Critical Thinking Solutions: Number System | Mathematics (Maths) Class 9 PDF Download

 Page 1


S o l u t i o n s : N u m b e r S y s t e m C r i t i c a l T h i n k i n g ( C l a s s 9 )
1 Conceptual Understanding Solutions
Here are detailed solutions to the conceptual questions.
1.1 Exploring Rational and Irrational Num b ers
1. Can a n um b er b e b oth rational and irrational?
Solution: A n um b er cannot b e b oth rational and irrational. By definition, a rational
n um b er can b e expressed as
p
q
, where p and q are in tegers and q?= 0 . An irrational
n um b er cannot b e expressed in this form and has a non-terminating, non-rep eating
decimal expansion. Since these definitions are m utually exclusiv e, a n um b er cannot
satisfy b oth conditions sim ultaneously .
2. If
v
2 is irrational, is
v
2+2 rational or irrational?
Solution: Assume
v
2+2 is rational, sa y equal to
a
b
, where a and b are in tegers, b?= 0 .
Then:
v
2+2 =
a
b
=?
v
2 =
a
b
-2 =
a-2b
b
.
Since a and 2b are in tegers,
a-2b
b
is rational. This implies
v
2 is rational, whic h con-
tradicts the fact that
v
2 is irrational. Th us,
v
2+2 is irrational.
1.2 Real Num b ers and the Num b er Line
1. Plotting
v
3 and
v
5 on a n um b er line without a calculator.
Solution: T o plot
v
3 , use a geometric construction:
• Dra w a n um b er line wit h p oin t O at 0 and A at 1.
• A t A , dra w a p erp endicular line and mark p oin t B at a distance of 1 unit (forming
a righ t triangle OAB ).
• The h yp oten use OB =
v
1
2
+1
2
=
v
2 .
• Extend the construction b y marking a p oin t C at 1 unit from B along the p erp en-
dicular.
• The h yp oten use OC =
v
(
v
2)
2
+1
2
=
v
3 .
• Use a compass to t ransfer this length to the n um b er line.
F or
v
5 , extend the construction or estimate: since 2
2
= 4 and 3
2
= 9 ,
v
5˜ 2.236 lies
b et w een 2 and 3, closer to 2. Divide the in terv al [2, 3] in to 10 equal parts (eac h 0.1)
and estimate
v
5 around 2.2–2.3.
2. Wh y do real n um b ers form a con tin uous n um b er line?
Solution: Real n um b ers include all rational and irrational n um b ers, filling ev ery p oin t
on the n um b er line. The completeness prop ert y of real n um b ers ensures there are no
gaps: for an y t w o distinct real n um b ers, there exists another real n um b er b et w een them
(e.g., their a v erage). This densit y and completeness mak e the n um b er line con tin uous.
1
Page 2


S o l u t i o n s : N u m b e r S y s t e m C r i t i c a l T h i n k i n g ( C l a s s 9 )
1 Conceptual Understanding Solutions
Here are detailed solutions to the conceptual questions.
1.1 Exploring Rational and Irrational Num b ers
1. Can a n um b er b e b oth rational and irrational?
Solution: A n um b er cannot b e b oth rational and irrational. By definition, a rational
n um b er can b e expressed as
p
q
, where p and q are in tegers and q?= 0 . An irrational
n um b er cannot b e expressed in this form and has a non-terminating, non-rep eating
decimal expansion. Since these definitions are m utually exclusiv e, a n um b er cannot
satisfy b oth conditions sim ultaneously .
2. If
v
2 is irrational, is
v
2+2 rational or irrational?
Solution: Assume
v
2+2 is rational, sa y equal to
a
b
, where a and b are in tegers, b?= 0 .
Then:
v
2+2 =
a
b
=?
v
2 =
a
b
-2 =
a-2b
b
.
Since a and 2b are in tegers,
a-2b
b
is rational. This implies
v
2 is rational, whic h con-
tradicts the fact that
v
2 is irrational. Th us,
v
2+2 is irrational.
1.2 Real Num b ers and the Num b er Line
1. Plotting
v
3 and
v
5 on a n um b er line without a calculator.
Solution: T o plot
v
3 , use a geometric construction:
• Dra w a n um b er line wit h p oin t O at 0 and A at 1.
• A t A , dra w a p erp endicular line and mark p oin t B at a distance of 1 unit (forming
a righ t triangle OAB ).
• The h yp oten use OB =
v
1
2
+1
2
=
v
2 .
• Extend the construction b y marking a p oin t C at 1 unit from B along the p erp en-
dicular.
• The h yp oten use OC =
v
(
v
2)
2
+1
2
=
v
3 .
• Use a compass to t ransfer this length to the n um b er line.
F or
v
5 , extend the construction or estimate: since 2
2
= 4 and 3
2
= 9 ,
v
5˜ 2.236 lies
b et w een 2 and 3, closer to 2. Divide the in terv al [2, 3] in to 10 equal parts (eac h 0.1)
and estimate
v
5 around 2.2–2.3.
2. Wh y do real n um b ers form a con tin uous n um b er line?
Solution: Real n um b ers include all rational and irrational n um b ers, filling ev ery p oin t
on the n um b er line. The completeness prop ert y of real n um b ers ensures there are no
gaps: for an y t w o distinct real n um b ers, there exists another real n um b er b et w een them
(e.g., their a v erage). This densit y and completeness mak e the n um b er line con tin uous.
1
S o l u t i o n s : N u m b e r S y s t e m C r i t i c a l T h i n k i n g ( C l a s s 9 )
1.3 Decimal Expansions
1. Compare the decimal expansions of 1/7 and 1/6 .
Solution: T o determine if a decimal is terminating or non-terminating, factorize the
denominator after simplifying the fraction:
• F or1/6 , the denominator is6 = 2×3 . Since it has no factors other than 2 and 5, the
decimal ma y terminate, but compute: 1÷6 = 0.1666... , whic h is non-terminating
rep eating.
• F or 1/7 , the denominator is prime. Compute: 1÷7 = 0.142857142857... , whic h is
non-terminating rep eat ing.
A decimal terminates if the denominator (in lo w est terms) has only 2 and 5 as prime
factors.
2. Predict the decimal expansion of 5/12 .
Solution: The denominator 12 = 2
2
×3 . Since it includes a factor of 3, the decimal
is non-terminating rep eating. Compute: 5÷12 = 0.41666... , confirming a rep eating
decimal.
2 Analytical Problems Solutions
2.1 Problem 1: Comparing Num b ers
Arrange
v
2 , 3/2 , and p .
Solution: Use appro ximations:
v
2˜ 1.414 , 3/2 = 1.5 , p˜ 3.142 . Compare:
• 1.414 < 1.5 < 3.142 .
• Th us, the order is
v
2 < 3/2 < p .
T o confirm without decimals, note
v
2
2
= 2 , (3/2)
2
= 2.25 , and p
2
˜ 9.869 , so
v
2 < 3/2 .
Since p > 3 > 1.5 , the orde r holds.
2.2 Problem 2: Rationalizing Denominators
Simplify
v
5+
v
3
v
5-
v
3
.
Solution: Multiply n umerator and denominator b y the conjugate
v
5+
v
3 :
(
v
5+
v
3)(
v
5+
v
3)
(
v
5-
v
3)(
v
5+
v
3)
=
(
v
5+
v
3)
2
(
v
5)
2
-(
v
3)
2
=
5+2
v
15+3
5-3
=
8+2
v
15
2
= 4+
v
15.
Rationalizing simplifies computations b y eliminating irrational denominators. F or a+
v
b ,
m ultiply b y a-
v
b to get a rational denominator.
2.3 Problem 3: Exploring P atterns
Sequence: 1 ,
v
2 ,
v
3 , 2 ,
v
5 , …
1. P attern and Nature: The sequence alternates: rational (1 =
v
1 ), irrational (
v
2 ),
irrational (
v
3 ), rational (2 =
v
4 ), irrational (
v
5 ). P attern:
v
1,
v
2,
v
3,
v
4,
v
5,... ,
or
v
n for n = 1,2,3,... . Rational when n is a p erfect square; irrationa l otherwise.
2
Page 3


S o l u t i o n s : N u m b e r S y s t e m C r i t i c a l T h i n k i n g ( C l a s s 9 )
1 Conceptual Understanding Solutions
Here are detailed solutions to the conceptual questions.
1.1 Exploring Rational and Irrational Num b ers
1. Can a n um b er b e b oth rational and irrational?
Solution: A n um b er cannot b e b oth rational and irrational. By definition, a rational
n um b er can b e expressed as
p
q
, where p and q are in tegers and q?= 0 . An irrational
n um b er cannot b e expressed in this form and has a non-terminating, non-rep eating
decimal expansion. Since these definitions are m utually exclusiv e, a n um b er cannot
satisfy b oth conditions sim ultaneously .
2. If
v
2 is irrational, is
v
2+2 rational or irrational?
Solution: Assume
v
2+2 is rational, sa y equal to
a
b
, where a and b are in tegers, b?= 0 .
Then:
v
2+2 =
a
b
=?
v
2 =
a
b
-2 =
a-2b
b
.
Since a and 2b are in tegers,
a-2b
b
is rational. This implies
v
2 is rational, whic h con-
tradicts the fact that
v
2 is irrational. Th us,
v
2+2 is irrational.
1.2 Real Num b ers and the Num b er Line
1. Plotting
v
3 and
v
5 on a n um b er line without a calculator.
Solution: T o plot
v
3 , use a geometric construction:
• Dra w a n um b er line wit h p oin t O at 0 and A at 1.
• A t A , dra w a p erp endicular line and mark p oin t B at a distance of 1 unit (forming
a righ t triangle OAB ).
• The h yp oten use OB =
v
1
2
+1
2
=
v
2 .
• Extend the construction b y marking a p oin t C at 1 unit from B along the p erp en-
dicular.
• The h yp oten use OC =
v
(
v
2)
2
+1
2
=
v
3 .
• Use a compass to t ransfer this length to the n um b er line.
F or
v
5 , extend the construction or estimate: since 2
2
= 4 and 3
2
= 9 ,
v
5˜ 2.236 lies
b et w een 2 and 3, closer to 2. Divide the in terv al [2, 3] in to 10 equal parts (eac h 0.1)
and estimate
v
5 around 2.2–2.3.
2. Wh y do real n um b ers form a con tin uous n um b er line?
Solution: Real n um b ers include all rational and irrational n um b ers, filling ev ery p oin t
on the n um b er line. The completeness prop ert y of real n um b ers ensures there are no
gaps: for an y t w o distinct real n um b ers, there exists another real n um b er b et w een them
(e.g., their a v erage). This densit y and completeness mak e the n um b er line con tin uous.
1
S o l u t i o n s : N u m b e r S y s t e m C r i t i c a l T h i n k i n g ( C l a s s 9 )
1.3 Decimal Expansions
1. Compare the decimal expansions of 1/7 and 1/6 .
Solution: T o determine if a decimal is terminating or non-terminating, factorize the
denominator after simplifying the fraction:
• F or1/6 , the denominator is6 = 2×3 . Since it has no factors other than 2 and 5, the
decimal ma y terminate, but compute: 1÷6 = 0.1666... , whic h is non-terminating
rep eating.
• F or 1/7 , the denominator is prime. Compute: 1÷7 = 0.142857142857... , whic h is
non-terminating rep eat ing.
A decimal terminates if the denominator (in lo w est terms) has only 2 and 5 as prime
factors.
2. Predict the decimal expansion of 5/12 .
Solution: The denominator 12 = 2
2
×3 . Since it includes a factor of 3, the decimal
is non-terminating rep eating. Compute: 5÷12 = 0.41666... , confirming a rep eating
decimal.
2 Analytical Problems Solutions
2.1 Problem 1: Comparing Num b ers
Arrange
v
2 , 3/2 , and p .
Solution: Use appro ximations:
v
2˜ 1.414 , 3/2 = 1.5 , p˜ 3.142 . Compare:
• 1.414 < 1.5 < 3.142 .
• Th us, the order is
v
2 < 3/2 < p .
T o confirm without decimals, note
v
2
2
= 2 , (3/2)
2
= 2.25 , and p
2
˜ 9.869 , so
v
2 < 3/2 .
Since p > 3 > 1.5 , the orde r holds.
2.2 Problem 2: Rationalizing Denominators
Simplify
v
5+
v
3
v
5-
v
3
.
Solution: Multiply n umerator and denominator b y the conjugate
v
5+
v
3 :
(
v
5+
v
3)(
v
5+
v
3)
(
v
5-
v
3)(
v
5+
v
3)
=
(
v
5+
v
3)
2
(
v
5)
2
-(
v
3)
2
=
5+2
v
15+3
5-3
=
8+2
v
15
2
= 4+
v
15.
Rationalizing simplifies computations b y eliminating irrational denominators. F or a+
v
b ,
m ultiply b y a-
v
b to get a rational denominator.
2.3 Problem 3: Exploring P atterns
Sequence: 1 ,
v
2 ,
v
3 , 2 ,
v
5 , …
1. P attern and Nature: The sequence alternates: rational (1 =
v
1 ), irrational (
v
2 ),
irrational (
v
3 ), rational (2 =
v
4 ), irrational (
v
5 ). P attern:
v
1,
v
2,
v
3,
v
4,
v
5,... ,
or
v
n for n = 1,2,3,... . Rational when n is a p erfect square; irrationa l otherwise.
2
S o l u t i o n s : N u m b e r S y s t e m C r i t i c a l T h i n k i n g ( C l a s s 9 )
2. Next Three Num b ers: Con tin ue with
v
6 ,
v
7 ,
v
8 , as the sequence follo ws
v
n .
3. New Sequence: Example: 1,
v
2/2,
v
3/3,2/2,
v
5/5,... . P attern:
v
n/n . All terms
are irrational except when n is a p erfect square (e.g.,
v
4/4 = 1 ).
3 Real-W orld Application Solutions
3.1 A ctivit y: Designing a Num b er Line Mo del
Solution:
• Placemen t: Place rational n um b ers lik e 1/2 = 0.5 exactly . F or
v
2 , use the geometric
construction from Section 1.2.1. F or p , appro ximate at 3.14.
• Challenges: Irrational n um b ers ha v e non-rep eating decimals, making precise plotting
di?icult without to ols. Use appro ximations or constructions.
• F eedbac k: Ensure the mo del is clear with lab eled p oin ts and a consisten t scale.
3.2 Scenario: Budgeting with Decimals
Estimate items for ?100 at ?
v
2 , ?
v
3 , ?p .
Solution: Use appro ximations:
v
2˜ 1.414 ,
v
3˜ 1.732 , p˜ 3.142 . Divide budget:
•
v
2 : 100÷1.414˜ 70.72 , so 70 items.
•
v
3 : 100÷1.732˜ 57.74 , so 57 items.
• p : 100÷3.142˜ 31.83 , so 31 items.
Appro ximations simplify calculations but ma y sligh tly o v erestimate or underestimate
quan tities, affecting precision.
4 Higher-Order Thinking Skills (HOTS) Solutions
4.1 Pro of and Reasoning
Pro v e the sum of a rational and irrational n um b er is irrational.
Solution: Let r =
a
b
(rational, a,b?Z,b?= 0 ) and s (irrational). Assume r+s = t , where
t is rational. Then:
s = t-r = t-
a
b
=
tb-a
b
.
Since t is rational, sa y t =
c
d
, then s =
c
d
b-a
b
=
c-ad
db
, whic h is rational. This con tradicts s
b eing irrational. Th us, r+s is irrational. No coun terexample exists.
4.2 Exploring Irrational Num b ers
Is
v
2×
v
3 rational or irrational?
Solution: Compute:
v
2×
v
3 =
v
6 . Since 6 is not a p erfect square,
v
6 is irrational.
Generalization: The pro duct of t w o irrat ional n um b ers can b e rational (e.g.,
v
2×
v
2 = 2 )
or irrational (e.g.,
v
2×
v
3 =
v
6 ). It dep ends on whether the pro duct forms a p erfect
square or rational n um b er.
3
Page 4


S o l u t i o n s : N u m b e r S y s t e m C r i t i c a l T h i n k i n g ( C l a s s 9 )
1 Conceptual Understanding Solutions
Here are detailed solutions to the conceptual questions.
1.1 Exploring Rational and Irrational Num b ers
1. Can a n um b er b e b oth rational and irrational?
Solution: A n um b er cannot b e b oth rational and irrational. By definition, a rational
n um b er can b e expressed as
p
q
, where p and q are in tegers and q?= 0 . An irrational
n um b er cannot b e expressed in this form and has a non-terminating, non-rep eating
decimal expansion. Since these definitions are m utually exclusiv e, a n um b er cannot
satisfy b oth conditions sim ultaneously .
2. If
v
2 is irrational, is
v
2+2 rational or irrational?
Solution: Assume
v
2+2 is rational, sa y equal to
a
b
, where a and b are in tegers, b?= 0 .
Then:
v
2+2 =
a
b
=?
v
2 =
a
b
-2 =
a-2b
b
.
Since a and 2b are in tegers,
a-2b
b
is rational. This implies
v
2 is rational, whic h con-
tradicts the fact that
v
2 is irrational. Th us,
v
2+2 is irrational.
1.2 Real Num b ers and the Num b er Line
1. Plotting
v
3 and
v
5 on a n um b er line without a calculator.
Solution: T o plot
v
3 , use a geometric construction:
• Dra w a n um b er line wit h p oin t O at 0 and A at 1.
• A t A , dra w a p erp endicular line and mark p oin t B at a distance of 1 unit (forming
a righ t triangle OAB ).
• The h yp oten use OB =
v
1
2
+1
2
=
v
2 .
• Extend the construction b y marking a p oin t C at 1 unit from B along the p erp en-
dicular.
• The h yp oten use OC =
v
(
v
2)
2
+1
2
=
v
3 .
• Use a compass to t ransfer this length to the n um b er line.
F or
v
5 , extend the construction or estimate: since 2
2
= 4 and 3
2
= 9 ,
v
5˜ 2.236 lies
b et w een 2 and 3, closer to 2. Divide the in terv al [2, 3] in to 10 equal parts (eac h 0.1)
and estimate
v
5 around 2.2–2.3.
2. Wh y do real n um b ers form a con tin uous n um b er line?
Solution: Real n um b ers include all rational and irrational n um b ers, filling ev ery p oin t
on the n um b er line. The completeness prop ert y of real n um b ers ensures there are no
gaps: for an y t w o distinct real n um b ers, there exists another real n um b er b et w een them
(e.g., their a v erage). This densit y and completeness mak e the n um b er line con tin uous.
1
S o l u t i o n s : N u m b e r S y s t e m C r i t i c a l T h i n k i n g ( C l a s s 9 )
1.3 Decimal Expansions
1. Compare the decimal expansions of 1/7 and 1/6 .
Solution: T o determine if a decimal is terminating or non-terminating, factorize the
denominator after simplifying the fraction:
• F or1/6 , the denominator is6 = 2×3 . Since it has no factors other than 2 and 5, the
decimal ma y terminate, but compute: 1÷6 = 0.1666... , whic h is non-terminating
rep eating.
• F or 1/7 , the denominator is prime. Compute: 1÷7 = 0.142857142857... , whic h is
non-terminating rep eat ing.
A decimal terminates if the denominator (in lo w est terms) has only 2 and 5 as prime
factors.
2. Predict the decimal expansion of 5/12 .
Solution: The denominator 12 = 2
2
×3 . Since it includes a factor of 3, the decimal
is non-terminating rep eating. Compute: 5÷12 = 0.41666... , confirming a rep eating
decimal.
2 Analytical Problems Solutions
2.1 Problem 1: Comparing Num b ers
Arrange
v
2 , 3/2 , and p .
Solution: Use appro ximations:
v
2˜ 1.414 , 3/2 = 1.5 , p˜ 3.142 . Compare:
• 1.414 < 1.5 < 3.142 .
• Th us, the order is
v
2 < 3/2 < p .
T o confirm without decimals, note
v
2
2
= 2 , (3/2)
2
= 2.25 , and p
2
˜ 9.869 , so
v
2 < 3/2 .
Since p > 3 > 1.5 , the orde r holds.
2.2 Problem 2: Rationalizing Denominators
Simplify
v
5+
v
3
v
5-
v
3
.
Solution: Multiply n umerator and denominator b y the conjugate
v
5+
v
3 :
(
v
5+
v
3)(
v
5+
v
3)
(
v
5-
v
3)(
v
5+
v
3)
=
(
v
5+
v
3)
2
(
v
5)
2
-(
v
3)
2
=
5+2
v
15+3
5-3
=
8+2
v
15
2
= 4+
v
15.
Rationalizing simplifies computations b y eliminating irrational denominators. F or a+
v
b ,
m ultiply b y a-
v
b to get a rational denominator.
2.3 Problem 3: Exploring P atterns
Sequence: 1 ,
v
2 ,
v
3 , 2 ,
v
5 , …
1. P attern and Nature: The sequence alternates: rational (1 =
v
1 ), irrational (
v
2 ),
irrational (
v
3 ), rational (2 =
v
4 ), irrational (
v
5 ). P attern:
v
1,
v
2,
v
3,
v
4,
v
5,... ,
or
v
n for n = 1,2,3,... . Rational when n is a p erfect square; irrationa l otherwise.
2
S o l u t i o n s : N u m b e r S y s t e m C r i t i c a l T h i n k i n g ( C l a s s 9 )
2. Next Three Num b ers: Con tin ue with
v
6 ,
v
7 ,
v
8 , as the sequence follo ws
v
n .
3. New Sequence: Example: 1,
v
2/2,
v
3/3,2/2,
v
5/5,... . P attern:
v
n/n . All terms
are irrational except when n is a p erfect square (e.g.,
v
4/4 = 1 ).
3 Real-W orld Application Solutions
3.1 A ctivit y: Designing a Num b er Line Mo del
Solution:
• Placemen t: Place rational n um b ers lik e 1/2 = 0.5 exactly . F or
v
2 , use the geometric
construction from Section 1.2.1. F or p , appro ximate at 3.14.
• Challenges: Irrational n um b ers ha v e non-rep eating decimals, making precise plotting
di?icult without to ols. Use appro ximations or constructions.
• F eedbac k: Ensure the mo del is clear with lab eled p oin ts and a consisten t scale.
3.2 Scenario: Budgeting with Decimals
Estimate items for ?100 at ?
v
2 , ?
v
3 , ?p .
Solution: Use appro ximations:
v
2˜ 1.414 ,
v
3˜ 1.732 , p˜ 3.142 . Divide budget:
•
v
2 : 100÷1.414˜ 70.72 , so 70 items.
•
v
3 : 100÷1.732˜ 57.74 , so 57 items.
• p : 100÷3.142˜ 31.83 , so 31 items.
Appro ximations simplify calculations but ma y sligh tly o v erestimate or underestimate
quan tities, affecting precision.
4 Higher-Order Thinking Skills (HOTS) Solutions
4.1 Pro of and Reasoning
Pro v e the sum of a rational and irrational n um b er is irrational.
Solution: Let r =
a
b
(rational, a,b?Z,b?= 0 ) and s (irrational). Assume r+s = t , where
t is rational. Then:
s = t-r = t-
a
b
=
tb-a
b
.
Since t is rational, sa y t =
c
d
, then s =
c
d
b-a
b
=
c-ad
db
, whic h is rational. This con tradicts s
b eing irrational. Th us, r+s is irrational. No coun terexample exists.
4.2 Exploring Irrational Num b ers
Is
v
2×
v
3 rational or irrational?
Solution: Compute:
v
2×
v
3 =
v
6 . Since 6 is not a p erfect square,
v
6 is irrational.
Generalization: The pro duct of t w o irrat ional n um b ers can b e rational (e.g.,
v
2×
v
2 = 2 )
or irrational (e.g.,
v
2×
v
3 =
v
6 ). It dep ends on whether the pro duct forms a p erfect
square or rational n um b er.
3
S o l u t i o n s : N u m b e r S y s t e m C r i t i c a l T h i n k i n g ( C l a s s 9 )
4.3 Creativ e Problem Design
Solution: Question: If a and b are irrational, is a+b+ab rational or irrational?
Answ er: Assume a =
v
2 , b =
v
3 . Then:
a+b+ab =
v
2+
v
3+
v
2×
v
3 =
v
2+
v
3+
v
6.
Supp ose this is rational, sa y r . Then
v
6 = r-
v
2-
v
3 . Squaring b oth sides leads
to a con tradiction, as the left is irrational and the righ t in v olv es rational and irrational
terms mismatc hed. Th us, it’s irrational. This question encourages exploring op erations
on irrational n um b ers.
5 Self-Assessmen t and Reflection Solutions
1. Challenge: Answ ers v ary . Example: Irrational n um b ers are c hallenging due to their
non-rep eating decimals, making visualization di?icult.
2. Impro v emen t: These problems clarify the distinction b et w een rational and irrational
n um b ers and their prop erties through practical applications.
3. Relev ance: Num b er systems are vital in finance (e.g., precise calculations in banking),
measuremen ts (e.g., irrational n um b ers lik e p in engineering), and tec hnology (e.g.,
binary systems in computing).
4