Integers, Fractions, Decimals and Rational Numbers

What to remember

• Integers include negative numbers, zero and positive numbers; on the number line a number is larger than every number to its left. Same signs add and keep the sign; different signs subtract and take the sign of the larger absolute value.
• To add or compare fractions, make the denominators equal using the LCM; to divide by a fraction, multiply by its reciprocal.
• A rational number is p/q with q not 0 and p, q integers; every terminating or recurring decimal is rational. Between any two rational numbers there are infinitely many rational numbers.

Integers

Integers are ..., -3, -2, -1, 0, 1, 2, 3, ... The set is written Z (from the German "Zahlen").

Number line: positive integers lie to the right of zero, negative integers to the left. Of two integers, the one on the right is greater. So -2 > -5.

Absolute value (modulus): the distance of the number from zero. |-7| = 7 and |7| = 7. It is never negative.

Opposite (additive inverse): the opposite of 5 is -5. A number plus its opposite is 0.

Rules of operations:

Properties of integers:

• Closure: closed under addition, subtraction and multiplication, but not under division (1 / 2 is not an integer).
• Commutative: a + b = b + a; a x b = b x a. Subtraction and division are not commutative.
• Associative: (a + b) + c = a + (b + c); same for multiplication. Not for subtraction or division.
• Identity: 0 for addition; 1 for multiplication.
• Distributive: a x (b + c) = a x b + a x c.
• Division by zero is undefined. Zero divided by a non-zero number is 0.

Order of operations (BODMAS): Brackets, Orders (powers and roots), Division, Multiplication, Addition, Subtraction. Division and multiplication have equal rank and are done left to right; addition and subtraction also.

Example: 8 + 6 x (-2) = 8 - 12 = -4.

Teaching integers

• Use real situations: temperature above and below zero, a lift going up and down floors, profit and loss, money in hand and debt, height above and depth below sea level.
• Use the number line and coloured counters (red for negative, yellow for positive); a pair of one positive and one negative counter cancels to zero.
• Common errors: thinking -5 > -2 because 5 > 2; thinking subtraction always gives a smaller number; sign mistakes in subtraction of a negative.

Fractions

A fraction is a part of a whole. In a/b, a is the numerator and b the denominator (b is not 0).

Types:

Conversion: 1 3/4 = (1 x 4 + 3)/4 = 7/4; and 17/5 = 3 2/5 (17 = 3 x 5 + 2).

Lowest form: divide the numerator and denominator by their HCF. 12/18 = 2/3.

Comparing fractions: For like fractions, the larger numerator gives the larger fraction. For unlike fractions, convert to a common denominator using the LCM, or cross-multiply. Example: 3/4 and 5/6: LCM of 4 and 6 is 12; 9/12 < 10/12, so 3/4 < 5/6. For unit fractions, the larger denominator gives the smaller fraction (1/5 < 1/3).

Operations:

• Addition and subtraction: use the LCM as the common denominator. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
• Multiplication: multiply numerators and denominators. 2/3 x 3/4 = 6/12 = 1/2. The word "of" means multiplication: 1/2 of 20 = 10.
• Division: multiply by the reciprocal. 3/4 / 3/8 = 3/4 x 8/3 = 2.
• Reciprocal (multiplicative inverse) of a/b is b/a. The reciprocal of 0 does not exist; the reciprocal of 1 is 1.

Multiplying by a proper fraction gives a smaller result; dividing by a proper fraction gives a larger result.

Example: 2 1/2 + 1 1/3 = 5/2 + 4/3 = 15/6 + 8/6 = 23/6 = 3 5/6.

Decimals

A decimal shows a fraction with denominator 10, 100, 1000 and so on. Place values after the decimal point: tenths, hundredths, thousandths. In 4.375: 3 is in the tenths place, 7 in hundredths and 5 in thousandths.

• Adding zeros to the right of the last decimal digit does not change the value: 2.5 = 2.50.
• Comparing: compare the whole part first, then the tenths, hundredths, and so on. 0.8 > 0.75 because 8 tenths > 7 tenths.
• Addition and subtraction: line up the decimal points.
• Multiplication: multiply as whole numbers, then place the point so that the decimal places in the answer equal the sum of the decimal places in the factors. 0.3 x 0.4 = 0.12.
• Multiplying by 10, 100, 1000: move the decimal point to the right by 1, 2, 3 places. Dividing: move to the left.
• Division by a decimal: multiply both numbers by a power of 10 to make the divisor a whole number. 4.5 / 0.5 = 45 / 5 = 9.

Fraction to decimal: divide the numerator by the denominator. 3/8 = 0.375. 1/4 = 0.25; 1/5 = 0.2; 1/8 = 0.125.

Decimal to fraction: 0.25 = 25/100 = 1/4.

Types of decimals

A fraction in lowest form p/q has a terminating decimal if the prime factors of q are only 2 and/or 5. So 7/20 terminates (20 = 2^2 x 5), but 1/3 and 5/12 recur.

Converting a recurring decimal to a fraction: 0.333... = 1/3. Let x = 0.666...; 10x = 6.666...; subtract: 9x = 6, so x = 2/3.

Rational numbers

A rational number is a number that can be written as p/q where p and q are integers and q is not 0. The set is written Q (from "quotient").

• Every integer is rational (5 = 5/1). Every fraction and every terminating or recurring decimal is rational.
• Zero is rational (0 = 0/5).
• Standard form: the denominator is positive and p, q have no common factor except 1. -6/-8 = 3/4 in standard form; 4/-6 = -2/3.
• Equivalent rational numbers: multiply or divide numerator and denominator by the same non-zero number.
• Positive rational: numerator and denominator have the same sign. Negative rational: opposite signs.
• Comparing: make the denominators positive and equal and compare the numerators. -2/3 and -3/4: -8/12 and -9/12, so -2/3 > -3/4.
• Additive inverse of p/q is -p/q. Multiplicative inverse of p/q is q/p (p not 0).
• Rational numbers between two rational numbers: there are infinitely many. One way is to find the mean: between 1/4 and 1/2 is (1/4 + 1/2)/2 = 3/8. Another way is to convert to a larger common denominator: between 1/5 and 2/5, we get 5/25 and 10/25, so 6/25, 7/25, 8/25, 9/25 lie between them.
• On the number line, each rational number has a point, but not every point on the line is a rational number (irrational numbers also lie there).

Properties of rational numbers: closed under addition, subtraction and multiplication, and (for non-zero divisors) division; commutative and associative for addition and multiplication; 0 is the additive identity and 1 the multiplicative identity; distributive property holds.

Number system hierarchy

Natural numbers are a part of whole numbers; whole numbers are a part of integers; integers are a part of rational numbers; rational numbers are a part of real numbers. N is in W, W is in Z, Z is in Q, Q is in R.

Percentage connection and conversions

Percent means "out of 100". 1/2 = 50%; 1/4 = 25%; 3/4 = 75%; 1/5 = 20%; 1/8 = 12.5%. To convert a decimal to percent, multiply by 100; 0.35 = 35%.

Worked examples

• 1. Evaluate (-12) + 7 - (-5): = -12 + 7 + 5 = 0.
• 2. Find 2/3 of 45: 2/3 x 45 = 30.
• 3. A rod 7 1/2 m long is cut into pieces of 3/4 m. Number of pieces = 15/2 / 3/4 = 15/2 x 4/3 = 10.
• 4. Write 0.125 as a fraction: 125/1000 = 1/8.
• 5. Subtract 3/4 from 5/6: 10/12 - 9/12 = 1/12.

Teaching fractions and decimals

• Introduce a fraction through paper folding, a roti or chocolate bar shared equally, and set of objects (3 of 4 marbles).
• Use fraction strips, circles and number lines; show equivalence by folding.
• Teach decimals with money (rupees and paise), measurement of length (metres and centimetres) and place value charts.
• Common errors: adding numerators and denominators (1/2 + 1/3 = 2/5 is wrong); thinking 0.25 > 0.3 because 25 > 3; thinking a bigger denominator gives a bigger fraction.

Exam traps

• Adding fractions: add the numerators only after the denominators are made equal.
• -3 is smaller than -2; the number with the larger absolute value is smaller among negatives.
• Division by zero is undefined; zero divided by a non-zero number is zero.
• Subtraction and division are not commutative on integers or rational numbers.
• The reciprocal of 0 does not exist; the reciprocal of -1 is -1.
• 0.5 is a terminating decimal; 0.333... is recurring; both are rational.
• Integers are not closed under division; rational numbers (excluding division by 0) are.
• Decimal multiplication: count the decimal places of both factors and add them.

One-liners

• 1. Every integer is a rational number.
• 2. The absolute value of a number is never negative.
• 3. The product of two negative integers is positive.
• 4. Division by zero is undefined.
• 5. Integers are not closed under division.
• 6. The reciprocal of a/b is b/a.
• 7. The additive identity is 0; the multiplicative identity is 1.
• 8. A fraction terminates in decimal form if the denominator has only 2 and 5 as prime factors.
• 9. 1/8 = 0.125.
• 10. Between two rational numbers there are infinitely many rational numbers.
• 11. pi is irrational, not rational.
• 12. BODMAS gives the order of operations.

Practice questions

1. The value of (-9) + 5 is

   1. -4
   2. 4
   3. -14
   4. 14
   Answer

   A. -4

   Different signs: 9 - 5 = 4, with the sign of the larger absolute value (negative).

2. The value of (-3) x (-4) is

   1. 7
   2. -7
   3. 12
   4. -12
   Answer

   C. 12

   The product of two negatives is positive.

3. The value of 3 - (-4) is

   1. -1
   2. 7
   3. -7
   4. 1
   Answer

   B. 7

   Subtracting -4 is adding 4: 3 + 4 = 7.

4. The value of 8 + 6 x (-2) is

   1. 28
   2. 4
   3. -28
   4. -4
   Answer

   D. -4

   Multiply first: 6 x (-2) = -12; then 8 - 12 = -4.

5. The absolute value of -7 is

   1. 7
   2. 0
   3. -7
   4. 1/7
   Answer

   A. 7

   Absolute value is the distance from zero and is never negative.

6. Which is greater, -2 or -5?

   1. -5
   2. Cannot be said
   3. They are equal
   4. -2
   Answer

   D. -2

   -2 lies to the right of -5 on the number line.

7. The additive inverse of -8 is

   1. 1/8
   2. -1/8
   3. 8
   4. 0
   Answer

   C. 8

   A number plus its additive inverse is 0: -8 + 8 = 0.

8. The set of integers is not closed under

   1. Division
   2. Addition
   3. Multiplication
   4. Subtraction
   Answer

   A. Division

   1 divided by 2 is not an integer.

9. The value of (-12) + 7 - (-5) is

   1. 10
   2. 0
   3. -10
   4. -24
   Answer

   B. 0

   -12 + 7 + 5 = 0.

10. The value of (-36) / (-9) is

    1. -4
    2. 27
    3. 4
    4. -27
    Answer

    C. 4

    The quotient of two negative integers is positive: 36 / 9 = 4.

11. The temperature falls from 5 degrees to -3 degrees. The fall in temperature is

    1. 3 degrees
    2. 2 degrees
    3. -8 degrees
    4. 8 degrees
    Answer

    D. 8 degrees

    5 - (-3) = 8.

12. How many integers lie between -3 and 4?

    1. 5
    2. 6
    3. 8
    4. 7
    Answer

    B. 6

    The integers are -2, -1, 0, 1, 2, 3.

13. The value of (-5) x (-4) + (-20) / 4 is

    1. 25
    2. -25
    3. 15
    4. -15
    Answer

    C. 15

    20 + (-5) = 15.

14. The sum of 1/2 and 1/3 is

    1. 5/6
    2. 2/5
    3. 2/6
    4. 1/5
    Answer

    A. 5/6

    1/2 + 1/3 = 3/6 + 2/6 = 5/6.

15. Two-thirds of 45 is

    1. 20
    2. 35
    3. 15
    4. 30
    Answer

    D. 30

    2/3 x 45 = 30.

16. The value of 3/4 divided by 3/8 is

    1. 2
    2. 1/2
    3. 9/32
    4. 4
    Answer

    A. 2

    3/4 x 8/3 = 2.

17. The value of 5/6 - 3/4 is

    1. 2/12
    2. 1/2
    3. 2/10
    4. 1/12
    Answer

    D. 1/12

    10/12 - 9/12 = 1/12.

18. The value of 2 1/2 + 1 1/3 is

    1. 3 2/5
    2. 3 5/6
    3. 4 1/6
    4. 3 1/3
    Answer

    B. 3 5/6

    5/2 + 4/3 = 15/6 + 8/6 = 23/6 = 3 5/6.

19. A rod 7 1/2 m long is cut into pieces of 3/4 m. The number of pieces is

    1. 5
    2. 12
    3. 10
    4. 9
    Answer

    C. 10

    15/2 divided by 3/4 = 15/2 x 4/3 = 10.

20. The improper fraction 17/5 as a mixed number is

    1. 2 3/5
    2. 3 2/5
    3. 4 2/5
    4. 3 1/5
    Answer

    B. 3 2/5

    17 = 3 x 5 + 2.

21. Which fraction is the greatest?

    1. 3/4
    2. 2/3
    3. 7/12
    4. 5/6
    Answer

    D. 5/6

    In twelfths: 10, 9, 8 and 7; so 5/6 is greatest.

22. The reciprocal of -3/7 is

    1. 3/7
    2. 7/3
    3. -7/3
    4. -3/7
    Answer

    C. -7/3

    The reciprocal of a/b is b/a, keeping the sign.

23. The value of 0.3 x 0.4 is

    1. 0.12
    2. 0.012
    3. 0.7
    4. 1.2
    Answer

    A. 0.12

    3 x 4 = 12, with two decimal places: 0.12.

24. 3/8 expressed as a decimal is

    1. 0.375
    2. 0.38
    3. 0.35
    4. 0.425
    Answer

    A. 0.375

    3 divided by 8 = 0.375.

25. The value of 4.5 divided by 0.5 is

    1. 0.9
    2. 2.25
    3. 90
    4. 9
    Answer

    D. 9

    Multiply both by 10: 45 / 5 = 9.

26. Which is the greatest decimal?

    1. 0.75
    2. 0.8
    3. 0.078
    4. 0.708
    Answer

    B. 0.8

    Compare the tenths: 8 is the largest.

27. The decimal 0.125 as a fraction is

    1. 1/5
    2. 125/10
    3. 1/8
    4. 1/4
    Answer

    C. 1/8

    125/1000 = 1/8.

28. Which fraction has a terminating decimal?

    1. 7/20
    2. 2/7
    3. 5/12
    4. 1/3
    Answer

    A. 7/20

    20 = 2^2 x 5, so the decimal terminates (0.35).

29. The decimal form of 2/3 is

    1. Non-terminating non-recurring
    2. Non-terminating recurring
    3. Terminating
    4. Irrational
    Answer

    B. Non-terminating recurring

    2/3 = 0.666..., a repeating decimal.

30. The value of 0.0345 x 1000 is

    1. 345
    2. 3.45
    3. 0.345
    4. 34.5
    Answer

    D. 34.5

    Multiplying by 1000 moves the point three places right.

31. The standard form of 4/-6 is

    1. 4/6
    2. 2/3
    3. -2/3
    4. -4/6
    Answer

    C. -2/3

    The denominator must be positive; divide by 2: -2/3.

32. A rational number between 1/4 and 1/2 is

    1. 3/8
    2. 1/8
    3. 3/4
    4. 5/8
    Answer

    A. 3/8

    The mean (1/4 + 1/2)/2 = 3/8.

33. Which statement is correct?

    1. pi is rational
    2. Every rational number is an integer
    3. Zero is not rational
    4. Every integer is a rational number
    Answer

    D. Every integer is a rational number

    Any integer n can be written as n/1.

34. Which of the following is irrational?

    1. 0.75
    2. pi
    3. -5
    4. 1/3
    Answer

    B. pi

    pi is non-terminating and non-recurring.

35. Which rational number is greater?

    1. They are equal
    2. Cannot be compared
    3. -2/3
    4. -3/4
    Answer

    C. -2/3

    -8/12 > -9/12.

36. A teacher uses red and yellow counters for negative and positive integers, where a red-yellow pair makes zero. This activity best teaches

    1. Adding integers
    2. Place value
    3. Decimal fractions
    4. Prime factors
    Answer

    A. Adding integers

    A pair of opposite counters cancels to zero, showing addition of integers.

37. A child writes 1/2 + 1/3 = 2/5. The error is that the child

    1. Divided instead of added
    2. Used the LCM correctly
    3. Changed to decimals
    4. Added numerators and denominators separately
    Answer

    D. Added numerators and denominators separately

    Fractions need a common denominator before adding.

38. Consider the statements about division. 1. Division by zero is undefined. 2. Zero divided by a non-zero number is zero.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are correct.

39. Consider the statements about integers. 1. Subtraction is commutative on integers. 2. Integers are closed under subtraction.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    3 - 5 is not equal to 5 - 3.

40. Consider the statements about numbers. 1. Every integer is a rational number. 2. Every rational number is an integer.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    1/2 is rational but not an integer.

41. Consider the statements about decimals. 1. 0.333... is a rational number. 2. A non-terminating non-recurring decimal is a rational number.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Such decimals are irrational.

42. Consider the statements about positive numbers. 1. Multiplying by a proper fraction gives a smaller number. 2. Dividing by a proper fraction gives a larger number.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    For example, 8 x 1/2 = 4 and 8 / (1/2) = 16.

43. Consider the statements about reciprocals and order. 1. The reciprocal of 0 is 0. 2. -5 is greater than -2.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    D. Neither 1 nor 2

    The reciprocal of 0 does not exist, and -2 is greater than -5.

44. Which rational number has no multiplicative inverse?

    1. 1
    2. 0
    3. -1
    4. 1/2
    Answer

    B. 0

    The reciprocal of 0 would need division by zero.

45. Match the fraction with its type. 1. 3/5 2. 7/4 3. 1 3/4

    1. 1 - mixed; 2 - improper; 3 - proper
    2. 1 - proper; 2 - improper; 3 - mixed
    3. 1 - proper; 2 - mixed; 3 - improper
    4. 1 - improper; 2 - proper; 3 - mixed
    Answer

    B. 1 - proper; 2 - improper; 3 - mixed

    3/5 is proper, 7/4 is improper, 1 3/4 is a mixed number.