used to indicate both
operation and direction from a reference point or zero
Positive and negative numbers are called signed
numbers
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A positive number is indicated with either no sign or a plus sign (+)
A negative number is indicated with a minus sign (–)
Example: A Celsius temperature reading of 20 degrees above zero is
written as +20ºC or 20ºC; a temperature reading 20 degrees below zero
is written as –20ºC
ADDITION OF SIGNED
NUMBERS
• Procedure for adding two or more
numbers with the same signs
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Add the absolute values of the numbers
If all the numbers are positive, the sum is
positive
If all the numbers are negative, prefix a
negative sign to the sum
ADDITION OF SIGNED
NUMBERS
• Example: Add 9 + 5.6 + 2.1
o All the numbers have the same sign so
add and assign a positive sign to the
answer
9 + 5.6 + 2.1 = +16.7 or 16.7
ADDITION OF SIGNED
NUMBERS
• Example: Add (–6.053) + (–0.072) +
(–15.763) + (–0.009)
o All the numbers have the same sign (–) so add
and assign a negative sign to the answer
(–6.053) + (–0.072) + (–15.763) + (–0.009)
= –21.897
ADDITION OF SIGNED
NUMBERS
• Procedure for adding a positive and a
negative number:
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Subtract the smaller absolute value from
the larger absolute value
The answer has the sign of the number
having the larger absolute value
ADDITION OF SIGNED
NUMBERS
• Example: Add +10 and (–4)
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Different signs, so subtract and assign the
sign of the larger absolute value
10 – 4 = 6
Prefix the positive sign to the difference
(+10) + (– 4) = +6 or 6
ADDITION OF SIGNED
NUMBERS
• Example: Add (–10) and +4
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Different signs, so subtract and assign the
sign of larger absolute value
10 – 4 = 6
Prefix the negative sign to the difference
(–10) + (+ 4) = –6
ADDITION OF SIGNED
NUMBERS
• Procedure for adding combinations of two
or more positive and negative numbers:
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Add all the positive numbers
Add all the negative numbers
Add their sums, following the procedure for
adding signed numbers
ADDITION OF SIGNED
NUMBERS
• Example: Add (–12) + 7 + 3 + (–5) + 20
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Add all the positive numbers and all the
negative numbers
30 + (–17)
Add the sums using the procedure for adding
signed numbers
30 + (–17) = +13 or 13
SUBTRACTION OF SIGNED
NUMBERS
• Procedure for subtracting signed
numbers:
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Change the sign of the number subtracted
(subtrahend) to the opposite sign
Follow the procedure for addition of signed
numbers
SUBTRACTION OF SIGNED
NUMBERS
• Example: Subtract 8 from 5
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Change the sign of the subtrahend to the
opposite sign
8 to –8
Add the signed numbers
5 + (–8) = –3
SUBTRACTION OF SIGNED
NUMBERS
• Example: Subtract –10 from 4
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Change the sign of the subtrahend to the
opposite sign
–10 to 10
Add the signed numbers
4 + (10) = 14
MULTIPLICATION OF SIGNED
NUMBERS
• Procedure for multiplying two or more signed
numbers
o Multiply the absolute values of the numbers
 If all numbers are positive, the product is positive
o Count the number of negative signs
 An odd number of negative signs gives a negative product
 An even number of negative signs gives a positive product
MULTIPLICATION OF SIGNED
NUMBERS
• Example: Multiply 3(–5)
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Multiply the absolute values
Since there is an odd number of negative
signs (1), the product is negative
3(–5) = –15
MULTIPLICATION OF SIGNED
NUMBERS
• Example: Multiply (–3)(–1)(–2)(–
3)(–2)(–1)
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Multiply the absolute values
Since there is an even number of
negative signs (6), the product is positive
(–3)(–1)(–2)(–3)(–2)(–1) = +36 or 36
DIVISION OF SIGNED
NUMBERS
• Procedure for dividing signed numbers
o Divide the absolute values of the numbers
o Determine the sign of the quotient
 If both numbers have the same sign (both negative
or both positive), the quotient is positive
 If the two numbers have unlike signs (one positive
and one negative), the quotient is negative
DIVISION OF SIGNED
NUMBERS
• Example: Divide –20 ÷ (–4)
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Divide the absolute values
Since there is an even number of negative
signs (2), the quotient is positive
–20 ÷ (–4) = +5 or 5
DIVISION OF SIGNED
NUMBERS
• Example: Divide 24 ÷ (–8)
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Divide the absolute values
Since there is an odd number of negative
signs (1), the quotient is negative
24 ÷ (–8) = –3
POWERS OF SIGNED
NUMBERS
• Determining values with positive exponents
o Apply the procedure for multiplying signed
numbers to raising signed numbers to powers
 A positive number raised to any power is positive
 A negative number raised to an even power is positive
 A negative number raised to an odd power is negative
POWERS OF SIGNED
NUMBERS
• Example: Evaluate
4
2
• Since 2 is positive, the answer is
positive
4
2
= (2)(2)(2)(2) = +16 or 16
POWERS OF SIGNED
NUMBERS
• Example:
3
(–4)
• Since a negative number is raised to an
odd power, the answer is negative
3
(–4)
= (–4)(–4)(–4) = –64
NEGATIVE EXPONENTS
• Two numbers whose product is 1 are
multiplicative inverses or reciprocals of
each other
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For example:
A number with a negative exponent is equal to the
reciprocal of the number with a positive exponent:
POWERS OF SIGNED
NUMBERS
• Determining values with negative
exponents
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Invert the number (write its reciprocal)
Change the negative exponent to a positive
exponent
POWERS OF SIGNED
NUMBERS
• Example:
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–2
(–5)
Write the reciprocal of (–5)–2 and change
the negative exponent –2 to a positive
exponent +2
Simplify
ROOTS OF SIGNED NUMBERS
• A root of a number is a quantity that is
taken two or more times as an equal
factor of the number
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Roots are expressed with radical signs
An index is the number of times a root is to be
taken as an equal factor
The square root of a negative number has no
solution in the real number system
ROOTS OF SIGNED NUMBERS
• The expression
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is a radical
The 3 is the index and 64 is the radicand
Use the following chart to determine the sign of
a root based on the index and radicand
Index
Radicand
Root
Even
Positive (+)
Positive (+)
Even
Negative (–)
No Solution
Odd
Positive (+)
Positive (+)
Odd
Negative (–)
Negative (–)
ROOTS OF SIGNED NUMBERS
• Example: Determine the indicated
roots for the following problems:
5
4
81
32
COMBINED OPERATIONS
• The same order of operations applies
to terms with exponents as in arithmetic
• Parentheses
• Powers and roots
• Multiply and divide from left to right
• Add and subtract from left to right
COMBINED OPERATIONS
• Example: Evaluate 50 + (–2)[6 +
3
(–2) (4)]
50 + (–2)[6 + (–2)3(4)]
Powers or exponents first
= 50 + (–2)[6 + (–8)(4)]
Multiplication in []
= 50 + (–2)[6 + –32]
Evaluate the brackets
= 50 + (–2)(–26)
Multiply
= 50 + (52)
Add
50 + (–2)[6 + (–2)3(4)] = 102
SCIENTIFIC NOTATION
• In scientific notation, a number is
written as a whole number or
decimal between 1 and 10
multiplied by 10 with a suitable
exponent
SCIENTIFIC NOTATION
• Examples:
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In scientific notation, 146,000 is written as
1.46 × 105
In scientific notation, 0.00003 is written as
3 × 10–5
The number –3.8 × 10-4 is written as a
whole number as –0.00038
SCIENTIFIC AND ENGINEERING
NOTATION
• Example: Multiply (5.7 ×
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3
10 )(3.2
×
9
10 )
Multiply the decimals
5.7 × 3.2 = 18.24
Multiply the powers of 10s using the rules for exponents
(103)(109) = 1012
Combine both parts
18.24 × 1012