2.1 Place Value, Integers & Number Sense

Place Value: How Numbers Are Built

Every digit in a number has a value based on its position. Moving left from the ones place, each position is 10 times larger; moving right past the decimal point, each position is 10 times smaller.

Worked example. In 4,072.356:

• 4 is in the thousands place → 4,000
• 0 is in the hundreds place → 0
• 7 is in the tens place → 70
• 2 is in the ones place → 2
• 3 is in the tenths place → 0.3
• 5 is in the hundredths place → 0.05
• 6 is in the thousandths place → 0.006

So 4,072.356 = 4,000 + 70 + 2 + 0.3 + 0.05 + 0.006.

Reading and Writing Numbers

To write a number in words, group the digits to the left of the decimal in sets of three (ones, thousands, millions). Read each group as a normal number, then say the group name.

• 12,045 → "twelve thousand forty-five"
• 0.025 → "twenty-five thousandths"
• 3,600,108 → "three million six hundred thousand one hundred eight"

A common trap: the digit 0 in the hundreds place of 4,072 is skipped when reading aloud, but the place itself still matters for the value of the number.

Comparing and Ordering Numbers

To compare numbers, align them by place value (line up the decimal points) and compare digit by digit from the leftmost place.

Worked example. Order 0.45, 0.405, and 0.45 from least to greatest.

1. Pad with trailing zeros: 0.450, 0.405, 0.450.
2. Compare tenths: all are 4. Compare hundredths: first and third have 5, second has 0.
3. Least is 0.405. The other two tie at 0.450, so 0.45 = 0.450.

Trailing zeros after a decimal do not change value: 0.45 = 0.450 = 0.4500.

Integers

An integer is any whole number and its opposite, plus zero: ..., -3, -2, -1, 0, 1, 2, 3, ...

The opposite of a number is the same distance from zero on the other side. The opposite of 6 is -6; the opposite of -6 is 6.

The absolute value of a number is its distance from zero, always non-negative. |-6| = 6, |6| = 6, |0| = 0.

On a number line, numbers increase to the right and decrease to the left. A temperature of -2°C is to the left of 3°C, so -2 < 3.

Adding and Subtracting Signed Numbers

Rules:

• Same signs: add the magnitudes, keep the sign. (-4) + (-7) = -11.
• Different signs: subtract the smaller magnitude from the larger, keep the sign of the larger. (-4) + 9 = 5; 4 + (-9) = -5.
• Subtracting: change subtraction to addition by taking the opposite of the second number, then follow addition rules. 3 - (-5) = 3 + 5 = 8.

Common trap — the double negative. Subtracting a negative is adding a positive:

• 10 - (-4) = 10 + 4 = 14, not 6.
• -3 - (-7) = -3 + 7 = 4.

Nursing example. A patient's temperature moves from 1.2°C above baseline to 0.8°C below baseline. The change is (-0.8) - (1.2) = -2.0°C, a drop of 2.0 degrees.

Multiplying and Dividing Signed Numbers

Sign rules:

• Same signs → positive result. (-4) × (-3) = 12; (-12) ÷ (-4) = 3.
• Different signs → negative result. (-4) × 3 = -12; 12 ÷ (-4) = -3.
• Zero times anything is 0.

With more than two factors, count the negatives: an even number of negatives gives a positive, an odd number gives a negative. (-2)(-3)(-1) = -6 (three negatives → negative).

Nursing example. A weight change of -0.4 kg per day for 3 days: (-0.4) × 3 = -1.2 kg total change.

Number Sense Quick Checks

• Adding a negative always decreases the value.
• Subtracting a negative always increases the value.
• Multiplying by -1 flips the sign.
• The absolute value is never negative; |-5| is 5, never -5.

These foundations reappear every time you read a lab value, track an intake/output total, or convert a temperature. Lock in the sign rules before moving to fractions and decimals.

Quick Practice

Try these before the quiz:

• -10 + 6 = -4 (different signs, subtract magnitudes, keep sign of larger: 10 - 6 = 4, sign negative).
• (-5) × (-2) × (-1) = -10 (three negatives → odd → negative).
• 0 - 7 = -7 (subtracting a positive from zero gives a negative).
• |8| + |-3| = 8 + 3 = 11.

If you can do these four without hesitation, your signed-number foundation is solid for the NEX.