Place value, ordering and negative numbers
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Place Value in Integers and Decimals
- Each digit in a number has a place value depending on its position.
- In a whole number, the place values from right to left are ones, tens, hundreds, thousands, and so on.
- In a decimal, the place values after the decimal point are tenths, hundredths, thousandths, and so on.
- For example, in 3.45, the digit 3 is in the ones place, 4 is in the tenths place, and 5 is in the hundredths place.
- The value of a digit is the digit multiplied by its place value. For example, in 3.45, the 4 represents 4 × 0.1 = 0.4.
- Zero can be used as a placeholder to show that a place value is empty. For example, in 2.05, the 0 shows there are no tenths.
The Number Line
- A number line shows the relationship between negative numbers, positive numbers, and zero.
- On a number line, numbers increase as you move to the right and decrease as you move to the left.
- Zero is in the middle, with positive numbers to the right and negative numbers to the left.
- A negative number with greater magnitude is considered less. For example, −8 is less than −5.
- The number line helps compare and order numbers, including negative numbers and decimals.
Positive and Negative Numbers
- A negative number is a real number that is less than zero. It is often written with a minus sign in front, e.g. −3.
- A positive number is a number that is greater than zero. It may be written with a plus sign, e.g. +3.
- Zero is neither positive nor negative; it is a neutral number.
- Every real number other than zero is either positive or negative.
- The sign of a number refers to whether it is positive or negative.
- Negative numbers can represent a loss or deficiency, such as a debt or a temperature below zero.
Ordering Numbers
- To order numbers, compare their positions on the number line: numbers farther to the right are greater.
- For positive numbers, the greater the magnitude, the greater the number. For example, 8 > 5.
- For negative numbers, the greater the magnitude, the smaller the number. For example, −8 < −5.
- Any positive number is greater than any negative number. For example, 1 > −100.
- Zero is greater than any negative number and less than any positive number.
- When ordering decimals, compare digits from left to right, using place value.
Symbols for Comparison
- = means is equal to. For example, 2 + 3 = 5.
- ≠ means is not equal to. For example, 2 + 3 ≠ 6.
- < means is less than. For example, −4 < −1.
- > means is greater than. For example, 3 > −2.
- ≤ means is less than or equal to. For example, x ≤ 5 means x can be 5 or any number less than 5.
- ≥ means is greater than or equal to. For example, x ≥ −2 means x can be −2 or any number greater than −2.
Adding and Subtracting Negative Numbers
- Adding a positive number moves to the right on the number line. For example, −3 + 5 = 2.
- Adding a negative number is the same as subtracting its positive counterpart. For example, 4 + (−2) = 4 − 2 = 2.
- Subtracting a positive number moves to the left on the number line. For example, 2 − 5 = −3.
- Subtracting a negative number is the same as adding its positive counterpart. For example, 4 − (−2) = 4 + 2 = 6.
- The opposite of an opposite is the original value: −(−3) = 3.
- When adding numbers with different signs, find the difference between their magnitudes and take the sign of the number with the larger magnitude. For example, −7 + 4 = −3.
Multiplying and Dividing Negative Numbers
- The product or quotient of two numbers with the same sign is positive. For example, (−3) × (−4) = 12 and (−12) ÷ (−3) = 4.
- The product or quotient of two numbers with different signs is negative. For example, (−3) × 4 = −12 and 12 ÷ (−3) = −4.
- Multiplying or dividing by zero is undefined for division; multiplication by zero gives zero.
- These rules ensure that arithmetic with negative numbers reflects the common-sense idea of opposites.
Negative Numbers in Context
- Temperature: Temperatures below 0 °C or 0 °F are negative. For example, −5 °C is colder than 0 °C.
- Finance: A bank balance can be negative if you owe money, representing a debt. For example, a balance of −£20 means you owe £20.
- Sport: Goal difference, points difference, and run differential can be negative. For example, a goal difference of −3 means a team has scored 3 fewer goals than it has conceded.
- Science: Latitudes south of the equator and longitudes west of the prime meridian are negative. Elevations below sea level are negative.
- Electrical circuits: A battery connected in reverse polarity applies a negative voltage. For example, a 6-volt battery in reverse applies −6 volts.
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Übungsfragen
Gratis-Vorschau — 8 von 64 Fragen. Registriere dich, um alle zu sehen.
1.Which of the following statements about zero is correct?
Easy- AZero is neither positive nor negative.
- BZero is positive.
- CZero is negative.
- DZero is both positive and negative.
2.On a number line, which number is farthest to the left?
Easy- A-8
- B-5
- C0
- D3
3.Which symbol correctly compares -3 and -7?
Easy- A-3 > -7
- B-3 < -7
- C-3 = -7
- D-3 ≤ -7
4.Which list shows the numbers in ascending order?
Easy- A-5, -2, 0, 3, 7
- B-2, -5, 0, 3, 7
- C0, -2, -5, 3, 7
- D7, 3, 0, -2, -5
5.Which of the following are integers? (Select all that apply.)
Medium- A5
- B-3
- C0
- D2.5
- E-7.1
6.The number -(-3) is equal to 3.
EasyTrue or false?
7.Match each symbol with its meaning.
Medium- =
- ≠
- <
- >
- is equal to
- is not equal to
- is less than
- is greater than
8.Arrange these numbers in ascending order (smallest first).
Medium- -3.5
- -2.1
- 0
- 1.7
- 4.2
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