1.1 Real numbers and their properties

Measurements, counts, calculated values, and model outputs all belong to a common number system.

Real number. A number represented by a point on the real number line. The set of all real numbers is denoted by \(\mathbb{R}\).

The numbers used in measurement and calculation are real numbers, denoted by \(\mathbb{R}\). They include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. A rational number can be written as \(a/b\), where \(a\) and \(b\) are integers and \(b\ne0\). Irrational numbers, including \(\sqrt2\) and \(\pi\), cannot be written in that form.

Rational number. A number that can be written as \(a/b\), where \(a\) and \(b\) are integers and \(b\ne0\).

Worked Example: Classifying a measured value

A recorded turbidity reading of 2.75 can be written as \(275/100=11/4\). It is therefore rational and real. It is not an integer.

Try It. Classify \(-6\), \(\sqrt7\), and \(4/9\) using the smallest applicable number set.

Check Your Work \(-6\) is an integer, \(\sqrt7\) is irrational, and \(4/9\) is rational.

For real numbers \(a\), \(b\), and \(c\), addition and multiplication are commutative and associative.

Commutative, associate, and distributive laws.

Let \(a\), \(b\), and \(c\) be real numbers. Then

Commutative law \[a + b = b + a,\qquad a \times b = b \times a,\]

Associate law \[a + (b+c) = (a+b) + c,\qquad a \times (b \times c) = (a \times b) \times c,\]

Distributive law \[a \times (b+c)=a \times b + a \times c,\qquad (b + c) \times a = b \times a + c \times a.\]

Worked Example: Applying the distributive property

Simplify \(3(8-2y)\):

\[3(8-2y)=24-6y.\]

Every term inside the parentheses is multiplied by 3.

Try It. Simplify \(-4(2-3x)\).

Check Your Work \(-8+12x\).

The commutative property changes order, while the associative property changes grouping. Neither property permits arbitrary rearrangement of subtraction or division. For instance, \(8-3\) is not equal to \(3-8\). The properties of negatives follow from multiplication by \(-1\):

Properties of negative numbers.

For any real numbers \(a\), we have

\[(-1)a=-a,\qquad -(-a)=a,\qquad (-a)(-b)=ab,\]

\[-(a+b)=-a-b,\qquad -(a-b)=-a+b.\]

These identities are especially important when a negative sign appears outside a long formula. A reliable practice is to regard the sign as a factor of \(-1\) and distribute it explicitly.

Worked Example: Interpreting a signed water-level change

A reservoir level changes by \(-0.18\) m in the morning and by \(+0.07\) m in the afternoon. The net change is

\[-0.18+0.07=-0.11\ \text{m}.\]

The negative result means that the final level is 0.11 m below the initial level.

Try It. A wet well level changes by \(-0.24\) m and then by \(+0.31\) m. Find and interpret the net change.

Check Your Work \(-0.24+0.31=0.07\) m, so the final level is 0.07 m above the initial level.

1.1.1 Translating signs carefully

The negative sign can indicate a negative number, subtraction, or multiplication by \(-1\). These meanings are related but should not be blurred. In \(-3^2\), the exponent applies before the negative sign, so the value is \(-(3^2)=-9\). In \((-3)^2\), the parentheses make \(-3\) the base, so the value is 9. Parentheses are therefore essential when a negative value is substituted into a power.

Engineering sign conventions should be stated before calculation. A negative elevation can mean below a chosen datum. A negative change in tank depth can mean that depth decreased. Neither represents a negative physical length. The sign communicates direction relative to a reference.

Worked Example: Evaluating a negative squared correction

A calibration calculation contains \(-0.4^2\). Because the exponent is evaluated first,

\[-0.4^2=-(0.4^2)=-0.16.\]

If the intended base were \(-0.4\), the expression would need to be written \((-0.4)^2=0.16\).

Try It. Evaluate \(-1.5^2\) and \((-1.5)^2\).

Check Your Work \(-1.5^2=-2.25\), while \((-1.5)^2=2.25\).

Applied Problem: Tracking a reservoir level. A reservoir begins the day at 6.42 m relative to its operating datum. Withdrawal lowers the level by 0.38 m, inflow raises it by 0.21 m, and evaporation lowers it by another 0.03 m. Determine the final level and the net change.

Check Your Work The final level is \(6.42-0.38+0.21-0.03=6.22\) m. The net change is \(6.22-6.42=-0.20\) m, so the reservoir finishes 0.20 m below its starting level.

1.1.2 Practice Problems

  1. Simplify \(5(3-2x)\).
    Check Your Work \(15-10x\).
  2. Name the smallest applicable set for 0.
    Check Your Work The whole numbers.
  3. Is \(0.125\) rational? Explain.
    Check Your Work Yes. \(0.125=1/8\).
  4. Simplify \(-(7-y)\).
    Check Your Work \(-7+y\).
  5. A measured flow is 12.4 L/s. Identify the smallest familiar number set containing 12.4.
    Check Your Work It is rational because \(12.4=62/5\).
  6. A tank level falls 0.35 m and then rises 0.12 m. Find the net change.
    Check Your Work \(-0.35+0.12=-0.23\) m. The level falls by 0.23 m overall.
  7. A pressure correction is written \(-(1.8-0.6)\). Simplify it.
    Check Your Work \(-(1.8-0.6)=-1.2\).
  8. A flow measurement is \(\sqrt{50}\) L/s. Is the exact value rational or irrational?
    Check Your Work \(\sqrt{50}=5\sqrt2\), which is irrational.