Calculate with signed numbers

Calculation

\(\displaystyle -4-(-7)\)

Which rewrite preserves the calculation?

Keep addition and subtraction meanings separate from product signs

For addition, combine signed changes. Two negative addends give a more negative total, not a positive sum

For opposite signs, compare magnitudes and keep the sign of the larger magnitude

Subtracting a number means adding its opposite

A rule about two negative factors does not apply to two negative addends

Combine two negative changes

Subtract the complete negative quantity

Extend a multiplication pattern consistently

Multiplying a negative number by a positive integer adds equal negative changes

Reducing the second factor by one increases each product by four

A negative times a negative must be positive to continue this pattern and preserve distribution

The second product must be eight to cancel negative eight

Keep a at negative three first and decrease b one step at a time through zero. Then try other positive, negative and zero values of a

Which claim agrees with the non-zero cases and the zero boundary?

Apply sign and magnitude separately

Evaluate the expression

The two non-zero operands have matching signs, so the result is positive

Multiply the positive magnitudes

Apply the positive sign to the magnitude

Check by dividing the product by the non-zero second factor

Multiply two negative integers

Keep a different-sign product negative

Use multiplication to explain division signs

A quotient is the number that multiplies the divisor to make the dividend

For non-zero operands, matching signs give a positive quotient and different signs give a negative quotient

An integer dividend and divisor do not always produce an integer quotient

Check a signed quotient by multiplication

Evaluate the expression

The two non-zero operands have different signs, so the result is negative

Divide the positive magnitudes

Apply the negative sign to the magnitude

Check by multiplying the quotient by the divisor

Divide two negative integers

Separate a zero dividend from a zero divisor

Any product with a zero factor is zero. Zero divided by a non-zero number is zero

Division by zero is undefined. No number multiplied by zero produces a non-zero dividend

Zero divided by zero is also undefined: every number times zero is zero, so there is no unique quotient

Use the zero-factor rule

Divide a zero dividend by a non-zero divisor

Quotients

\(\displaystyle (-8)\div0\quad\text{and}\quad0\div0\)

Which statement is correct?

A sign belongs to the entire fraction or mixed number

Keep the denominator positive and put the sign on the complete value

A negative mixed number is not a negative whole part plus a positive fraction

For addition and subtraction, first use equal-sized parts, then combine the signed numerators

Add two negative fractions

Calculate exactly. Give a fraction in simplest form

For two non-zero addends with the same sign, add their magnitudes and keep that sign

Use the lowest common denominator, 12, and scale both numerators with their denominators

Add the numerators and keep the common denominator

Keep the denominator positive and reduce any common factors

Combine opposite signs using a common denominator

Add the opposite before combining fractional parts

The first operand stays negative. Only the second complete operand is replaced by its opposite when subtraction changes to addition

Equal and opposite values cancel even when they are written with different denominators

Keep the sign while converting a mixed number

Calculate exactly. Give a fraction in simplest form

Subtract the complete second quantity by adding its opposite

Write the first mixed number as an improper fraction, keeping its sign

Use the lowest common denominator, 4, and scale both numerators with their denominators

Subtract the second numerator from the first and keep the common denominator

Keep the denominator positive and reduce any common factors

Check by adding the original subtracted quantity back

Subtract a negative fraction from a negative mixed number

Identify exact cancellation

Use whole-term signs and fraction operations together

The integer sign rules also apply to non-zero rational factors and divisors

Cancel common factors or simplify the final fraction. Do not add denominators or cancel terms across addition

For division, keep the dividend and multiply by the reciprocal of the non-zero divisor

Multiply a negative mixed number and a negative fraction

Calculate exactly. Give a fraction in simplest form

The non-zero operands have the same sign, so the result is positive

Write the first mixed number as an improper fraction, keeping its sign

Cancel the common factor 3 from the first numerator and second denominator

Cancel the common factor 2 from the second numerator and first denominator

Multiply the numerators and denominators

Keep the denominator positive and reduce any common factors

Reduce a different-sign product

Reciprocate only the divisor

Calculate exactly. Give a fraction in simplest form

The non-zero operands have the same sign, so the result is positive

Keep the dividend and multiply by the reciprocal of the divisor

Cancel the common factor 3 from the first numerator and second denominator

Cancel the common factor 5 from the second numerator and first denominator

Multiply the numerators and denominators

Keep the denominator positive and reduce any common factors

Check that the quotient multiplied by the original divisor gives the dividend

Divide by a negative fraction

Combine signed counts of equal place-value units

For decimal addition and subtraction, align decimal points. Appending zeros preserves each value

Subtracting a negative decimal adds its positive opposite

Do not apply multiplication sign rules to addition

Add negative decimal amounts

Calculate exactly. Give a decimal or integer answer without rounding

For two non-zero addends with the same sign, add their magnitudes and keep that sign

Align decimal points and append zeros so the place values match

Combine the signed counts of the same-sized units

This result is exact, so no rounding is needed

Compare magnitudes with aligned places

Subtract a larger negative magnitude

Calculate exactly. Give a decimal or integer answer without rounding

Subtract the complete second quantity by adding its opposite

Align decimal points and append zeros so the place values match

Combine the signed counts of the same-sized units

This result is exact, so no rounding is needed

Check by adding the original subtracted quantity back

Retain thousandths and the subtraction sign

Account for both decimal scales

Two negative factors give a positive product. Hundredths times tenths produce thousandths before simplification

The number of decimal places describes the scale, not a command to pad or round the final exact value

For division, scale both operands by the same non-zero power of ten

Scaling only the divisor would change the quotient

Multiply two negative decimals exactly

Calculate exactly. Give a decimal or integer answer without rounding

The non-zero operands have the same sign, so the result is positive

The first factor has 2 decimal places and the second has 1

Multiply the signed integer numerators and account for both place-value scales

This result is exact, so no rounding is needed

Keep a negative product and its place-value scale

Scale a negative dividend and positive divisor together

Calculate exactly. Give a decimal or integer answer without rounding

The non-zero operands have different signs, so the result is negative

Multiply both dividend and divisor by 100 to obtain integers without changing the quotient

The divisor must be non-zero, and only the divisor is reciprocated in a fraction method

This result is exact, so no rounding is needed

Check that the quotient multiplied by the original divisor gives the dividend

Divide two negative decimals

Use valid regrouping and exact equivalent forms

Addition and multiplication may be reordered and regrouped, including signed operands

A useful fraction can avoid lengthy decimal working. A recurring fraction must not be replaced by a rounded decimal in an exact calculation

For subtraction, first rewrite subtracted terms as their opposites before reordering

Subtraction and division cannot simply be reordered as if they were commutative

Construct the useful intermediate sum

Use the intermediate sum in the complete calculation

Split a decimal multiplier into an integer and a scale

Zero point four is four tenths. Multiplying by four and then dividing by ten preserves the exact value

The intermediate product and final answer have different values. Keep the division by ten visible

Find the numerator before applying the decimal scale

Complete the place-value scaling

Keep operation order and repeated negative factors explicit

Evaluate brackets first, then powers as repeated multiplication, then multiplication and division from left to right, then addition and subtraction from left to right

Brackets show when a negative value is the complete repeated factor

Pairs of negative factors produce a positive product, and an unpaired negative factor changes its sign. General exponent laws are studied separately

Calculate the product before the sum

Estimate magnitude without losing the sign

Before a long calculation, choose nearby convenient values and state how you rounded them

Here each operand was rounded to the nearest integer. The exact product is negative sixty point seven six, close to the estimate

An estimate checks scale and sign; it does not prove every digit or replace an exact requested answer

Rounding a small non-zero divisor to zero would make an estimate unusable. Choose a finer place or another convenient non-zero value instead

Calculation

\(\displaystyle (-29.7)\times4.1\)

Round each operand to the nearest integer, then multiply the rounded values. Enter the estimate, not the exact product

Round the two operands before multiplying, and retain the negative sign

Requested calculation

\(\displaystyle (-29.7)\times4.1\)

Copied result

\(\displaystyle -12.17\)

A calculator display has been copied as negative twelve point one seven. Which conclusion is justified?

Choose when a digital tool is useful

This independent investigation is not automatically marked. Record your estimate, chosen tool, entry and check in your own working

Calculate negative two point seven five times three point six plus one point eight. Predict the sign and an approximate value before using a calculator or spreadsheet

Use the negative-number entry key or brackets so the tool receives negative two point seven five as one factor. Record the expression you actually entered

Check the display by exact fraction or decimal scaling and an inverse operation. Explain why a tool is useful for checking but unnecessary for a simple fact such as negative six times four

If the sign or scale disagrees with your estimate, inspect the entry before trusting the display

Review the digital calculation independently

One possible estimate uses negative three times four plus two

The estimate is rough because each input was changed

Undo the addition and then the multiplication to check the original first factor

A correct tool entry should agree with negative eight point one. The saved working explains the result; the display alone does not

Keep signs, quantities and checks together

Addition combines signed quantities; subtraction adds the opposite of the second complete quantity

Matching non-zero signs give a positive product or quotient, and different signs give a negative product or quotient

A zero factor gives zero, but a zero divisor is never allowed

Use common denominators for fractional sums and differences, and reciprocate only a non-zero divisor

Use exact decimal place-value scales, valid regrouping and the established operation order

Estimate first when useful, check by an inverse operation and inspect a digital entry rather than trusting its display